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+{"text":"\\section{Introduction}\n\n\n\n\n\nThis paper is part of a program aimed to understand equivariant and $K$-theoretic invariants for Artin groups.\nThese groups are ubiquitious in geometric group theory, as they comprehend many different subfamilies of groups (braid groups, right-angled, spherical type, extra-large, free groups...), provide examples and counterexamples for many interesting phenomena, and at the same time they are not well understood from a global point of view: it is not even known if every Artin group is torsion-free. See \\cite{Par09} for a good survey. In this context, Azzali  \\emph{et al.} have recently computed explicitly both sides of Baum-Connes for pure braid groups \\cite{ABGRW}; in joint work of the second author with J. Gonz\\'alez-Meneses \\cite{FlGo18} it is computed the minimal dimension of a model of ${\\underline{\\underline{E}}} G$ for braid groups; J. and virtually-cyclic dimensions of mapping class groups (which include in particular certain Artin groups) have been recently investigated by Aramayona \\emph{et al.}  (\\cite{AJT18}, \\cite{ArMa}) and by Petrosyan and Nucinkis \\cite{NuPe18}.\n\n \n\nIn the present paper we consider the case of Artin groups of dihedral type, denoted by $A_n$, which are the groups defined by\n$$A_n=\\gp{a,b}{\\mathrm{prod}(a,b;n)=\\mathrm{prod}(b,a;n)}.$$\nHere  $\\mathrm{prod}(x,y;n)$ denotes the word of length $n$ that alternates $x$ and $y$ and starts with $x$.\n These groups   are one-relator and torsion-free, they  have small geometric dimension and its group-theoretic structure is quite well understood. These features makes them strongly appropriate for computations, and in fact have been recently studied from different angles, as growth series \\cite{MaMa06}, systems of equations \\cite{CHR20} or geodesics \\cite{Wal09}.\n\nWe compute here Bredon homology groups of the classifying space of these groups with respect to the family of virtually cyclic groups. Bredon homology was first described by Glen Bredon in the sixties \\cite{Bre67}, and it is a $G$-equivariant homology theory that takes into account the action over the target space of the subgroups of $G$ that belong to a predefined family. Since their appearance, Bredon homology groups have played a prominent role in Homotopy Theory and Group Theory, particularly in relation with finiteness conditions, dimension theory for groups and classifying spaces, and in the framework of the Isomorphism Conjectures. In fact, our choice of coefficients in the $K$-theory of a ring $G$ has been done with an eye in possible applications of the computations to the left-hand side of Farrell-Jones Conjecture (see below), although we believe that the methods of the present paper may be useful in a more general context, switching the $K$-theoretic coefficients to a general coefficient module.\n\nOur main technical result is the following:\n\n\n\n\\noindent\\textbf{Theorem \\ref{Thm:Bredon}.} Let $A_n$ be an Artin group of dihedral type and $R$ a ring, $R$ a ring, $K_q(R[-])$ the covariant module over the orbit category $O_{\\mathcal F}(G)$ that sends every (left) coset  $G\/H$ to the $K$-theory group $K_q(R[H])$. Then we have the following:\n\\begin{enumerate}\n  \\setlength{\\itemindent}{-2em}\n\\item $H_i^{vc}({\\underline{\\underline{E}}} A_n,K_q (R[-]))=\\{0\\}$ for $i\\geq 4$.\n\\item $H_3^{vc}({\\underline{\\underline{E}}} A_n,K_q (R[-]))=\\begin{cases}\\bigoplus_{[H]\\neq [Z(A_n)]} K_q(R) & \\text{$n$ odd}\\\\\n\\ker g_2^2 & \\text{$n$ even.}\n\\end{cases}$\n\n\\item $H_2^{vc}({\\underline{\\underline{E}}} A_n,K_q (R[-]))= \\ker g_2^1$.\n\n\\item $H_1^{vc}({\\underline{\\underline{E}}} A_n,K_q (R[-]))= \\emph{coker } g_2^1 = \\begin{cases}(\\bigoplus_{[H]\\neq [Z(A_n)]}N_q^{[H]})\\oplus T_1(K_q(R))\\oplus T_2(K_q(R)) & \\text{$n$ odd}\\\\\n\\bigoplus_{[H]} N_q^{[H]}\n  & \\text{$n$ even.}\n\\end{cases}$%\n\\item $H_0^{vc}({\\underline{\\underline{E}}} A_n,K_q (R[-]))= \\emph{ coker } g_2^0= \\begin{cases}(\\bigoplus_{[H]\\neq [Z(A_n)]} N_q^{[H]})\\oplus K_q(R)\\oplus\\overline{C}(K_q(R)) & \\text{$n$ odd}\\\\\n(\\bigoplus_{[H]} N_q^{[H]})\\oplus K_q(R)\n  & \\text{$n$ even.}\n  \\end{cases}$%\n\\end{enumerate}\n\n\nHere $g^j_i$ stands for a homomorphism in the Degrijse-Petrosyan exact sequence  {\\cite[Section 7]{DP}}, $[H]$ for the commensurability class of a {non-trivial} cyclic subgroup $H$, and $N_q^{[H]}$, $T_i(K_q(R))$ and $\\overline{C}(K_q(R))$ for groups that depend on $H$ and the Bass-Heller-Swan decomposition of $K_q(R[\\Z])$. See Section \\ref{Sect:BredonArtin} for details.\n\n\n\n\nThe main tool used in the proof of Theorem \\ref{Thm:Bredon} is an exact sequence in Bredon homology \\cite{DP}, which is in turn the Mayer-Vietoris sequence associated to the push-out that defines the L\\\"{u}ck-Weiermann model for ${\\underline{\\underline{E}}} G$ (\\cite{LW}, see also Section \\ref{Sect:Prelim} below). The knowledge about the group-theoretic structure of the groups includes a complete understanding of the commensurators, which is crucial in the computations. Using the theorem, we are able to describe with precision the Bredon homology of $A_n$ with coefficients in the $K$-theory of several rings, both regular and non-regular (see Section \\ref{Sect:concrete}). \n\nNext we describe the implications of our work in relation with the Farrell-Jones Conjecture. Recall that given a group $G$, a ring $R$ and $n\\in\\mathbb{Z}$, Farrell-Jones stated in \\cite{FaJo93} the existence of an assembly map:\n\n$$H_n^G({\\underline{\\underline{E}}} G,\\mathbf{K} (R))\\rightarrow K_n(RG),$$\nwhere ${\\underline{\\underline{E}}} G$ is the classifying space of $G$ with respect to the family of virtually cyclic groups,\n$H_*^G(-,\\mathbf{K} (R))$ is the $G$-homology theory defined in Section 1 of \\cite{FaJo93} and $K_n(RG)$ stands for the $n$-th group of algebraic $K$-theory of the group ring $RG$.\nThe Farrell-Jones conjecture predicts that the assembly map is an isomorphism.\nThe conjecture has been verified for a big family of groups (see \\cite{LR05} for an excellent survey), and no counterexample has been found so far.\nThe philosophy in this context is, for a group for which the assembly map is known to be an isomorphism, to perform computations in the topological side in order to extract information about the algebraic $K$-theory of the group ring. It is remarkable that the latter are difficult to compute, and at the same time encode fundamental invariants of manifolds, including obstructions to the existence of cobordisms and information about groups of pseudoisotopies. Explicit calculations in this context can be found in \\cite{BuSa16}, \\cite{DQR11}, \\cite{KLL21} or \\cite{SaVe18}, for example.\n\nThe left-hand side of the conjecture can be approached by means of a $G$-equivariant version of the Atiyah-Hirzebruch spectral sequence, which  converges to the Farrell-Jones $K$-homology, and whose $E_2$-page is  the Bredon homology of the classifying space ${\\underline{\\underline{E}}} G$ with coefficients in the $K$-theory of the group rings of the virtually cyclic subgroups of $G$; example of such calculations in the Farrell-Jones framework can be found in \\cite{BJV14} \\cite{LuRo14}. In this setting, Theorem \\ref{Thm:Bredon} and the examples of Section \\ref{Sect:concrete} can be interpreted as an explicit computation of such $E_2$-page, in the case of Artin groups of dihedral type. We remark that in the case of $R$ regular, the left-hand side of the conjecture for these Artin groups can be deduced from \\cite[Lemma 16.12]{Luc21}, using previously a splitting result of L\\\"{u}ck-Steimle \\cite{LS16} (see the end of Section \\ref{Sect:concrete} for details), so our results provide new information in this context for a non-regular $R$; in this sense, we expect that Examples 5.3-5.5 will be useful. It is worth to point out that the Atiyah-Hirzebruch spectral sequence collapses at most at the $E_5$-page in this context, so an (at least partial) computation of the differentials may not be completely out of sight.\n\nWe finish by pointing out that Artin groups of dihedral type are free-by-cyclic, and the Farrell-Jones conjecture has been recently verified for this class of groups \\cite{BFW21}. Hence, all the computations in the left-hand side can be read in terms of algebraic $K$-theory of $RG$.\n\n\n\n\\textbf{Summary of contents}. In Section \\ref{Sect:Prelim} we recall the main definitions about classifying spaces for families and Bredon homology, with special emphasis in the L\\\"{u}ck-Weiermann model and its associated Mayer-Vietoris sequence. In Section \\ref{Sect:Artin} the main properties of Artin groups that will be used on the rest of the paper are studied. Then, in Section \\ref{Sect:BredonArtin}, we carefully analyze the homomorphisms in the exact sequence and prove Theorem \\ref{Thm:Bredon}; and in final Section \\ref{Sect:concrete} we apply our results to describe different concrete examples.\n\n\n\\section{Preliminaries}\n\\label{Sect:Prelim}\n\nIn this section we state some notions of $G$-equivariant homotopy that will frequently appear in the rest of the paper.\nThe exposition will be sketchy and very focused to our goals; the reader interested in a thorough treatment of the subject is referred to \\cite{TDieck} for the theory of $G$-$CW$-complexes and actions on them, to \\cite{Luc05} for the theory of classifying spaces and to the first part of \\cite{MiVa03} for Bredon homology.\n\n\\subsection{Classifying spaces for families}\n\\label{Sect:classify}\nIn this section we will briefly recall the notion of classifying space for a family of subgroups, which is the central object in the topological side of the Isomorphism Conjectures. Then we will review L\\\"{u}ck-Weiermann model and the definition of commensurator, which will be crucial in our computations.\n\n\\begin{defn}\n\nLet $G$ be a discrete group, and $\\mathcal{F}$ be a family of subgroups of $G$  closed under passing to subgroups and conjugation. A $G$-CW-complex $X$ is a \\emph{classifying space for the family} $\\mathcal{F}$ if for every $H\\in \\mathcal{F}$ the fixed-point set $X^H$ is contractible, and empty otherwise.\n\n\\end{defn}\n\nThe classifying space for the family $\\mathcal{F}$ is usually denoted by $E_{\\mathcal{F}}G$.\nMoreover,  two models for $E_{\\mathcal{F}}G$ are $G$-homotopy equivalent. A point is always a model for $E_{\\mathcal{F}}G$ if $G\\in \\mathcal{F}$, and the closeness under subgroups implies that $E_{\\mathcal{F}}G$ is always a contractible space.\n\nIf there is a family ${\\mathcal{F}}$ of subgroups of $G$ with the previous closeness properties and a subgroup $H\\leqslant G$, we denote by ${\\mathcal{F}}\\cap H$ the family whose elements are the intersections $F\\cap H$, with $F\\in {\\mathcal{F}}$.\nThe family ${\\mathcal{F}}\\cap H$ of subgroups of $H$ is again closed under $H$-conjugation and taking subgroups.\nIn these conditions the action of $H$ over $E_{{\\mathcal{F}}}G$ by restriction turns $E_{{\\mathcal{F}}}G$ into a model for $E_{{\\mathcal{F}}\\cap H}H$.\n\n\nThe most important families of subgroups in this context are the trivial family $\\mathcal F_{\\{1\\}}$, the family $\\mathcal{F}_{Fin}$ of finite groups and the family $\\mathcal{F}_{vc}$ of virtually cyclic groups of $G$; the classifying spaces for these families are respectively denoted by $EG$, $\\underline{E} G$ and ${\\underline{\\underline{E}}} G$. Observe that $\\mathcal F_{\\{1\\}}\\subseteq \\mathcal{F}_{Fin}\\subseteq \\mathcal{F}_{vc}$, and that $\\mathcal F_{\\{1\\}}=\\mathcal{F}_{Fin}$ if and only if $G$ is torsion-free.\nIt is  also a standard argument to show that torsion-free virtually cyclic groups are cyclic (see for example Lemma 3.2 in \\cite{Mac96}).\nThen, for torsion-free groups, $\\mathcal F_{vc}$ is the set of cyclic subgroups.\n\nFrom now on we will describe the model of ${\\underline{\\underline{E}}} G$ developed by L\\\"{u}ck-Weiermann in \\cite{LW}, for the special families we are interested (the construction is indeed more general). Given a group $G$, and two subgroups $H$ and $K$,  we consider the equivalence relation generated by $H\\sim K$ if $H\\cap K$ has finite index in both $H$ and $K$.\nObserve that if $H$ is in ${\\mathcal F}_{vc}\\setminus \\mathcal{F}_{Fin}$ then $H\\sim K$ if and only if $K$ is virtually cyclic and $H\\cap K$ is infinite.\nAlso remark  that if $H\\in \\mathcal{F}_{Fin}$, then $H\\sim K$ if and only if $K\\in \\mathcal{F}_{Fin}$.\nThe equivalence class of $H$ will be denoted by $[H]$.\nThis equivalence relation is preserved by conjugation, it can be defined $g^{-1}[H]g$ as $[g^{-1}Hg]$ for any $H\\leqslant G$ and for any $g\\in G$.\n\nNow we can define the notion of \\emph{commensurator}, central in this model and  in our paper:\n\n\\begin{defn}\n\\label{defn:comm}\nGiven an equivalence class $[H]$ of the relation $\\sim$, the \\emph{commensurator} of $[H]$ in $G$ is defined as the subgroup\n$$ \\textrm{Comm}_G[H]=\\{g\\in G\\:|\\ g^{-1}[H]g=[H]\\}.$$\n\\end{defn}\n\n\nIn \\cite{LW}, it is also defined a family of subgroups of $\\textrm{Comm}_G[H]$ as:\n$${\\mathcal{F}} [H]:=\\{K<{ \\textrm{Comm}_G[H]} \\: |\\ K\\sim H \\text{ or } |K|<\\infty \\}.$$\nIf $H$ is virtually cyclic, it is easy to check that this family is closed under taking subgroups and conjugation in {$\\textrm{Comm}_G[H]$}. {We remark that the commensurator of $[H]$ is sometimes denoted by $N_G[H]$ in the literature}.\n\nNow we have all the ingredients needed for building L\\\"{u}ck-Weiermann model:\n\n\\begin{thm}\n\\label{maintheorem}{\\rm (\\cite{LW}, Theorem 2.3)} We denote by $I$ a complete set of representatives of the $G$-orbits (under conjugation) of equivalence classes $[H]$ of infinite virtually cyclic subgroups of $G$, and we choose, for every $[H]\\in I$, models for the classifying spaces $\\underline{E}  \\Comm_G[H]$ and $E_{{\\mathcal{F}} [H]}\\Comm_G[H]$.\nWe also choose a model for $\\underline{E}  G$. Consider the $G$-pushout:\n$$\n\\xymatrix{  \\coprod_{[H]\\in I}G\\times_{\\Comm_G[H]}\\underline{E}  \\Comm_G[H] \\ar[r]^{\\hspace{2cm} i} \\ar[d]^{\\coprod_{[H]\\in I}id_G\\times_{\\Comm_G[H]}f_{[H]}}  & \\underline{E} G \\ar[d] \\\\\n\\coprod_{[H]\\in I}G\\times_{\\Comm_G[H]}E_{{\\mathcal{F}} [H]}\\Comm_G[H]  \\ar[r] & X }\n$$\nwhere  $f_{[H]}$ is a cellular $\\Comm_G[H]$-map for every $[H]\\in I$ and $i$ is the inclusion.\n In these conditions, $X$ is a model for ${\\underline{\\underline{E}}} G$.\n\\end{thm}\n\n\nIn practice, this theorem implies that the existence of good models for the proper classifying space of the commensurators and $G$, and also of the classifying spaces with respect to the families ${\\mathcal{F}} [H]$ will lead to the knowledge of good models for ${\\underline{\\underline{E}}} G$. Moreover, the push-out implies the existence of a long exact sequence in Bredon homology, and dimensional consequences that we will analyze in next section.\n\n\n\n\\subsection{Bredon homology}\n\\label{Sect:Bredon}\n\n{In this subsection we will briefly review the main definitions concerning Bredon homology. We follow the topological concise approach from \\cite{San08}, which we use in our computations}.\n\n{Consider a discrete group $G$, $\\mathcal{F}$ a family of groups which is closed under conjugation and taking subgroups.  Let $O_{\\mathcal{F}}(G)$ be the \\emph{orbit category}  whose objects are the homogeneous spaces $G\/K$, $K\\subset G$ with $K\\in\\mathcal{F}$, and whose morphisms are the $G$-equivariant maps.\nThen a \\emph{left Bredon module} $N$ over $O_{\\mathcal{F}}(G)$ is a covariant functor $$N:O_{\\mathcal{F}}(G)\\rightarrow \\textbf{Ab},$$ where $\\textbf{Ab}$ is the category of abelian groups}.\n\n{Let $N$ be  a left Bredon module and $X$ a $G$-CW-complex, and assume that all the stabilizers of the $G$-action belong to the family $\\mathcal{F}$. Then the \\emph{Bredon chain complex} $(C_n^{\\mathcal{F}}(X,N),\\Phi_n)$ can be defined in the following way.\nFor every $d\\geq 0$, consider a set $\\{e_i^d\\}_{i\\in I}$ of representatives of orbits of $d$-cells in $X$, and denote by $\\stab(e_i^d)$ the stabilizer of $e_i^d$.\nThen we define the \\emph{n-th group of Bredon chains} as $C_n^{\\mathcal{F}}(X,N)=\\bigoplus_{i\\in I} N(G\/\\stab(e_i^d))$}.\n\n{Consider now a $(d-1)$-face of $e_i^d$, which can be given as $ge$ for a certain $(d-1)$-cell $e$.\nThen we have an inclusion of stabilizers $g^{-1}\\stab(e_i^d)g\\subseteq \\stab(e)$.\nAs $g^{-1}\\stab(e_i^d)g$ and $stab(e_i^d)$ are isomorphic, the previous inclusion induces an equivariant $G$-map $f\\colon G\/\\stab(e_i^d)\\rightarrow G\/\\stab(e)$.\nIn turn, as $N$ is a functor, we have an induced homomorphism $N(f)\\colon N(G\/\\stab(e_i^d))\\rightarrow N(G\/\\stab(e))$.\nTaking into account that the boundary of $e_i^d$ can be written as $\\partial e^d_i=\\sum_{j=1}^n g_j e_j^{d-1}$ for certain $g_j\\in G$ and using linear extension to all representatives of equivariant $d$-cells, we obtain a differential $\\Phi_d\\colon C_d^{\\mathcal{F}}(X,N)\\rightarrow C_{d-1}^{\\mathcal{F}}(X,N)$ for every $d>0$.\nSo we have the following definition:}\n\n\\begin{defn}\n{The homology groups of the chain complex $(C_i^{\\mathcal{F}}(X,N),\\Phi_i)$ will be denoted by $H_i^{\\mathcal{F}}(X,N)$ and called \\emph{Bredon homology groups} of $X$ with coefficients in $N$ with respect to the family $\\mathcal{F}$}.\n\nWe define $H_i^{\\mathcal{F}}(G,N)$, the {\\emph Bredon homology groups} of $G$ with coefficients in $N$  with respect to the family $\\mathcal{F}$ as $H_i^{{\\mathcal{F}}}(E_{\\mathcal{F}}G,M)$.\n\\end{defn}\n\n{These groups are preserved under $G$-equivariant homotopy equivalence}.\n\n\n\\textbf{Notation}. When $\\mathcal{F}$ is the family of finite groups, we use indistinctly the notations $\\underline{E}G$ or $E_{\\mathcal{F}}G$, and similarly when $\\mathcal{F}$ is the family of virtually cyclic groups and ${\\underline{\\underline{E}}} G$ and $E_{\\mathcal{F}}G$ notations for the corresponding classifying space. If $\\mathcal{F}$ is the family that only contains the trivial group, the superindex in the Bredon homology will be supressed, as it is ordinary homology in this case.\nIt is worth noticing that there is an algebraic definition of $H_*^{\\mathcal{F}}(G,M)$, however there is an isomorphism $H_*^{\\mathcal{F}}(G,M)\\simeq H_*^{\\mathcal{F}}(E_{\\mathcal{F}}G,M)$ between the algebraic and the topological definitions of Bredon homology \\cite[page 15]{MiVa03}.\nTo not overload the paper with unnecessary notation, we commonly denote this homology groups by $H_*^{\\mathcal{F}}(G,M)$, although as said above, we mainly deal with the topological definition.\n\nThe cyclic group of order $n$ will be denoted by $C_n$. When we want to consider its ring structure we might use  $\\Z \/{\\bf n}$. When $n$ is prime, we might use $\\mathbb{F}_n$ to emphasize its field structure.\n\n\n\n\\section{Artin groups of dihedral type}\n\\label{Sect:Artin}\nIn this section we present the main features of the Artin groups of dihedral type that we will need in the remaining of the paper. We start with the definition of the groups:\n\n\\begin{defn}\n\\label{Defn:Artindih}\nLet $n\\geq 1$. By $\\mathrm{prod}(x,y;n)$ we denote the word of length $n$ that alternates $x$ and $y$ and starts with $x$.\nFor example, $\\mathrm{prod}(x,y;3)=xyx$ and $\\mathrm{prod}(x,y;4)=xyxy$. With this notation, a {\\it dihedral Artin group of type $n$} is the group $$A_n=\\gp{a,b}{\\mathrm{prod}(a,b;n)=\\mathrm{prod}(b,a;n)}.$$\n\\end{defn}\nThe name ``dihedral\" comes from the associated Coxeter group, $$\\gp{a,b}{a^2=b^2=1, \\mathrm{prod}(a,b;n)=\\mathrm{prod}(b,a;n)}$$ which is the dihedral group of order $2n$. Dihedral Artin groups are torsion-free, even more $A_n\\cong F_{n-1}\\rtimes \\Z$, where $F_k$ is a free group of rank $k$. To see this, one can check that the kernel of $A_n\\to \\Z$, $a,b\\mapsto 1$ is free on rank $n-1$. In particular, Dihedral Artin groups satisfy the Farrell-Jones conjecture \\cite{BFW21}.\n\nWe are interested on understanding the commensurators of the virtually cyclic subgroups of $A_n$, for that we will use that $A_n$ is also a central extension of a virtually free group.\n\nAs indicated above an important ingredient of our calculations will be the description of some commensurators inside these Artin groups of subgroups from the family of virtually cyclic groups (which in this case turn to be just cyclic groups). We start by observing that for any  virtually cyclic subgroup $H$ of a group $G$, and any $h\\in H$ of` infinite order, we have that\n$$\\Comm_G[H]=\\Comm_G(\\gen{h})=\\{g\\in G\\mid gh^mg^{-1}=h^n \\text{ for some }n,m\\in \\Z-\\{0\\}\\}.$$\n\nWe will be interested in computing commensurators up to isomorphism, and we will use the fact that commensurators of conjugated subgroups are conjugated.\nLet us denote by $\\pi$ the natural\nprojection map $\\pi\\colon A_{n}\\to \\ol{A_{n}}\\mathrel{\\mathop\\mathchar\"303A}\\mkern-1.2mu= A_{n}\/Z(A_{n})$. Then, for every $g\\in A_{n}$,\n$$\\pi(\\Comm_{A_{n}}[\\gen{g}])\\leqslant \\Comm_{\\ol{A_{n}}}[\\gen{\\pi(g)}],$$\nand\n$$Z(A_{n})\\leqslant \\Comm_{A_{n}}[\\gen{g}].$$\n\nWe will prove the following.\n\\begin{lem}\\label{lem:commensurators}\nLet $A_n$ be a dihedral Artin group and $g\\in A_n$ of infinite order. Then\n\\begin{enumerate}\n\\item If $\\langle g\\rangle \\cap  Z(A_n)\\neq \\{1\\}$ then $\\Comm_{A_n}[\\langle g\\rangle]= A_n$ and $Z(A_n)\\in [H]$,\n\\item If $\\langle g\\rangle \\cap Z(A_n)= \\{1\\}$ then $\\Comm_{A_n}[\\langle g\\rangle]\\cong \\mathbb{Z}^2$ and there is $\\langle g'\\rangle \\in [\\langle g\\rangle]$  that is a direct factor of $\\Comm_{A_n}[\\langle g\\rangle]$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n\nIt is a well-known fact that the centers of the dihedral Artin groups have different shape depending on the parity of $n$. Hence, we should divide the study of the commensurators of their cyclic subgroups in two subcases, even and odd.\n\n\n{\\bf Case $n$ even: }\nIn the group $$A_{2n}=\\gp{a,b}{\\mathrm{prod}(a,b;2n)=\\mathrm{prod}(b,a;2n)},$$\nit is known that $Z(A_{2n})=\\gen{(ab)^{n}}$.\nMoreover, $A_{2n}\\cong \\gp{x,y}{x^{-1}y^nx=y^n}$ via the isomorphism defined by $x\\to b$, $y\\mapsto ba$.\nTherefore $A_{2n}\/Z(A_{2n})=\\gp{x,y}{x^{-1}y^nx=y^n,y^n =1}=\\gp{\\ol{x},\\ol{y}}{\\ol{y}^n=1}\\cong C_{\\infty}*C_{n}$.\n\n\nSince $\\ol{A_{2n}}$ is hyperbolic (even more, virtually free), the commensurator of an infinite order element  is virtually cyclic. (See for example \\cite[Theorem 2]{Arz} bearing in mind that infinite cyclic subgroups of hyperbolic groups are quasi-convex).\n\nSuppose first that $\\pi(g)$ has infinite order. Then $\\Comm_{\\overline{A_{2n}}}[\\langle \\pi(g)\\rangle]$ is infinite and virtually cyclic and hence $\\pi (\\Comm_{A_{2n}}[\\gen{g}])$ is infinite and virtually cyclic.\nSince all infinite virtually  cyclic subgroups of $\\ol{A_{2n}}$ are infinite cyclic  we have that  $\\pi (\\Comm_{A_{2n}}[\\gen{g}])$ is infinite cyclic. Therefore,\n$\\Comm_{A_{2n}}[\\gen{g}]$ is a central extension of $\\Z$ by $\\Z$ and hence $\\Comm_{A_n}[\\gen{g}]\\cong \\Z^2$.\nWe can take $g'$ as a pre-image of a generator of $\\pi (\\Comm_{A_{2n}}[\\gen{g}])$.\n\nSuppose now that $\\pi(g)$ has finite order.\nThus $\\langle g \\rangle \\cap Z(A_{2n})$ is non-trivial and infinite,  and thus $[\\langle g \\rangle]= [Z(A_{2n})]$ and $\\Comm_{A_{2n}}[\\gen{g}]=\\Comm_{A_{2n}}Z(A_n)=A_{2n}.$\n\n\n\n{\\bf Case $n$ odd:}\nIn the group $$A_{2n+1}=\\gp{a,b}{\\mathrm{prod}(a,b;2n+1)=\\mathrm{prod}(b,a;2n+1)}$$ we have that $Z(A_{2n+1})=\\gen{(ab)^{2n+1}}$.\nMoreover, $A_{2n+1}\\cong \\gp{x,y}{xy^n=y^{n+1}x^{-1}}$ via the isomorphism defined by $x\\to b$, $y\\mapsto ab$.\nTherefore\n\\begin{align*}\nA_{2n+1}\/Z(A_{2n+1})&=\\gp{x,y}{xy^nx=y^{n+1},y^{2n+1}=1}=\\gp{x,y}{(xy^n)^2=1=y^{2n+1}}\\\\\n&=\\gp{\\ol{z},\\ol{y}}{\\ol{z}^2=1=\\ol{y}^{2n+1}}\\cong C_{2}*C_{2n+1},\n\\end{align*}\nwhere $\\ol{z}$ denotes the class of $xy^nZ(A_{2n+1})$.\n\n\n\nSince $\\ol{A_{2n+1}}$ is hyperbolic and infinite virtually cyclic subgroups of $\\ol{A_{2n+1}}$ are infinite cyclic we get, arguing as above, that if $\\pi(g)$ has infinite order then $\\Comm_{A_{2n+1}}[\\gen{g}]\\cong \\Z^2$.\n\nSuppose now that $\\pi(g)$  has finite order. Arguing as above, $\\Comm_{A_{2n+1}}[\\gen{g}] = \\Comm_{A_{2n+1}}[Z(A_{2n+1})] = A_{2n+1}$.\n\\end{proof}\n\n\n\nWe  recall  the ordinary homology of these groups, which will be important in the remaining sections of the paper.\n\n\\begin{prop}\n\\label{OrdHom}\nFor $n\\geq 2$.\n\nIf $n$ is even, we have  $H_0(A_n)=\\mathbb{Z}$, $H_1(A_{n})=\\mathbb{Z}\\oplus \\mathbb{Z}$, $H_2(A_{n})=\\mathbb{Z}$ and $H_i(A_{n})=0$ for $i>2$.\n\n If $n$ is odd, we have $H_0(A_n)=\\mathbb{Z}$, $H_1(A_{n})=\\mathbb{Z}$ and $H_i(A_{n})=0$ for $i>1$.\n\n\\end{prop}\n\n\\begin{proof}\nAs dihedral Artin groups are one-relator groups, the Cayley complex (associated to the one-relator presentation) gives a 2-dimensional model for $K(A_n,1)$ (see \\cite{Lyndon} or \\cite{CMW04} for more general Artin groups).\n As $A_n$ is not free, this implies that $\\textrm{gd }A_n=\\textrm{cd }A_n=2$ for every $n$, and then in particular $H_i(A_n)=0$ for $i\\geq 3$. Moreover, as $K(A_n,1)$ is connected, $H_0(A_n)=\\mathbb{Z}$ for every $n$.\nThe formulae for $H_1$ follow from performing abelianization to the groups. Finally, the results for the Schur multiplier $H_2$ are a consequence of Theorem 3.1 in \\cite{CE09}.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Bredon homology of Artin groups of dihedral type}\n\\label{Sect:BredonArtin}\n\nIn this section we start with the computation of homological invariants of the Artin groups of dihedral type, which is the main goal of this paper. The computation of the Farrell-Jones homology in the case of a general ring is usually very complicated. A powerful tool to undertake this computation is the $G$-equivariant version of the Atiyah-Hirzebruch spectral sequence, which is a spectral sequence of the 1st and 4th quadrant. In our case of interest, the $E_2$-page of this sequence is given by Bredon homology of the classifying space ${\\underline{\\underline{E}}} G$, and the $E_{\\infty}$-page encodes the Farrell-Jones homology. Let us make the last statement more precise.\n\nLet $G$ be a discrete group, $\\mathcal{F}$ its family of virtually cyclic subgroups, $R$ a ring. For every $q\\in\\mathbb{Z}$, denote by $K_q(R[-])$ the covariant module over the orbit category $O_{\\mathcal F}(G)$ that sends every (left) coset  $G\/H$ to the $K$-theory group $K_q(R[H])$. In these conditions, the $E_2$-page of the Atiyah-Hirzebruch  spectral sequence that we will use is defined as $E_2^{p,q}=H_p^{\\mathcal{F}}({\\underline{\\underline{E}}} G,K_q(R[-]))$, for every $p\\geq 0$ and $q\\in\\mathbb{Z}$, and converges to $H^G_{p+q}({\\underline{\\underline{E}}} G,\\mathbf{K}(R))$ the $(p+q)$-th group of Farrell-Jones homology of ${\\underline{\\underline{E}}} G$ with coefficients in $R$. An excellent source for more information about this sequence is \\cite{LR05}.\n\nAlthough the path {to} the computation is very clear, in general it is difficult to obtain explicit formulae for the Bredon\nhomology of these classifying spaces. Different reasons for this are the complexity of the models for ${\\underline{\\underline{E}}} G$ and the fact that the exact values of $K_q(R[H])$ are only known for very special instances of $R$ and $H$. In fact, taking $R=\\mathbb{Z}$ and $H$ the trivial group, the groups $K_q(\\mathbb{Z})$ are not completely listed, as their value in some cases depend on the solution of the Vandiver conjecture, which remains unsolved. See \\cite{Wei05}.\n\n\nIn this section we show that for Artin groups of dihedral type it is possible to describe to some extent the aforementioned Bredon homology groups, with coefficients in $K_q(R[-])$ for a general ring $R$ (see Theorem \\ref{Thm:Bredon}); of course, the result strongly depends on the concrete shape of the $K$-theory groups of the group rings involved. The key to our computations is the description of the commensurators of the virtually cyclic subgroups of the groups $A_n$ (see previous section), and mainly the Mayer-Vietoris sequence associated to the push-out of L\\\"{u}ck-Weiermann model (Theorem \\ref{maintheorem}), that was explicitly stated by Degrijse-Petrosyan in Section 7 of \\cite{DP}, for the cohomological case. We offer here the homological version, which is the one needed in our context. To avoid confusions, we denote Bredon homology with respect to the family of virtually cyclic groups as $H_*^{vc}$ from now on.\n\n\\begin{prop}{\\rm (\\cite[Proposition 7.1]{DP}, see also \\cite[Theorem 2.3]{LW})}\n\\label{mainMV} Let  $M$ be a left Bredon module over $O_{{\\mathcal{F}}_{vc}}(A_n)$.\nThere is an exact sequence:\n\n$$\\ldots \\stackrel{}{\\rightarrow} H^{vc}_{i+1}(A_n,M) \\stackrel{g_1^{i+1}}{\\rightarrow} \\bigoplus_{[H]\\in I}H_i^{Fin\\cap \\Comm_{A_n}[H]}(\\Comm_{A_n}[H],M) \\stackrel{g_2^i}{\\rightarrow} $$\n\n$$\\stackrel{g_2^i}{\\rightarrow} (\\bigoplus_{[H]\\in I}H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M))\\oplus H_i^{Fin}(A_n,M) \\stackrel{g_3^i}{\\rightarrow} H^{vc}_i(A_n,M) \\rightarrow \\ldots$$\n\n\\end{prop}\n\nBefore we undertake our computations of the homology groups, we will compute their geometric dimension ${\\underline{\\underline{\\textrm{gd}}}\\ }$ with respect to the family of virtually cyclic groups.\n\n\\begin{prop}\n\\label{gd}\n\nThe geometric dimension of $A_n$ with respect of the family of virtually cyclic groups is 3.\n\n\\end{prop}\n\n\\begin{proof}\nAs seen in Lemma \\ref{lem:commensurators}, $A_n=\\langle a,b\\,|\\,\\textrm{prod}(a,b;n)=\\textrm{prod}(b,a;n) \\rangle$, contains subgroups isomorphic to $\\mathbb{Z}\\oplus\\mathbb{Z}$ (for example $\\Comm_{A_n}(\\langle a\\rangle))$. According to Example 5.21 in \\cite{LW}, this implies that ${\\underline{\\underline{\\textrm{gd}}}\\ } A_n\\geq 3.$\n\n\nOn the other hand, as the Artin groups $A_n$ are one-relator, Corollary 3 in \\cite{Deg16} implies that ${\\underline{\\underline{\\textrm{gd}}}\\ } A_n\\leq 3$, and then ${\\underline{\\underline{\\textrm{gd}}}\\ } A_n=3$.\n\\end{proof}\n\nAs stated, the strategy to compute $H_i^{vc}(A_n,M)$ goes through describing the different elements of the exact sequence of Proposition \\ref{mainMV}.\nWe first understand the terms $H_i^{\\mathcal{F}[H]}$ on Subsection \\ref{HiF[H]}.\nThen, in Subsection \\ref{Ktheory}, we give a more concrete description of the different homology groups when we take coefficients in the $K$-theory.\nFinally, in Subsection \\ref{morphism} {we study the homomorphisms} $g_1$ and $g_2$ of the exact sequence of Proposition \\ref{mainMV}  and prove our main theorems.\n\n\\subsection{Computing {the homology of the commensurators}}\\label{HiF[H]}\nLet us start the main calculations of this section.\nIt is clear from the previous Mayer-Vietoris sequence that the computations of the ordinary homology of the commensurators\n(which includes the homology of $A_n$, as the center of $A_n$ is virtually cyclic)\n and the homology of the commensurators with coefficients in $\\mathcal{F}[H]$ will give us valuable information about the Bredon homology of $A_n$ with respect to the family of virtually cyclic subgroups,\nso we will perform these calculations in the sequel.\nWe start with the ordinary homology, which is straightforward and depends on the shape of the commensurators:\n\n\\begin{itemize}\n\n\\item {If} $\\textrm{Comm}_{A_n}[H]\\simeq\\mathbb{Z}\\oplus \\mathbb{Z}$, we have $H_0(\\textrm{Comm}_{A_n}[H])=H_2(\\textrm{Comm}_{A_n}[H])=\\mathbb{Z}$, $H_1(\\textrm{Comm}_{A_n}[H])=\\mathbb{Z}\\oplus \\mathbb{Z}$, $H_i(\\textrm{Comm}_{A_n}[H])=0$ for $i>2$.\n\n\\item {If} $\\textrm{Comm}_{A_n}[H]\\simeq A_n$, see Proposition \\ref{OrdHom}.\n\n\n\\end{itemize}\n\n\nWe concentrate now in the case of $\\mathcal{F}[H]$, which we recall is the family of subgroups of $\\Comm_{A_n}[H]$ that are either finite or commensurable with $H$. Concretely, we intend to compute $H_i^{\\mathcal{F}[H]}(\\textrm{Comm}_{A_n}[H],M)$ for every non-trivial cyclic subgroup $H$ of $A_n$.\nWe have two cases, either $H\\cap Z(A_n)$ is trivial or not. We will use the following Convention throughout the paper without explictly refering to it.\n\n\\begin{conv}\nFor $[H]$ a class of infinite cyclic subgroups, we can always take a representative $H$ that is normal in $\\Comm_{A_n}[H]$.\n\n\nIf $H\\cap Z(A_n)$ is trivial, then Lemma \\ref{lem:commensurators} states  that $H\\unlhd \\Comm_{A_n}[H]$. Moreover, by  Lemma \\ref{lem:commensurators} we assume that  $H$ is a direct factor of  $\\Comm_{A_n}[H]$.\n\n\nIf $H\\cap Z(A_n)$ is non-trivial, then $[H]=[Z(A_n)]$ and we will assume in this case that we choose $Z(A_n)$ as the representative of $[H]$, and thus  $H\\unlhd \\Comm_{A_n}[H]$ again.\n\nWith this convention,  the projection $\\pi\\colon\\textrm{Comm}_{A_n}[H]\\rightarrow \\textrm{Comm}_{A_n}[H]\/H$ is well-defined and $\\Comm_{A_n}[H]\/H$ is isomorphic to $C_\\infty$, $C_\\infty*C_n$ or $C_2*C_{2n+1}$.\n\\end{conv}\n\n\nLet now $M$ be a module over the orbit category of $\\textrm{Comm}_{A_n}[H]$ with respect to $\\mathcal{F}[H]$, and $\\pi^{-1}M$ the induced module over the orbit category of $\\Comm_{A_n}[H]\/H$ with respect to the family of finite groups.\nThat is, for $K\\leqslant \\textrm{Comm}_{A_n}[H]\/H$ finite {it is defined}\n$$\\pi^{-1}M((\\Comm_{An}[H]\/H)\/K)\\mathrel{\\mathop\\mathchar\"303A}\\mkern-1.2mu= M(\\Comm_{A_n}[H]\/\\pi^{-1}(K)).$$ {Observe that this assignation gives rise to a natural transformation of functors $\\pi^{-1}M\\rightarrow M$. In the next result, which is essentially \\cite[Lemma 4.2]{DP}, but stated for homology instead of cohomology, we will see that this natural transformation induces an isomorphism in Bredon homology}. It will be a powerful tool in our computations.\n\\begin{prop}\n\\label{induced}\nLet $H$ be an infinite cyclic subgroup of $A_n$ normal in its commensurator.\nFor every $n\\geq 0$, and every module $M$ over the orbit category of $\\Comm_{A_n}[H]$ there is an isomorphism\n{$$H_i^{Fin}(\\Comm_{A_n}[H]\/H,\\pi^{-1} M)\\simeq H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M).$$}\nMoreover, every model for $\\uE(\\Comm_{A_n}[H]\/H)$ is a model for $E_{\\mathcal{F}[H]}\\Comm_{A_n}H$, with the action induced by the quotient map $\\Comm_{A_n}[H]\\to \\Comm_{A_n}[H]\/H$.\n\\end{prop}\n\\begin{proof}\nThe argument here is taken from the proof of \\cite[Lemma 4.2]{DP}. We write it here to make our paper more self-contained.\n\nThe projection $\\pi\\colon \\Comm_{A_n}[H]\\to \\Comm_{A_n}[H]\/H$ maps the family $\\mathcal{F}[H]$ onto the family $Fin$ of finite subgroups of the quotient $\\textrm{Comm}_{A_n}[H]\/H$.\nMoreover, the pre-image $\\pi^{-1}(K)$ for any finite group $K$ of $\\Comm_{A_n}[H]\/H$ lies in $\\mathcal{F}[H]$.\nTherefore, $\\uE \\Comm_{A_n}[H]\/H$ is a model for $E_{\\mathcal{F}[H]}\\textrm{Comm}_{A_n}[H]$, with the action induced by the projection $\\pi$.\n\nConsider now the spectral sequence associated to the short exact sequence\n$1\\to H\\to \\Comm_{A_n}[H]\\stackrel{\\pi}{\\to} \\Comm_{A_n}[H]\/H\\to 1$ for homology \\cite{Mar02}.\nFor every module $M$ over the orbit category of $\\Comm_{A_n}[H]$ we have\n$$E^{p,q}_2 (M)= H_p^{{Fin}}(\\Comm_{A_n}[H]\/H, H_q^{\\mathcal{F}[H]\\cap \\pi^{-1}(-)}(\\pi^{-1}(-), M)),  $$\nwhich converges to $E^{p,q}_{\\infty} (M)=H_{p+q}^{\\mathcal{F}[H]}(\\Comm_{A_n}[H], M)$.\n\nObserve that $E^{p,q}_2$ is trivial for $q\\geq 1$, as for every finite subgroup $K<\\Comm_{A_n}[H]\/H$, $\\pi^{-1}(K)$ belongs to the family  $\\mathcal{F}[H]\\cap \\pi^{-1}(K)$, and then $H_q^{\\mathcal{F}[H]\\cap \\pi^{-1}(-)}(\\pi^{-1}(-),M)$ is zero. Thus, as $E^{i,0}_2 (M)=H_i^{Fin}(\\Comm_{A_n}[H]\/H,\n\\pi^{-1}M)$\nin the 0-th row of the sequence, we have  {$$H_i^{Fin}(\\Comm_{A_n}[H]\/H,\\pi^{-1} M)\\simeq H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M).$$} for every $i\\geq 0.$\nWe are done.\n\\end{proof}\n\n The family of finite subgroups is easier to deal with than the family $\\mathcal{F}[H]$,  taking account of the previous proposition, the next step is to construct the corresponding classifying spaces for proper actions of the commensurators in $A_n$ modulo the corresponding subgroups. We have the following:\n\n\\begin{itemize}\n\n\n\\item If $\\textrm{Comm}_{A_n}[H]\\simeq\\mathbb{Z}\\oplus \\mathbb{Z}$, then there is a representative of the class $[H]$ that can be identified with one of these copies of $\\Z$; hence, we may assume that the inclusion $H\\hookrightarrow \\textrm{Comm}_{A_n}[H]=\\mathbb{Z}\\oplus \\mathbb{Z}$ is the inclusion of the first factor, and so $\\textrm{Comm}_{A_n}[H]\/H$ is isomorphic to $\\mathbb{Z}$. Then a model for $\\underline{{E}}(\\textrm{Comm}_{A_n}[H]\/H)$ is the straight line, and the action is by shifting.\n\n\\item If $\\textrm{Comm}_{A_n}[H]\\simeq A_n$, then $Z(A_n)$ is a representative of the class $[H]$.\nWe have seen that $\\textrm{Comm}_{A_n}[H]\/H=A_n\/Z(A_n)$ is an amalgamated product of two cyclic groups, depending its concrete shape on the parity of $n$.\nIn this case a tree model for ${\\uE}(\\textrm{Comm}_{A_n}[H]\/H)$ can be explicitly constructed.\n\nIf $n$ is even, we have that $A_{n}\/Z(A_{n})\\cong C_{\\infty}*C_{n}$.\nDenote $C_\\infty*C_{n}$ by $\\overline{A_n}$.\nLet $s$ be a generator of $C_\\infty$.\nThen Bass-Serre theorem guarantees that the graph with vertex set $\\overline{A_n}\/C_n$, edge set $\\overline{A_n}$ and incidence maps $\\iota(g) = gC_\\infty$ and $\\tau(g)=gsC_\\infty$ is a $\\overline{A_n}$-equivariant oriented tree.\n\n\nIf $n$ is odd, we have that $A_{n}\/Z(A_{n})\\cong C_{2}*C_{n}$. Denote $C_2*C_{n}$ by $\\overline{A_n}$.\nThen Bass-Serre theorem guarantees that the graph with vertex set $\\overline{A_n}\/C_2 \\sqcup \\overline{A_n}\/C_n$, edge set $\\overline{A_n}$ and incidence maps $\\iota(g) = gC_2$ and $\\tau(g)=gC_n$ is a $\\overline{A_n}$-equivariant oriented tree.\n\nNote that in both cases, the isotropy groups are the finite subgroups of $\\overline{A_n}$ and they fix exactly a vertex; and therefore these are  models for $\\uE \\overline{A_n}$.\n\\end{itemize}\n\n\n\n\nAs the classifying spaces of the commensurators that we have described are all 1-dimensional, we will use the following result of Mislin:\n\n\n\\begin{lem}[\\cite{MiVa03}, Lemma 3.14]\n\\label{1dim}\nSuppose that for a family $\\mathcal{F}$ of subgroups of a group $G$ there is a tree model $T$ for $E_{\\mathcal{F}}G$. Let $S_e$ be the stabilizer of the edge $e\\in T$ and $S_v$ the stabilizer of a vertex. Let $N$ be a coefficient module. Then $H_i^{\\mathcal{F}}(G,N)=0$ for $i>1$ and there is an exact sequence:\n\n$$ 0\\rightarrow H_1^{\\mathcal{F}}(G,N)\\rightarrow \\bigoplus_{[e]} N(G\/S_e) \\rightarrow \\bigoplus_{[v]} N(G\/S_v)\\rightarrow H_0^{\\mathcal{F}}(G,N)\\rightarrow 0,$$ where $[e]$ and $[v]$ run over the $G$-orbits of edges and vertices of $T$, respectively.\n\\end{lem}\n\n\\begin{rem}\n\\label{diff}\nIt is interesting to remark that the middle map $\\bigoplus_{[e]} N(G\/S_e) \\rightarrow \\bigoplus_{[v]} N(G\/S_v)$ is induced by the (formal) border operation defined by $\\partial[e]=[v_1]-[v_2]$, being $v_1$ and $v_2$ vertices of a representative $e$ of $[e]$. The tree is assumed to be oriented.\n\\end{rem}\n\nIn particular we obtain:\n\n\\begin{cor}\n\\label{zerohomology}\nFor every $n\\geq 2$, $i\\geq 2$, $H\\leqslant  A_n$ infinite cyclic and every module $M$ over the orbit category with respect to the family $\\mathcal{F}[H]$, we have $H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M)=0$.\n\\end{cor}\n\\begin{proof}\nRecall that by Proposition \\ref{induced} $H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M)\\simeq H_i^{Fin}(\\Comm_{A_n}[H]\/H,\\pi^{-1}M)$. As $\\uE\\Comm_{A_n}[H]\/H$ is 1-dimensional, the result follows by Lemma \\ref{1dim}.\n\\end{proof}\n\nSo it remains to compute $H_0^{\\mathcal{F}[H]}(\\textrm{Comm}_{A_n}[H],M)$ and $H_1^{\\mathcal{F}[H]}(\\textrm{Comm}_{A_n}[H],M)$. We assume here a general coefficient module $M$, but the reader may keep in mind that our case of interest is $M(A_n\/-\n)=K_q (R[-])$, $q\\in\\Z$.\n\n\n\\begin{prop}\\label{homology0and1}\nLet $A_n =\\langle a, b \\,\\textrm{ }|\\textrm{ } \\mathrm{prod}(a,b;n) = \\mathrm{prod}(b,a;n)\\rangle$\n be a dihedral Artin group, let $H\\leqslant A_n$ be an infinite cyclic group that is normal in its commensurator, let $\\pi\\colon \\Comm_{A_n} [H]\\to \\Comm_{A_n}[H]\/H$ be the natural projection.\nLet $M$ be a module over the orbit category of $\\Comm_{A_n}[H]$.\nThen\n\\begin{enumerate}\n\\item[(i)] If $H\\cap Z(A_n)$ is trivial, then\n$$H_0^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M)=H_1^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M)=M(\\Comm_{A_n}[H]\/H).$$\n\\item[(ii)] If $H=Z(A_{n})$, $n=2k+1$  and if $f\\colon M(A_{n}\/Z(A_n))\\to M(A_n\/\\langle b(ab)^k\\rangle ) \\oplus M(A_n\/\\langle ab \\rangle)$ {is} induced by the  natural projections,  we have\n$$H_0^{\\mathcal{F}[H]}(A_{n},M)= \\mathrm{coker} f\\quad \\text { and } \\quad H_1^{\\mathcal{F}[H]}(A_{n},M)= \\ker  f.$$\n\n\\item[(iii)] If $H=Z(A_n)$ and $n=2k$, we have\n $$H_0 ^{\\mathcal{F}[H]}(A_{n},M)=M(A_{n}\/\\langle ab \\rangle)\\quad \\text{  and  }\\quad H_1^{\\mathcal{F}[H]}(A_{n},M)= M(A_{n}\/Z(A_n)).$$\n\\end{enumerate}\nNote that in cases (ii) and (iii), $A_n=\\Comm_{A_n}[H]$.\n\\end{prop}\n\n\\begin{proof}\nIn the case $H\\cap Z(A_n)$ is trivial, the classifying space $\\uE\\Comm_{A_n}[H]\/H$ is the real line,  there is one orbit of edges and one orbit of vertices, and the action is free. Then by Remark \\ref{diff}, the central map in that exact sequence is trivial and by Lemma \\ref{1dim},\n$$H_1^{Fin}(\\Comm_{A_n}[H]\/H,\\pi^{-1}M)= \\pi^{-1}M((\\Comm_{A_n}[H]\/H)\/\\{1\\})$$ and\n$$H_0^{Fin}(\\Comm_{A_n}[H]\/H,\\pi^{-1}M)=\\pi^{-1}M((\\Comm_{A_n}[H]\/H)\/\\{1\\}).$$\nUsing  Proposition \\ref{induced}, case (i) follows.\n\nLet us discuss the case $H = Z(A_n)$. Now $\\Comm_{A_n}[H]\/H=A_n\/Z(A_n)$ is an amalgamated product of two finite cyclic groups or of a finite cyclic and an infinite cyclic group, depending on the parity of $n$. In both cases, there is a tree model for the classifying space for proper actions of the commutator modulo $H$. Denote $\\Comm_{A_n}[H]\/H$ by $\\overline{A_n}$ for brevity.\n\nWe consider first the case $n=2k+1$.\nHere $\\overline{A_{n}}=S*L$, with $S=\\langle b(ab)^k Z(A_{n})\\rangle \\cong C_2$ and $L=\\langle ab Z(A_{n}) \\rangle \\cong C_{n}$.\nThe Bass-Serre tree has two orbits of vertices, with stabilizers conjugated to $S$ or $L$, and one free orbit of edges.\nThen, by Lemma \\ref{1dim}\n\n\\begin{equation}\n\\label{MVodd}\nH_1^{Fin}(\\overline{A_n},\\pi^{-1} M)\\hookrightarrow \\pi^{-1}M(\\overline{A_{n}}\/\\{1\\})\\stackrel{f}{\\rightarrow}\n \\end{equation}\n$$\\stackrel{f}{\\rightarrow} \\pi^{-1}M(\\overline{A_{n}}\/S)\\oplus \\pi^{-1}M(\\overline{A_{n}}\/L)\\twoheadrightarrow H_0^{Fin}(\\overline{A_{n}},\\pi^{-1}M).\n$$\n Hence, we obtain that $$H_0^{Fin}(\\overline{A_{n}},\\pi^{-1}M)=(\\pi^{-1}M(\\overline{A_{n}}\/S)\\oplus \\pi^{-1}M(\\overline{A_{n}}\/L))\/\\textrm{Im }f$$ and $$H_1^{Fin}(\\overline{A_{n}},\\pi^{-1}M)=\\textrm{Ker }f.$$\n  Observe that the two components of $f$ are induced by the images of the projections $\\overline{A_{n}}\/\\{1\\}\\rightarrow \\overline{A_{n}}\/S$ and $\\overline{A_{n}}\/\\{1\\}\\rightarrow \\overline{A_{n}}\/L$ by the functor $\\pi^{-1}M$. The case (ii) follows by applying Proposition \\ref{induced}.\n\n\n\n\nConsider now the  case  $n=2k$.\nIn this situation $\\overline{A_{n}}=A_{n}\/Z(A_{n})$ is a free product $G=S\\ast L$, with $S=\\langle b Z(A_n) \\rangle \\simeq C_{\\infty}$ and $L=\\langle ab Z(A_n) \\rangle\\simeq C_n$.\nWe denote by $s$  the element  $b Z(A_n)$.\nThe Bass-Serre tree for this group has\none orbit of vertices with stabilizers conjugate to $L$ and one orbit of edges with trivial stabilizers.\n\nThe exact sequence of Proposition \\ref{1dim} has the following form:\n\\begin{equation}\n\\label{MVeven}\nH_1^{Fin}(\\overline{A_n},\\pi^{-1}M)\\hookrightarrow M(\\overline{A_{n}}\/\\{1\\}) \\stackrel{f}{\\rightarrow}  \\pi^{-1}M(\\overline{A_{n}}\/L) \\twoheadrightarrow H_0^{Fin}(\\overline{A_{n}},\\pi^{-1}M).\n\\end{equation}\nThe function $f$ is induced by  $g\\{1\\}\\in A_{2n}\/\\{1\\} \\mapsto gsL- gL$ by the functor $M$.\nAnd hence $f$ is trivial.\n\nHence, we deduce that $$H_1^{Fin}(\\overline{A_{n}},\\pi^{-1}M)= \\pi^{-1}M(\\overline{A_{n}}\/\\{1\\})$$ and $$H_0^{Fin}(\\overline{A_{n}},\\pi^{-1}M)=\\pi^{-1}M(\\overline{A_{n}}\/L).$$\nThe result now follows by applying Proposition \\ref{induced}.\n\\end{proof}\n\nObserve that taking into account that $\\overline{A_{2n}}$ is one-relator, this computation agrees with the result of Corollary 3.23 in \\cite{MiVa03}, in the case of $M=R_\\mathbb{C}$, the complex representation ring.\n\n\n\\subsection{ Coefficients in  $K$-theory groups}\n\\label{Ktheory}\nFrom the point of view of Farrell-Jones conjecture, the case of interest is the coefficients in the $K$-theory $K_q(R)$ of the group ring. We now describe all the homology groups of the exact sequence of Proposition \\ref{mainMV} with  coefficients in this module, except the groups $H^{vc}(A_n,K_q(R[-]))$ which will be studied in the next section.\n\nWe begin recalling the following definiton.\n\\begin{defn}\\label{defn:regular}\nA {\\it regular} ring is a\n commutative noetherian ring such that in the localization at every prime ideal, the Krull dimension of the maximal ideal is equal to the cardinal of a minimal set of generators.\n\\end{defn}\nExamples of regular rings include fields (of dimension zero) and Dedekind domains. If $R$ is regular then so is the polynomial ring $R[x]$, with dimension one greater than that of $R$.\n\n\n\n Before stating our results, we also need to recall the \\emph{Bass-Heller-Swan decomposition} in $K$-theory, which permits to {decompose} the $K$-theory of $R[\\Z]$. For thorough approaches to algebraic $K$-theory the reader is referred to \\cite{Bas68}, \\cite{Wei13} or \\cite{Luc21}.\n\n\\begin{thm}[\\cite{Sri91}, Theorem 9.8]\n\\label{BHS}\nGiven a ring $R$ and $q\\in\\Z$, there exists an isomorphism $$K_q(R[\\Z])\\simeq K_q(R)\\oplus K_{q-1}(R)\\oplus NK_q(R)\\oplus NK_q(R),$$ which is natural in the ring $R$.\n\\end{thm}\n\n{The additional terms $NK_q(R)$ are called the Nil-terms, and $NK_n(R)$ is defined as the kernel of the homomorphism induced in $K_q$ by the homomorphism $R[t]\\rightarrow R$ which sends $t$ to $1$. These terms vanish for a regular ring $R$, see \\cite[Section 9]{Sri91}.}\n\n{In our computations it will be important to understand the endomorphism $ind_n$ of $K_q(R[\\Z])$ induced by multiplication by $n$ in $\\Z$. The references for the sequel are \\cite[Section 2]{HaLu12}, \\cite{Sti82} and \\cite{Wei81}.}\n\nAccording to Bass-Heller-Swan-decomposition, $ind_n$ can be seen as a homomorphism\n$$ind_n\\colon K_q(R)\\oplus K_{q-1}(R)\\oplus NK_q(R)\\oplus NK_q(R)\\rightarrow K_q(R)\\oplus K_{q-1}(R)\\oplus NK_q(R)\\oplus NK_q(R).$$\n\\noindent{Here, by naturality of the decomposition, the image of $ind_n|_{K_q(R)}$ lies inside $K_q(R)$, the image of $ind_n|_{K_{q-1}(R)}$ lies inside $K_{q-1}(R)$ and analogously for the Nil-terms. Now, the restriction of  $ind_n$ to $K_q(R)$ is the identity and the restriction of  $ind_n$ to $K_{q-1}(R)$ is multiplication by $n$.  By Farrell's trick \\cite{Far77}, $ind_n$ admits a transfer $res_n$ such that $res_n\\circ ind_n$ is multiplication by $n$.\nThis implies the following:}\n\n{\\begin{prop} \\label{kernel}\nIn the previous notation, the kernel of $ind_n$ is isomorphic to a direct sum $T_1(K_q(R))\\oplus T_2(K_q(R))$, where $T_1(K_q(R))$ is the $n$-torsion subgroup of $K_{q-1}(R)$ and $T_2(K_q(R))$ is a subgroup of the  $n$-torsion subgroup of $NK_q(R)\\oplus NK_q(R).$\n\\end{prop}}\n\n{It should be remarked that the restriction of $ind_n|_{NK_q(R)}$ (sometimes call the \\emph{Frobenius map} in the literature) is related to the action of big Witt vectors in $NK_q(R)$, and is hard to describe in general. Anyhow, the previous proposition will be important when computing the Bredon homology of Artin groups of dihedral type.}\n\n\n\nThe following  propositions\n{particularize} our previous results when the coefficient module is $K$-theory.\n\n For the sake of clarity, we maintain the notation from Proposition \\ref{mainMV}, although we are aware that sometimes can be found redundant.\n\n\\begin{prop}\n\\label{Anordinary}\n\nLet us consider an Artin group $A_n$ of dihedral type, $q\\in\\mathbb{Z}$, $R$ a ring. Then we have:\n\n\\begin{itemize}\n\n\\item $H_0^{Fin}(A_n,K_q(R[-]))=K_q(R)$.\n\n\\item $H_1^{Fin}(A_n,K_q(R[-]))=K_q(R)\\oplus K_q(R)$ if $n$ is even\n\n\\item $H_1^{Fin}(A_n,K_q(R[-]))=K_q(R)$ if $n$ is odd.\n\n\\item $H_2^{Fin}(A_n,K_q(R[-]))=K_q(R)$ if $n$ is even\n\\item $H_2^{Fin}(A_n,K_q(R[-]))=0$ if $n$ is odd.\n\n\\item $H_i^{Fin}(A_n,K_q(R[-]))=0$ if $i\\geq 3$.\n\n\\end{itemize}\n\n\\end{prop}\n\n\\begin{proof}\n\nAs $A_n$ is torsion-free, Bredon homology with respect to the family of finite groups is in fact ordinary homology. The proposition is then obtained by applying Universal Coefficient Theorem \\cite[Theorem 3A.3]{Hat02} to the ordinary homology groups of $A_n$ (Proposition \\ref{OrdHom}), taking into account that the latter are torsion-free and that the $K$-theory groups are abelian.\n\\end{proof}\n\nWe consider now {the} Bredon homology of the commensurators.\n\n\\begin{prop}\n\\label{FinComm}\n\nLet us consider an Artin group $A_n$ of dihedral type, $q\\in\\mathbb{Z}$, $R$ a ring. Let $\\Comm_{A_n}[H]$ be the commensurator of a virtually cyclic group in $A_n$. Then:\n\n\\begin{itemize}\n\n\n\\item If $\\Comm_{A_n}[H]=\\mathbb{Z}\\oplus\\mathbb{Z}$, then\n\n$$H_i^{Fin\\cap \\Comm_{A_n}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=\\begin{cases} K_q(R) & \\text{ if $i=0,2$}\\\\\nK_q(R)\\oplus K_q(R) & \\text{ if i =1}\\\\\n0 & \\text{otherwise}. \\end{cases}$$\n\n\n\\item If $\\Comm_{A_n}[H]=A_n$, then $$H_i^{Fin\\cap \\Comm_{A_n}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=H_i^{Fin}(A_n,K_q(R))$$ and this case was described in the previous proposition.\n\n\n\\end{itemize}\n\n\\end{prop}\n\n\\begin{proof}\n\nGiven a commensurator $\\textrm{Comm}_{A_n}[H]$, its Bredon homology with respect to the family $Fin\\cap \\textrm{Comm}_{A_n}[H]$ is its ordinary homology. Then, {as in the previous proposition}, the result follows by applying the Universal Coefficient Theorem to the ordinary homology groups of the commensurators.\n\\end{proof}\n\nWe now undertake the remaining case. In item (3) we follow the notation of Proposition \\ref{kernel}.\nMoreover, we use the following notation.\n\\begin{nt}\\label{not: CKR}\nFor $n$ odd, we denote by $C(K_q(R))$ the cokernel of the homomorphism $$\\tilde{f}\\colon K_q(R[\\Z])\\to K_q(R[\\Z])\\oplus K_q(R[\\Z])$$ induced by\n$f\\colon R[\\Z]\\rightarrow R[\\Z]\\oplus  R[\\Z]$ induced in the first component by multiplication by 2 and in the second by multiplication by $n=2k+1$.\n\\end{nt}\nBy Bass-Heller-Swan decomposition, $C(K_q(R))$ this is a quotient of $K_q(R)\\oplus K_q(R)\\bigoplus(\\oplus_{i=1}^2 K_{q-1}(R))\\bigoplus (\\oplus_{i=1}^4 NK_q(R))$. Moreover, $\\tilde{f}$ restricts to the component corresponding to  $K_q(R)$ of the Bass-Heller-Swan decomposition of $K_q(R[\\Z])$ to the diagonal map to $K_q(R)\\oplus K_q(R)$, a component  of $K_q(R[\\Z])\\oplus K_q(R[\\Z])$.\nThus, $C(K_q(R))$ can be viewed as a quotient of $K_q(R)\\bigoplus(\\oplus_{i=1}^2 K_{q-1}(R))\\bigoplus (\\oplus_{i=1}^4NK_q(R))$ that can be identified in many cases, see Section \\ref{Sect:concrete}.\n\n\n\n\\begin{prop}\n\\label{FHcomm}\nLet $A_n$  be an Artin group of dihedral type, $q\\in\\mathbb{Z}$, $R$ a ring.\nConsider for every non-trivial virtually cyclic $H\\leqslant A_n$ and the family $\\mathcal{F}[H]$. Then:\n\n\n\n\\begin{enumerate}\n\n\n\\item If $i\\geq 2$, then $$H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=0.$$\n\n\\item If $H\\cap Z(A_n)$ is trivial, then $$H_0^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=H_1^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=K_q(R[\\mathbb{Z}]).$$\n\n\\item If $n$ is odd and $\\Comm_{A_n}[H]=A_n$, then $$H_0^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=C(K_q(R))$$ and $$H_1^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=T_1(K_q(R))\\oplus T_2(K_q(R)).$$\n\n\\item If $n$ is even and $\\Comm_{A_n}[H]=A_n$, then $$H_0^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=H_1^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=K_q(R[\\mathbb{Z}]).$$\n\n\\end{enumerate}\n\n\\end{prop}\n\n\n\\begin{proof}\n\nWe will check every item separately.\n\n\\begin{enumerate}\n\n\n\n\\item It follows straightforward from Corollary \\ref{zerohomology}.\n\n\\item The claim follows from item (i) in Proposition \\ref{homology0and1}, taking into account that $H$ is infinite cyclic an then $K_q(R[H])=K_q(R[\\mathbb{Z}])$.\n\n\\item Let $n=2k+1$. Consider the homomorphism $f\\colon M(A_{n}\/Z(A_n))\\to M(A_n\/\\langle b(ab)^k\\rangle ) \\oplus M(A_n\/\\langle ab \\rangle)$ defined in item (2) of Proposition \\ref{homology0and1}.\nTaking $M(A_n\/-)=K_q(R[-])$, we obtain a homomorphism $f_K:K_q(R[Z(A_n)])\\to K_q(R[\\langle b(ab)^k\\rangle])\\oplus K_q(R[\\langle ab \\rangle])$.\nObserve that the two components of $f_K$ are respectively induced by the inclusions $Z(A_n)\\hookrightarrow \\langle b(ab)^k\\rangle$ and $Z(A_n)\\hookrightarrow \\langle ab \\rangle$, which are both inclusions $\\Z\\hookrightarrow \\Z$ given respectively by multiplication by 2 and multiplication by $n$.\nThen, $f_K$ can be seen as the homomorphism $K_q(R[\\Z])\\rightarrow K_q(R[\\Z])\\oplus K_q(R[\\Z])$ induced by each multiplication in the corresponding component.\nAccording to Proposition \\ref{kernel}, the kernel of the first component of this homomorphism is equal to $T_1(K_q(R))\\oplus T_2(K_q(R))$, while its cokernel is $C(K_q(R))$ by definition.\nThe result now follows from item (2) in Proposition \\ref{homology0and1}.\n\n\n\n\\item  In this case $A_n=\\Comm_{A_n}[H]$. The claim follows from item (3) in Proposition \\ref{homology0and1}, taking into account that $H$ is infinite cyclic and  then $K_q(R[H])=K_q(R[\\mathbb{Z}])$.\n\\end{enumerate}\n\\end{proof}\n\n\\subsection{Understanding the homomorphisms of Proposition \\ref{mainMV}}\n\\label{morphism}\nIn our way to describe the Bredon homology of $A_n$ with respect to the family of virtually cyclic groups, we should describe to some extent the homomorphisms that appear in the Mayer-Vietoris sequence of Proposition \\ref{mainMV}.\n\n In the following we will use without explicit mention the previous three propositions, which identify the terms of the exact sequence. We also maintain the name of the homomorphisms in the sequence.\nNote that the superscript of the homomorphism $g^k_i$ specifies the degree of the homology in the source of $g^k_i$.\nMoreover,  when we need to refer to the $j$-th component of $g_i^k$, we will write $g_{ij}^k$.\nFor example, $g_{22}^1$ is the second component of the homomorphism $g_2^1$ defined over the first homology group.\n\nTo prove our results, we need to analyze in detail the following homomorphism\n\\begin{equation}\\label{g2}\n\\bigoplus_{[H]}H_i^{Fin\\cap \\Comm_{A_n}[H]}(\\Comm_{A_n}[H],M) \\stackrel{g_2^i}{\\rightarrow} \\end{equation}\n$$ \\stackrel{g_2^i}{\\rightarrow}\\left(\\bigoplus_{[H]}H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M)\\right)\\oplus H_i^{Fin}(A_n,M).\n$$\n\n\nThe homomorphism $g_{21}^i$ is induced by  the  vertical left arrow $\\sqcup_{[H]\\in I}\\mathrm{id}_{A_n}\\times_{\\Comm_{A_n}[H]}f_{[H]}$ of L\\\"{u}ck-Weiermann push-out  \\eqref{eq:Luck-Weiermann} below (see also Theorem \\ref{maintheorem}).\nIn turn, the homomorphism $g_{22}^i$ is induced by the inclusion in the upper horizontal arrow of the push-out  \\eqref{eq:Luck-Weiermann}\n\\begin{equation}\n\\label{eq:Luck-Weiermann}\n\\xymatrix{  \\coprod_{[H]\\in I}A_n\\times_{\\Comm_{A_n}[H]}\\uE \\Comm_{A_n}[H] \\ar[r]^{\\hspace{2cm} i} \\ar[d]^{\\coprod_{[H]\\in I}id_{A_n}\\times_{\\Comm_{A_n}[H]}f_{[H]}}  & \\uE A_n \\ar[d] \\\\\n\\coprod_{[H]\\in I}A_n\\times_{\\Comm_{A_n}[H]}E_{{\\mathcal{F}} [H]}\\Comm_{A_n}[H]  \\ar[r] & X. }\n\\end{equation}\n\n\n\n\nWe will decompose the homomorphism $g_2^i$ of Equation \\eqref{g2} into homomorphisms $g_{2[H]}^i$ which are the restriction of $g_2^i$ to the factor of the domain corresponding to $[H]$. We will write\n\n\\begin{equation}\n\\label{eq:g}\ng_{2[H]}^i\\colon H_i^{Fin\\cap \\Comm_{A_n}[H]}(\\Comm_{A_n}[H],M)\\to H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M)\\oplus H_i^{Fin}(A_n,M).\n\\end{equation}\nNote that we are making a slight abuse of notation, as the real codomain of $g_{2[H]}^i$ is the same as {that of} $g_2^i$, but we have chosen to write only the subgroup where the image of $g_{2[H]}^i$ lies.\nMoreover, when needed, we will further decompose $g^i_{2[H]}$ into $g^i_{21[H]}\\oplus g^i_{22[H]}$ indicating the different factors of the image of $g^i_{2[H]}$. Observe that this notation is coherent with the previous one.\n\n\nAs all the commensurators are torsion-free, $H_i^{Fin\\cap \\Comm_{A_n}[H]}(\\Comm_{A_n}[H],M)$ is ordinary homology with coefficients in $M(\\Comm_{A_n}[H]\/\\{1\\})$.\n\n\n\n\nWe now analyze the homomorphism $g_2^i$ on the case $M=K_q(R[-])$. Before that, we introduce the following notation.\n\\begin{nt}\nWe will denote\n$N_q^{[H]} =K_{q-1}(R)\\oplus NK_q(R)\\oplus NK_q(R)$, where the superindex means that this group is associated to a concrete commensurability class $[H]$.\n\\end{nt}\nWe now will give more detailed descriptions of the homomorphism $g_2^i$ and the cokernels introduced above.\n\n\\subsubsection{{\\bf The homomorphism $g_2^2$ when $n$ odd:}}\n\\label{NqH}\nIn this case we have that the codomain of $g_2^2$ is  $(\\bigoplus H_2^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M))\\oplus H_2^{Fin}(A_n,M)=\\{0\\}$, and therefore for all commensurability classes of infinite  cyclic subgroups $[H]$ the homomorphisms $g_{2[H]}^2$ are trivial.\n\n\\subsubsection{{\\bf The homomorphism $g_2^2$ when $n$ even:}}\nIn this case we have that the codomain of $g_2^2$ is  $(\\bigoplus H_2^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M))\\oplus H_2^{Fin}(A_n,M)=(\\bigoplus_{[H]}\\{0\\})\\oplus K_q(R)$.\nMoreover, by Proposition \\ref{Anordinary} and Proposition \\ref{FinComm},\nthe domain of all $g_{2[H]}^2$ are the same, {and} we have\n$$g_{2[H]}^2 \\colon K_q(R)\\to \\{0\\}\\oplus K_q(R).$$\nNote that $g_{22[H]}^2$ is induced by the inclusion given by the upper arrow in the push-out.\nWhen $H=Z(A_n)$, we have that $\\Comm_{A_n}[H]=A_n$ and therefore this inclusion is the identity.\nSince $g_{22[Z(A_n)]}^2$ is surjective, we have that $g_2^2$ is surjective.\n\n\n\n\n\\subsubsection{{\\bf The homomorphism $g_2^1$ when $n$ is odd:}}\n\\label{g21}\n{Before we describe this case, we need to make some considerations.\nRecall that we assume that $H$ is normal in $\\textrm{Comm}_{A_n}[H]$.\nObserve that in the models described in Section \\ref{HiF[H]} for $\\underline{E}(\\textrm{Comm}_{A_n} [H]\/H)$, the stabilizers of the edges are trivial for every $H$.\nThis means that if we consider the space $\\underline{E}(\\textrm{Comm}_{A_n} [H]\/H)$ as a model for $E_{{\\mathcal{F}} [H]}\\Comm_{A_n}[H]$ (see Proposition \\ref{induced}) the stabilizers of the edges are always isomorphic to $H$.\nOn the other hand, in any model of $\\underline{E}\\textrm{Comm}_{A_n} [H]$ the stabilizer of the edges should be trivial, as the action of $\\textrm{Comm}_{A_n} [H]$ is free.\nThen, the homomorphism induced by $f_{[H]}:\\underline{E}\\textrm{Comm}_{A_n} [H]\\rightarrow E_{{\\mathcal{F}} [H]}\\Comm_{A_n}[H]$ in the first chain group of the Bredon complex with coefficients in a module $M$ takes every copy of $M(\\textrm{Comm}_{A_n} [H]\/{1})$ to a copy of $M(\\textrm{Comm}_{A_n}[H]\/H)$ with the homomorphism induced by the inclusion of the trivial group in $H$. In particular, if $M=K_q(R[-])$ for some $q$, the corresponding homomorphism $K_q(R)\\rightarrow K_q(R[\\mathbb{Z}])$ is given by the inclusion of $K_q(R)$ in the corresponding piece of the Bass-Heller-Swan decomposition of $K_q(R[\\mathbb{Z}])$. This fact will be very useful in the sequel}.\n\nIn the following we use the notation of Proposition \\ref{kernel} when needed.\nWe have that $$g_{2[Z(A_n)]}^1\\colon K_q(R)\\to T_1(K_q(R))\\oplus T_2(K_q(R))\\oplus K_q(R)$$ and for $H$ nontrivial, $[H]\\neq [Z(A_n)]$\n$$g_{2[H]}^1\\colon  K_q(R)\\oplus K_q(R)\\to   K_q(R[\\Z]) \\oplus K_q(R).$$\n\n\n\n\nNote that there are infinitely many commensurability classes of infinity cyclic subgroups different from the $[Z(A_n)]$ and hence infinitely many terms of this  kind.\nWe now examine the cases $[H]\\neq [Z(A_n)]$ and $[H]=[Z(A_n)]$ separately.\n\nIf $[H]=[Z(A_n)]$ then we have $\\Comm_{A_n}[H]=A_n$.\n\n In order to describe $g^{1}_{21[Z(A_n)]}$, consider the homomorphism $$C_1^{Fin}(\\underline{E}\\textrm{Comm}_{A_n}[H],K_q)\\rightarrow C_1^{{\\mathcal{F}} [H]}(E_{{\\mathcal{F}} [H]}\\Comm_{A_n}[H],K_q)$$ at the level of Bredon chains that induces $g^{1}_{21[Z(A_n)]}$ in homology. According to the previous considerations, this homomorphism is given by the inclusion $i:K_q(R)\\hookrightarrow K_q(R[H])=K_q(R[\\Z])$ given by Bass-Heller-Swan decomposition. But by item (3) of Proposition \\ref{FHcomm}, the image of this homomorphism is trivial in $H_1^{{\\mathcal{F}} [H]}(E_{{\\mathcal{F}} [H]}\\Comm_{A_n}[H],K_q)$, and hence $g^{1}_{21[Z(A_n)]}$ is also trivial. In turn, $g^{1}_{22[Z(A_n)]}$ is the identity.\n\n\n\nIf $[H]\\neq[Z(A_n)]$ then $\\Comm_{A_n}[H]=\\Z^2$, and $\\Comm_{A_n}[H]\/H$ is isomorphic to $\\Z$.\nIn this case $g_{21 [H]}^1$ is given by {a} homomorphism $K_q(R)\\oplus K_q(R)\\rightarrow K_q(R[\\Z])$, where we assume that the first component of the domain corresponds to $H$ and the second to $Z(A_n)$. As the homomorphism $g_{21 [H]}^1$ is induced in homology by the quotient homomorphism $\\textrm{Comm}_{A_n}[H]\\rightarrow \\textrm{Comm}_{A_n}[H]\/H$, the previous results imply that the first component of $g_{21 [H]}^1$ is trivial, while the second, which corresponds to the center, identifies the copy of $K_q(R)$ in the Bass-Heller-Swan decomposition of $K_q(R[\\Z])$.\n\n\n\n On the other hand,  $g_{22[H]}^1\\colon (\\Z\\oplus\\Z)\\otimes K_q(R)\\rightarrow \\Z\\otimes K_q(R)$  is defined by the abelianization of the inclusion $H_1(\\Comm_{A_n}[H])\\rightarrow H_1(A_n)$ in the first component of the tensor product and by the identity in the second.\n\nNow since the image of $g^1_{2[Z(A_n)]}$ is precisely given by the copy of $K_q(R)$ that corresponds to $H_1(A_n,K_q)$, the previous computations imply that the cokernel of $g^1_2$ is then equal to $(\\bigoplus_{[H]\\neq [Z(A_n)]}N_q^{[H]})\\oplus T_1(K_q(R))\\oplus T_2(K_q(R))$.\n\n\\subsubsection{ {\\bf The homomorphism $g_2^1$ when $n$ is even:}}\n\n\nThis homomomorphism is defined in equation \\eqref{eq:g} and, according to Propositions {\\ref{Anordinary}, \\ref{FinComm} and \\ref{FHcomm}}, its first component is given by\n$$g_{2[H]}^1 \\colon K_q(R)\\oplus K_q(R)\\to K_q(R[\\Z]) \\oplus ( K_q(R)\\oplus K_q(R))$$\nfor every commensurability class $[H]$. Let us describe this component with more detail.\n\n\n\n\nWe consider first the case $[H] \\neq [Z(A_n)]$. Here, the same argument as in the odd case proves that the component of $g_{21[H]}^1$ given by the center is the inclusion $K_q(R)\\hookrightarrow K_q(R[\\Z])$ via Bass-Heller-Swan decomposition, and the other component is trivial.\n\nAlso when $[H]\\neq [Z(A_n)]$ the homomorphism $g_{22[H]}^1$ is identified (via Proposition \\ref{FinComm} and the Universal Coefficient Theorem) with a homomorphism $(\\Z\\oplus \\Z) \\otimes K_q(R)\\rightarrow (\\Z\\oplus\\Z)\\otimes K_q(R)$, which comes, as above, from tensoring with $K_q(R)$ the homomomorphism $H_1(\\Comm_{A_n}[H])\\rightarrow H_1(A_n)$ given by abelianization of the inclusion of the commensurator in $A_n$.\n\nNow we consider $[H]=[Z(A_n)]$. As in the previous case, the homomorphism $g_{21[H]}^1$ can be described as:\n$$H_1^{Fin}(A_n,K_q(-))\\to H_1^{{\\mathcal{F}} [H]}(A_n,\\pi^{-1} K_q(-)) \\cong H_1^{Fin}(A_n\/Z(A_n),\\pi^{-1} K_q(-)).$$\nNow recall from Proposition \\ref{homology0and1} that\n$$H_1^{Fin}(A_n\/Z(A_n),\\pi^{-1} K_q(-))=\\pi^{-1}K_q(R[\\{1\\}])=K_q(R[Z(A_n)])=K_q(R[\\Z]),$$\nwhere $Z(A_n)$ is interpreted here as the stabilizer of the unique $A_n$-class of edges in the model of $E_{{\\mathcal{F}} [H]}A_n$ described in Section \\ref{HiF[H]}.\nOn the other hand, the two copies of $K_q(R)$ in $H_1(A_n,K_q(R))$ come from taking values of the module $K_q(R[-])$ on the trivial group, interpreted as the stabilizer of two different $A_n$-classes of edges in a model of $EA_n$. Again taking into account our previous considerations about stabilizers, the two components of $g_{21[Z(A_n)]}^1 \\colon K_q(R)\\oplus K_q(R)\\to K_q(R[\\Z])$ induce inclusion of the $K_0(R)$ via Bass-Heller-Swan decomposition. On the other hand, it is clear that $g_{22[Z(A_n)]}^1$ is the identity.\n\nAs $H_1(A_n)$ is a free abelian group of rank 2 generated by the images of $a$ and $b$ under abelianization, we may assume that the two copies of $K_q(R)$ in the image of $g_{2[H]}^1$ correspond respectively to these two copies of $\\Z$, after tensoring with $K_q(R)$. Now observe that if $H=\\langle a\\rangle$, the image of the restriction of  $g_{22[H]}^1$ to the homology of its commensurator is exactly the first of the two copies of $K_q(R)$, while the image of the restriction of $g_{22[H]}^1$ to the homology of the commensurator of $H=\\langle b\\rangle$ is the other copy. As the restrictions of $g_{21[H]}^1$ to the homology of these commensurators are trivial, we obtain that $\\{0\\}\\oplus K_q(R)\\oplus K_q(R)$ lies in the image of $g_{2[H]}^1$. In fact, the description of $g_{21[H]}^1$ for $H=Z(A_n)$ implies that the copy of $K_q(R)$ inside $K_q(R[\\Z])$ is also in the image, and now it is easy to conclude that the image of $g_{2[H]}^1$ is in fact $(\\bigoplus_{[H]}) K_q(R))\\oplus K_q(R)\\oplus K_q(R)$, corresponding the big direct sum to the copies of $K_q(R)$ included in each copy of $K_q(R[\\Z])$, and the remaining two copies corresponding to the ordinary homology of $A_n$. In particular, we have that $\\textrm{coker } g_2^1=\\bigoplus_{[H]} N_q^{[H]}.$\n\n\n\n\n\n\\subsubsection{{\\bf The homomorphism $g_2^0$ for every $n$:}}\n\nSimilar considerations to those of the beginning of Section \\ref{g21} hold here.\nIf $x$ is a vertex of $\\underline{E}\\textrm{Comm}_{A_n}[H]$ such that the stabilizer of $f_{[H]}(x)$ is infinite cyclic (here $f_{[H]}$ is the function of \\eqref{eq:Luck-Weiermann}), then the induced map $K_q(R)\\rightarrow K_q(R[\\Z])$ induces the injection in the correspondent component of Bass-Heller-Swan decomposition. This is always the case except when $n$ is odd, $H=Z(A_n)$ and $f_{[H]}(x)$ has the shape $gC_{\\infty}$ if we consider $\\underline{E}(\\textrm{Comm}_{A_n} [H]\/H)$ as a $\\textrm{Comm}_{A_n} [H]\/H$-complex.\nThis will be enough to describe $g_2^0$ to the extent we need.\n\nWe begin describing $g_{2[H]}^0$ according to the different prossibilities of $[H]$ and $n$.\n\nWhen $H\\neq Z(A_n)$, the results of Propositions \\ref{Anordinary}, \\ref{FinComm} and \\ref{FHcomm} imply that $g_{2[H]}^0$ is defined in the following way:\n$$g_{2[H]}^0\\colon K_q(R)\\to K_q(R[\\Z])\\oplus K_q(R).$$\nBy the previous considerations, the homomorphism $g_{21[H]}^0$ identifies $K_q(R)$ as the corresponding direct summand of $K_q(R[\\mathbb{Z}])$ in the Bass-Heller-Swan decomposition, while $g^{0}_{22[H]}$ is induced by the inclusion $\\textrm{Comm}_{A_n} [H]\\hookrightarrow A_n$, counts the number of connected components of the classifying space, and then is the identity.\n\nWhen $n$ is even and $H=Z(A_n)$, we have that $H_0^{{\\mathcal{F}} [H]}(\\textrm{Comm}_{A_n} [H],K_q(-))=K_q(R[\\Z])$ by Proposition \\ref{FHcomm}, and this copy of $\\Z$ corresponds to a stabilizer which is isomorphic to $C_n$, a cyclic group of order $n$. Then, by the previous considerations, the homomorphism $g_{2[H]}^0$ behaves as in the case $n$ even and $H\\neq Z(A_n)$.\n\nIn the case $n$ odd and $H=Z(A_n)$, again by Proposition \\ref{FHcomm} we have that $ H_0^{{\\mathcal{F}} [H]}(\\textrm{Comm}_{A_n} [H],K_q(-))=C(K_q(R))$, where $C(K_q(R))$ was defined in Notation \\ref{not: CKR}.\nThen $g_{21[H]}^0$ is defined as a homomorphism\n$$g_{2[H]}^0\\colon K_q(R)\\to C(K_q(R))\\oplus K_q(R).$$\nRemark from the definition of $C(K_q(R))$ that this group is the direct sum of $K_q(R)$ with quotients of $\\oplus_{i=1}^2 K_{q-1}(R)$ and $\\oplus_{i=1}^4 NK_q(R)$,\nand that this copy of $K_q(R)$ comes from the identification of the respective copies of $K_q(R)$ that appear in the Bass-Heller-Swan decomposition of $K_q(R[\\langle b(ab)^k\\rangle])$ and $K_q(R[\\langle ab\\rangle])$.\nThese copies correspond respectively to the stabilizers $C_2$ and $C_{2k+1}$ in $\\underline{E}(\\textrm{Comm}_{A_n} [H]\/H)$ (as a model for the $\\textrm{Comm}_{A_n} [H]\/H$-action). Then, the previous considerations about stabilizers imply that $g_{21[H]}^0$ maps this $K_0(R)$ isomorphically to the mentioned copy of itself inside $C(K_q(R))$.\nIn turn, $g_{22[H]}^0$ is again the identity, arguing as in the even case.\n\n\nWith the previous information, we can describe $\\mathrm{coker}g_2^0$.\n\nNow if $n$ is even, observe that the source of $g_2^0$ is $\\oplus_{[H]} K_q(R)$ and the codomain is $(\\oplus_{[H]}K_q(R[\\Z]))\\oplus K_q(R)$.\nMoreover, the image of every $g_{2[H]}^0$ consists in a copy of $K_q(R)$ inside $K_q(R[\\Z])$ (again the copy that appears in the Bass-Heller-Swan decomposition), and the copy of $K_q(R)$ that corresponds to $H_0^{Fin}(A_n,M)$, which is fixed for any choice of $H$. Hence, the cokernel of the homomorphism $g_2^0$ is isomorphic to the quotient of a direct sum of copies of $K_q(R[\\Z])$ (indexed by $[H]$) by the identification of all the copies $K_q(R)$ which are the images of the homomorphisms $g^0_{22[H]}$. Observe that  the copies of $K_q(R[\\Z])$ are ``glued\" by the copy of $K_q(R)$ that corresponds to $H_0^{Fin}(A_n,M)$, and hence we have $$\\mathrm{ coker } g_2^0=(\\bigoplus_{[H]} N_q^{[H]})\\oplus K_q(R).$$\n\nWhen $n$ is odd, we only need to take into account that in the case of $H=Z(A_n)$ the role of $K_q(R[\\Z])$ in the codomain is played by $C(K_q(R))$.\nThen, using the same argument as in the previous case and denoting by $\\overline{C}(K_q(R))$ the quotient of $C(K_q(R))$ under the copy of $K_q(R)$ in the Bass-Heller-Swan decomposition, we obtain that: $$\\mathrm{ coker } g_2^0=(\\bigoplus_{[H]\\neq [Z(A_n)]} N_q^{[H]})\\oplus K_q(R)\\oplus\\overline{C}(K_q(R)).$$\n\nObserve that $g_2^0$ is a monomorphism for every $n$, as the inclusions of $K_q(R)$ in $K_q(R[\\Z])$ and $C(K_q(R))$ are so.\n\n\\vspace{0.5cm}\n\n\nNow we can describe the Bredon homology of $A_n$ with respect to the family of virtually cyclic groups.\nObserve that the (co)kernels of the statement have been previously described, and that the groups $N_q^{[H]}$ were defined in Section \\ref{NqH}.\n\n\n\n\\begin{thm}\n\\label{Thm:Bredon}\nLet $A_n$ be an Artin group of dihedral type. In the previous notation, we have the following:\n\\begin{enumerate}\n  \\setlength{\\itemindent}{-2em}\n\\item $H_i^{vc}(A_n,K_q (R[-]))=\\{0\\}$ for $i\\geq 4$.\n\\item $H_3^{vc}(A_n,K_q (R[-]))=\\begin{cases}\\bigoplus_{[H]\\neq [Z(A_n)]} K_q(R) & \\text{$n$ odd}\\\\\n\\ker g_2^2 & \\text{$n$ even.}\n\\end{cases}$\n\n\\item $H_2^{vc}(A_n,K_q (R[-]))= \\ker g_2^1$.\n\n\\item $H_1^{vc}(A_n,K_q (R[-]))= \\emph{coker } g_2^1= \\begin{cases}(\\bigoplus_{[H]\\neq [Z(A_n)]}N_q^{[H]})\\oplus T_1(K_q(R))\\oplus T_2(K_q(R)) & \\text{$n$ odd}\\\\\n\\bigoplus_{[H]} N_q^{[H]}\n  & \\text{$n$ even.}\n\\end{cases}$%\n\\item $H_0^{vc}(A_n,K_q (R[-]))= \\emph{ coker } g_2^0= \\begin{cases}(\\bigoplus_{[H]\\neq [Z(A_n)]} N_q^{[H]})\\oplus K_q(R)\\oplus\\overline{C}(K_qR) & \\text{$n$ odd}\\\\\n(\\bigoplus_{[H]} N_q^{[H]})\\oplus K_q(R)\n  & \\text{$n$ even.}\n  \\end{cases}$%\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\n\nThe proof is based on the sequence of Proposition \\ref{mainMV} and the previous homological computations. We will check the five items in a separate way.\n\n\\begin{enumerate}\n\\item By Proposition \\ref{Anordinary}, Proposition \\ref{FinComm}  and Proposition \\ref{FHcomm} the sequence of Proposition \\ref{mainMV} is identically trivial to the left of $g_3^3$.\n\n\\item As the exact sequence of Proposition \\ref{mainMV} is identically trivial to the left of $g_3^3$, the map $g_3^3$ is trivial and by exactness, $H_3^{vc}(A_n, K_q(R[-]))$ is isomorphic to the kernel of $g_2^2$. This completes the even case.\n\n\nFor the case $n$ odd, we claim that $g_1^3$ is an isomorphism.\nIndeed, from Proposition \\ref{Anordinary} and Proposition \\ref{FHcomm}, the term $(\\bigoplus H_2^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M))\\oplus H_2^{Fin}(A_n,M)=\\{0\\}$ and in particular  $\\textrm{Im}(g_2^2)=\\{0\\}$.\nSince $g_1^3$ is injective, the claim follows.\nBy Proposition \\ref{FinComm}, the specific description of the odd case follows.\n\n\\item There are two different arguments depending if $n$ is odd or even.\n\nFor $n$ odd, as the term  $(\\bigoplus H_2^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M))\\oplus H_2^{Fin}(A_n,M)$  is trivial, the statement is a direct consequence of the exactness of the sequence of Proposition \\ref{mainMV}.\n\nFor $n$ even, we have seen in the discussion above about $g_2^2$ that this map is surjective, and hence $g_3^2$ is the trivial map.\nThis implies that $H_2^{vc}(A_n, K_q(R[-]))$ is the image of $g_1^2$, or equivalently, the kernel of $g_2^1$.\n\n\\item The previous description of the homomorphisms in the Mayer-Vietoris sequence proves that $g_2^0$ is a monomorphism, and hence $g_3^1$ is surjective and $H_1^{vc}(A_n,K_q( R[-]))$ the cokernel of $g_2^1$, which has been described above in terms of the summands $N_q^{[H]}$.\n\n\\item Since the Mayer-Vietoris sequence {ends} at $H_0^{vc}(A_n ,K_q( R[-]))$ this term is equal to the image of $g_3^0$ which is equal to the cokernel of $g_2^0$ that was  described above.\n\\end{enumerate}\n\\end{proof}\n\n\nAs said in the introduction, this theorem opens the door to concrete computations of Bredon homology of $A_n$ with respect to the family of virtually cyclic groups, provided there is available information about the $K$-theory of the coefficient ring. In particular, when $R$ is a regular ring the absence of Nil-terms and negative K-theory groups make the calculations easier and more precise.\nFor instance, we have\n\\begin{cor}\\label{cor:K_0}\nLet $A_n$, $n>2$ be an Artin group of dihedral type. Let $R$ be a regular ring. Then $K_0(RA_n)=K_0(R)$.\n\\end{cor}\n\\begin{proof}\nAs $R$ is regular, $K_{i}(R)$ vanishes for $i<0$ and also the Nil-Terms of the Bass-Heller-Swan decomposition.\nIn particular, by Theorem \\ref{Thm:Bredon}, $H_0^{vc}(A_n, K_0(R[-]))= K_0(R)$.\nMoreover, as $K_{i}(R)$ vanishes for $i<0$ we see  by Theorem \\ref{Thm:Bredon}, that $H_j^{vc}(A_n,K_i(R[-]))=\\{0\\}$ for $i<0$.\nThen the $E_2$-page of the Atiyah-Hirzebruch spectral sequence is concentrated in the non-negative part of the 0th, 1st, 2nd and 3rd columns, and $E_{\\infty}^{0,0}=E_2^{0,0}= K_0(R)$.\n\\end{proof}\n\nIn particular, this implies that every finitely dominated $CW$-complex whose fundamental group is $A_n$ has the homotopy type of a finite $CW$-complex.\n\nIn the following section we compute these Bredon homology groups for different choices of the ring, including some non-regular ones.\n\n\\section{Computations of $H_i^{vc}(A_n,K_q (R[-]))$ for several coefficient rings}\n\\label{Sect:concrete}\n\nIn this section we use Theorem \\ref{Thm:Bredon} to describe $H_i^{vc}(A_n,K_q (R[-]))$ for some instances of the ring $R$, both regular ($\\mathbb{Z}, \\mathbb{F}_q$) and non-regular ($\\mathbb{Z}[\\mathbb{Z}\/{\\bf 2}], \\mathbb{Z}[\\mathbb{Z}\/{\\bf 2}\\times \\mathbb{Z}\/{\\bf 2}]$ and $\\mathbb{Z}[\\mathbb{Z}\/{\\bf 4}]$).\nWe recall that all these groups give information about the $E^2$-term of the corresponding Atiyah spectral sequence. We point out that in the regular cases many groups can be computed, because the $K$-theory of $\\Z$ is nearly known and the $K$-theory of $\\mathbb{F}_q$ is known (see the corresponding examples below).\nIn the non-regular framework, by contrast, it is very difficult to find concrete descriptions of $K_q(R)$ for $q>1$, or of the corresponding Nil-terms.\n\nRecall that the groups $H_i^{vc}(A_n,K_0 (R[-]))$ are trivial for $i\\geq 4$ and any ring $R$.\n\n\\begin{ex}\n\\label{KZ}\n\nFirst we compute the Bredon homology of $A_n$ with respect to the family of virtually cyclic subgroups, taking as coefficients $K_q(\\Z [-])$, for $q=0,1,2$.\n\nWe need in our computations the lower algebraic $K$-theory groups of the integers. The groups $K_q(\\Z)$ are known for all $q\\leq 7$ and all $q\\geq 8$ such that $q \\not\\equiv 0 \\mod 4$,  see  \\cite[page 2]{Wei05}.\nFor example,  the first values of $K_q(\\Z)$ are given in the following table:\n\\begin{center}\\begin{tabular}{c|c|c|c|c|c|c|c|c|c}\n\\hline\n$q$ & $<0$ &0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\n\\hline\n$K_q(\\Z)$ & 0 & $\\Z$ & $\\Z\/{\\bf 2}$ & $\\Z\/{\\bf 2}$ & $\\Z\/{\\bf 48}$ & 0 & $\\Z$ & 0 & $\\Z\/{\\bf 240}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nObserve that, as $\\Z$ is regular, the Nil-terms in the Bass-Heller-Swan decomposition are trivial, and then for every $i\\in \\mathbb{N}$, $K_i(\\Z[\\Z])=K_i(\\Z)\\oplus K_{i-1}(\\Z)$ and $N_q^{[H]}=K_{q-1}(\\Z)$. Moreover, also by regularity, $K_i(\\Z)=0$ if $i\\leq -1$.\n\nTaking into account of all these considerations, the previous theorem implies the following.\n\nFor $q=0$, we have:\n\n\\begin{enumerate}\n\n\\item $H_3^{vc}(A_n,K_0 (\\Z[-]))\\simeq H_2^{vc}(A_n,K_0 (\\Z[-])) \\simeq \\bigoplus_{\\aleph_0} \\Z$.\n\\item {$H_1^{vc}(A_n,K_0 (\\Z[-]))=0$.}\n\\item $H_0^{vc}(A_n,K_0 (\\Z[-]))=\\Z$.\n\n\\end{enumerate}\n\nNow for $q=1$,\n\n\\begin{enumerate}\n\n\n\n\\item $H_3^{vc}(A_n,K_1 (\\Z[-]))\\simeq H_2^{vc}(A_n,K_1 (\\Z[-])) \\simeq \\bigoplus_{\\aleph_0} \\Z \/{\\bf 2}$.\n\\item {$H_1^{vc}(A_n,K_1 (\\Z[-]))=\\bigoplus_{\\aleph_0} \\Z$}.\n\\item {$H_0^{vc}(A_n,K_1 (\\Z[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus \\Z \/{\\bf 2}$}.\n\n\\end{enumerate}\n\nAnd finally, for $q=2$, all these groups are isomorphic to $\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2}$.\n\n\nLet us briefly explain these computations.\nWe consider first $q=0$.\nFor $H_3^{vc}(A_n,K_0 (\\Z[-]))$, the odd case is immediate.\nIn the even case, $H_3^{vc}(A_n,K_1 (\\Z[-]))$ is the kernel of a homomorphism  $\\bigoplus_{\\aleph_0} \\Z\\rightarrow \\Z$, and then isomorphic to $\\bigoplus_{\\aleph_0} \\Z$.\nThis easy argument will be frequently used in the examples of this section without express mention.\n\nWe now deal with $H_2^{vc}(A_n,K_0 (\\Z[-]))$.\nWe only deal with the odd case, the even one is very similar.\nFirst observe that the kernel of $g_2^1$ is isomorphic to $\\bigoplus_{\\aleph_0} \\Z$.\nMore precisely, taking account the description of Section \\ref{g21} and the values of the $K_i(\\Z)$, $g_2^1$ (for $n$ odd) is defined in the following way:\n\n$$g_2^1: (\\bigoplus_{[H]\\neq [Z(A_n)]} \\Z^2)\\oplus \\Z\\rightarrow (\\bigoplus_{[H]\\neq [Z(A_n)]} \\Z)\\oplus \\Z.$$\n\nHere the kernel of $g_{21[H]}^1$ is isomorphic to $\\Z$ for every $[H]\\neq [Z(A_n)]$.\nAs there is an infinite number of such commensurators, we conclude that the kernel should be also infinite, and then $H_2^{vc}(A_n,K_1 (\\Z[-]))=\\bigoplus_{\\aleph_0} \\Z$.\n\nObserve that, as $K_{-1}(\\Z)=0$ and $\\Z$ is regular, $N_0^{[H]}=T_i(K_0\\Z)=\\overline{C}(K_0\\Z)=0$.\nNow the values of $H_1^{vc}(A_n,K_0 (\\Z[-]))$ and $H_0^{vc}(A_n,K_0 (\\Z[-]))$ are easily deduced from items (4) and (5) of Theorem \\ref{Thm:Bredon}.\n\nNow take $q=1$. For $H_3^{vc}(A_n,K_1 (\\Z[-]))$ and $H_2^{vc}(A_n,K_1 (\\Z[-]))$ it is argued as in the previous case, taking into account that every subgroup of an $\\mathbb{F}_2$-vector space is again an $\\mathbb{F}_2$-vector space.\nNow observe that $N_1^{[H]}=K_0(\\Z)=\\Z$ by the Bass-Heller-Swan decomposition, and regularity and the fact that $K_0(\\Z)=\\Z$ is torsion-free imply that $T_1(K_1\\Z)=0$, $T_2(K_1\\Z)=0$ and $\\overline{C}(K_1\\Z)=\\Z$.\nThe values of $H_1^{vc}(A_n,K_1 (\\Z[-]))$ and $H_0^{vc}(A_n,K_1 (\\Z[-]))$ follow again from items (4) and (5) of Theorem \\ref{Thm:Bredon}.\n\nFinally, for $q=2$, the values of the homology are immediately implied by regularity and the fact that $K_2(\\Z)=K_1(\\Z)=\\Z \/{\\bf 2}$.\n\n\\end{ex}\n\n\n\\begin{ex}\n\nNow we will compute the groups $H_i^{vc}(A_n,K_q (\\mathbb{F}_2[-]))$, for $0\\leq q\\leq 3$. First, the following table (see \\cite{Qui73}) includes the algebraic $K$-groups that are necessary in our computations:\n\n\\begin{center}\\begin{tabular}{c|c|c|c|c|c}\n\\hline\n$q$ & $<0$ & 0 & 1 & 2 & 3 \\\\\n\\hline\n$K_q(\\mathbb{F}_2)$ & 0 & $\\Z$ & 0 & 0 & $\\Z\/{\\bf 3}$  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\n\nAs $\\mathbb{F}_2$ is regular, the same considerations about the Nil-terms and the negative $K$-groups apply also in this case. Hence, again as a direct consequence of Theorem \\ref{Thm:Bredon} we have the following results. For $q=0$ and every $i\\in\\mathbb{N}$, $H_i^{vc}(A_n,K_0 (\\mathbb{F}_2[-]))=H_i^{vc}(A_n,K_0 (\\Z[-]))$, because $K_0(\\Z)=K_0(\\mathbb{F}_2)$ and then  $K_0 (\\mathbb{F}_2[-])=K_0 (\\Z[-])$ because both have the same Bass-Heller-Swan decomposition.\n\nFor $q=1$, we have $H_3^{vc}(A_n,K_1 (\\mathbb{F}_2[-]))=H_2^{vc}(A_n,K_1 (\\mathbb{F}_2[-]))=0$, as $K_1(\\mathbb{F}_2)=0$. On the other hand, as {$N_1^{[H]}=\\Z$} for every $H$, $H_1^{vc}(A_n,K_1 (\\mathbb{F}_2[-]))=H_0^{vc}(A_n,K_1 (\\mathbb{F}_2[-]))=\\bigoplus_{\\aleph_0}\\Z$.\n\n\nFor $q=2$, the triviality of $K_2(\\mathbb{F}_2)$ and $K_1(\\mathbb{F}_2)$ implies the triviality of $H_i^{vc}(A_n,K_2 (\\mathbb{F}_2[-]))$ for every $i$.\n\nFinally, for $q=3$, we have $H_2^{vc}(A_n,K_3 (\\mathbb{F}_2[-]))\\simeq H_3^{vc}(A_n,K_3 (\\mathbb{F}_2[-]))=\\bigoplus_{\\aleph_0}\\Z \/{\\bf 3}$.\nAs the groups $N_3^{[H]}$ are  trivial and $K_2(\\mathbb{F}_2)$ is so, we obtain that  $H_1^{vc}(A_n,K_3 (\\mathbb{F}_2[-]))=\\Z \/{\\bf 3}$ is trivial, and  $H_0^{vc}(A_n,K_3 (\\mathbb{F}_2[-]))=\\Z \/{\\bf 3}$.\n\\end{ex}\n\n\n\nWe follow with some non-regular examples, namely the rings $\\Z [\\Z \/{\\bf 2}]$, $\\Z [\\Z \/{\\bf 2}\\times \\Z \/{\\bf 2}]$ and $\\Z [\\Z \/{\\bf 4}]$, which we will respectively denote by $R_1$, $R_2$ and $R_3$.\nWe will compute the groups $H_i^{vc}(A_n,K_q (R_j[-]))$, for $0\\leq i\\leq 3$, $q=0,1$ and $1\\leq j\\leq 3$.\nIn order to do this we will need the values of their lower algebraic $K$-theory groups, as well as the Nil groups.\nAll these groups are displayed in the following table:\n\n\\begin{center}\\begin{tabular}{c|c|c|c|c|c}\n\\hline\n& $K_1$ & $K_0$ & $K_{-1}$ & $NK_0$ & $NK_1$ \\\\\n\\hline\n$\\Z [\\Z \/{\\bf 2}]$  & $(\\Z \/{\\bf 2})^2$ & $\\Z$ & 0 & 0 & 0 \\\\\n\\hline\n$\\Z [\\Z \/{\\bf 2}\\times \\Z \/{\\bf 2}]$  & $(\\Z \/{\\bf 2})^3$ & $\\Z$ & $\\Z^r$ & $\\bigoplus_{\\aleph_0} \\Z\/{\\bf 2}$ & $\\bigoplus_{\\aleph_0} \\Z\/{\\bf 2} $\\\\\n\\hline\n$\\Z [\\Z \/{\\bf 4}]$  & $\\Z \/{\\bf 2}\\times \\Z \/{\\bf 4}$ & $\\Z$ & $\\Z^s$ & $\\bigoplus_{\\aleph_0} \\Z\/{\\bf 2}$&$ \\bigoplus_{\\aleph_0} \\Z\/{\\bf 2}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nLet us briefly explain the values of the table.  By work of Oliver \\cite[Theorem 14.1-2]{Oli88}, the kernels $SK_1(R_j)$ of the determinant maps are trivial, and hence (see for example \\cite{Ste78}) $K_1(R_j)$ is the group of units of $R_j$. Now the values of $K_1$ follow from a theorem of Higman (\\cite{Hig40}, see also \\cite[II.4.1]{She78}). In turn, by \\cite[Proposition 6]{Cas73}, the reduced $K_0$ of these three rings is trivial, and then \\cite[Lemma 2.18]{Luc21} implies that $K_0(R_j)=\\Z$ for every $j$. The values of the third column follow from work of Carter \\cite[Theorem 1]{Car80}, being the figures $r$ and $s$ positive integers that depend on the Schur indexes. Finally, the Nil-terms of the two last columns were computed by Weibel in \\cite{Wei09}.\n\nWe are now ready to describe the homology of the dihedral Artin groups with respect to the family of virtually cyclic groups, referred to the $K$-theory $K_q$ of these group rings, $q=0,1$. As before, our main tool is Theorem \\ref{Thm:Bredon}.\n\n\n\n\n\\begin{ex}\nWe start with $R_1=\\Z [\\Z \/{\\bf 2}]$.\nWhen $q=0$, $H^{vc}_3(A_n,K_0(R_1[-]))=H^{vc}_2(A_n,K_0(R_1[-]))=\\bigoplus_{\\aleph_0} \\Z $.\nTaking into account that $N_0^{[H]}=0$ for every $H$ (because $K_{-1}(R_1)$ is trivial, and also the Nil-terms), we obtain that $H^{vc}_1(A_n,K_0(R_1[-]))=0$.\nOn the other hand, $H^{vc}_0(A_n,K_0(R_1[-]))=\\Z$ too.\n\nIf $q=1$, $H^{vc}_3(A_n,K_1(R_1[-]))=H^{vc}_2(A_n,K_1(R_1[-]))=\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2} $.\nMoreover, as $\\Z$ is torsion-free and then the correspondent $T_1(K_1(R_1))$ is trivial, we have $H^{vc}_1(A_n,K_1(R_1[-]))=\\bigoplus_{\\aleph_0}\\Z$.\nFinally, taking into account that $C(K_1(R_1))=\\Z$ in this case, $H^{vc}_0(A_n,K_1(R_1[-]))=\\Z \/{\\bf 2}\\oplus \\Z \/{\\bf 2}\\oplus(\\bigoplus_{\\aleph_0} \\Z)$.\n\n\\end{ex}\n\n\\begin{ex}\nWe continue by considering $R_2=\\Z [\\Z \/{\\bf 2}\\times \\Z \/{\\bf 2}]$.\nNow $H^{vc}_3(A_n,K_0(R_2[-]))=H^{vc}_2(A_n,K_0(R_2[-]))=\\bigoplus_{\\aleph_0} \\Z $, exactly as in the previous example. We have $N_0^{[H]}=K_{-1}(R_2)\\oplus NK_0(R_2)\\oplus NK_0(R_2)=\\Z^r\\oplus (\\bigoplus_{\\aleph_0}\\Z \/{\\bf 2})$ and then $H^{vc}_1(A_n,K_0(R_2[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})$.\n Observe that the extra term $T_2(K_0(R_2))$ is an $\\mathbb{F}_2$-vector space of at most countable dimension, and then it is included in the previous direct sum.\n Finally, $H^{vc}_0(A_n,K_0(R_2[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})$ as in the previous case, taking into account that $\\overline{C}(K_0(R_2))$ is a direct sum of free abelian groups and an $\\mathbb{F}_2$-vector space, both of at most countable dimension.\n\nWhen $q=1$, $H^{vc}_3(A_n,K_1(R_2[-]))=H^{vc}_2(A_n,K_0(R_2[-]))=\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2}$, because $K_1(R_2)=(\\Z \/{\\bf 2})^3$.\nNow $N_q^{[H]}=\\Z\\oplus \\bigoplus_{\\aleph_0}\\Z \/{\\bf 2}$, and then by similar reasons to the previous case, $H^{vc}_1(A_n,K_1(R_2[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})$.\nSimilarly $H^{vc}_0(A_n,K_1(R_2[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})$.\n\n\\end{ex}\n\n\\begin{ex}\nTo conclude, we consider $R_2=\\Z [\\Z \/{\\bf 4}]$.\nAs the groups $K_0$, $NK_0$ and $NK_1$ are all isomorphic as abelian groups to their counterparts in the previous example\nand $K_{-1}$ is also free abelian, $H^{vc}_3(A_n,K_0(R_3[-]))=H^{vc}_2(A_n,K_0(R_3[-]))=\\bigoplus_{\\aleph_0} \\Z $, $H^{vc}_1(A_n,K_0(R_3[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})$, and $H^{vc}_0(A_n,K_0(R_3[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})$.\n\nFor $q=1$, it is clear that\n$H^{vc}_3(A_n,K_1(R_3[-]))=(\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 4})$.\nNow, the analysis of $g_2^1$ in Section \\ref{g21} guarantees that $H^{vc}_2(A_n,K_1(R_3[-]))$ should contain a subgroup isomorphic to $(\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 4})$, and then be isomorphic to it, as the source of $g_2^1$ is also isomorphic to $(\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 4})$.\nWe have $N_1^{[H]}=K_{0}(R_3)\\oplus NK_0(R_2)\\oplus NK_0(R_2)=\\Z\\oplus (\\bigoplus_{\\aleph_0}\\Z \/{\\bf 2})$.\nBy an analogous reasoning to the case of $R_2$, $H^{vc}_1(A_n,K_1(R_3[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})$.\nFinally, $H^{vc}_0(A_n,K_1(R_3[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})\\oplus \\Z \/{\\bf 4}$, corresponding the extra term to $K_1(R_3)$.\n\n\\end{ex}\n\n\\begin{rem}\nObserve that the proposed method permits computations of the Bredon homology (with respect to the family of virtually cyclic groups) of $A_n$ with respect to the $K$-theory of any ring for which there is some knowledge of the algebraic $K$-groups.\nWe remark that the computation of the lower algebraic $K$-theory groups is a hot topic nowadays (see for example \\cite{LaOr07}, \\cite{HiJu2021} or \\cite{GJM18}), so it seems possible that good knowledge about the $E^2$-term of the Atiyah-Hirzebruch spectral sequence is achieved in these cases, even for more general families of Artin groups.\n\n\\end{rem}\n\n\n\n\n\nWhen the ring $R$ is regular, it is possible to take a shortcut in order to compute the left-hand side of the Farrell-Jones conjecture. Observe that Theorem 0.1 in \\cite{LS16}, that establishes a splitting $$H^G({\\underline{\\underline{E}}} G,\\mathbf{K} (R))\\simeq H^G(\\underline{E} G,\\mathbf{K} (R))\\oplus H^G({\\underline{\\underline{E}}} G, \\underline{E} G, \\mathbf{K} (R)),$$ for every group $G$ and ring $R$.\nNow if $R$ is regular and $G$ is torsion-free, the second term of the direct sum vanishes \\cite[Proposition 2.6]{LR05}. Likewise, when $G$ is torsion-free $\\underline{E}  G=EG$, the classical universal space for principal $G$-bundles, and then $H^G(\\underline{E}  G,\\mathbf{K}(R))=H(BG, \\mathbf{K}(R))$, being the latter ordinary homology. Hence, in the case $G=A_n$, it is enough to compute $H^(BA_n, \\mathbf{K}(R))$ to obtain the left-hand side of the Farrell-Jones conjecture for these groups. When the coefficients $\\mathbf{K}(R)$ are known, and taking into account that the Artin groups of dihedral type are one-relator, Lemma 16.21 in \\cite{Luc21} provides an accurate description of these homology groups, corresponding case i) of the lemma to $n$ even and case ii) to $n$ odd. For example, when $K=\\mathbb{F}_p$ for $p$ prime, the classical results of \\cite{Qui73} provide a complete knowledge of the groups $H^(BA_n, \\mathbf{K}(R))$ , while for $K=\\mathbb{Z}$ much information is available (\\cite[page 2]{Wei05}). When $R$ is not regular, however, this strategy does not work, as in this case groups in the left-hand side of Farrell-Jones. In this context, we expect that our results on Bredon homology of $A_n$ (znd in particular the last examples of this section) can bring some light over the groups $H^{A_n}({\\underline{\\underline{E}}} A_n,\\mathbf{K} (R))$, via appropriate computations in the equivariant Atiyah-Hirzebruch spectral sequence. Observe that, according to Theorem \\ref{Thm:Bredon}, this spectral sequence has four columns. This fact indicates that the sequence should collapse at most in the page $E_4$, and hence this page should provide the knowledge of Farrell-Jones groups. The analysis of the differentials, as well as the subsequent description of the $E_3$ and $E_4$-pages, seems a difficult and interesting future line of research.\n\n\n\n\\noindent{\\textbf{{Acknowledgments.}}}\n\nWe warmly thank D. Juan-Pineda, W. L\\\"{u}ck, N. Petrosyan, L.J. S\\'anchez-Salda\\~{n}a, V. Srinivasan and J. Stienstra and an anonymous referee for their useful comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nEntanglement entropy is a non-local observable which measures entanglement between two subsystems of a quantum system.\nIt has many applications in studies of phenomena in quantum gravity, quantum information, condensed matter and high energy physics.\nParticularly, entanglement entropy in the context of the gauge\/gravity duality is aimed to shed some light on understanding of quantum gravity into the bulk \\cite{Rangamani:2016dms}.\n\nFollowing the holographic prescription, the  entanglement entropy between a subsystem (region) $A \\in\\mathbb{R}^{d}$ that has a  $d-1$-dimensional boundary $\\partial A$ and a remaining part $B$  can be calculated by Ryu-Takayanagi formula \\cite{Ryu:2006bv,Ryu:2006ef,Hubeny:2007xt}\n\\begin{eqnarray}\\label{1.1}\nS =  \\frac{\\text{Area}(\\gamma_{A})}{4G^{d+2}_{N}},\n\\end{eqnarray}\nwhere $\\gamma_{A}$ is the minimal $d$-dimensional surface in $AdS_{d+2}$ space whose boundary coincides with the boundary of the region $A$ ($\\partial A = \\partial \\gamma_{A}$), $G^{d+2}_{N}$ is $d +2$-dimensional Newton constant.\n\nFor the classic  case with $AdS$ on the gravity side, which geometry is not supported by any scalar field,\n and conformal theory of QFT side the area of the surface $\\gamma_{A}$ is defined through the induced metric  by the relation\n\\begin{equation}\\label{1.1a}\nA = \\int d^{d}\\sigma \\sqrt{|\\det{G_{\\alpha\\beta}}|},\n\\end{equation}\nwhere  $G_{\\alpha\\beta}=g_{MN}\\partial_{\\alpha}X^{M}\\partial_{\\beta}X^{N}$ is the induced metric of $\\gamma_{A}$ and $g_{MN}$ is the metric of the background. Important examples of $AdS$ spacetimes include near-horizon geometries of $p$-branes.\nIn the form (\\ref{1.1a}) it can be applied to studies of entanglement entropy for non-dilatonic branes, namely D3, M2 and   M5 branes \\cite{Quijada:2017zif}.\n\nThe  generalization of the entangled functional (\\ref{1.1}) with (\\ref{1.1a}) for branes with non-conformal boundaries reads\n\\begin{eqnarray}\\label{RTncbr}\nS = \\int d^{8}\\sigma \\frac{1}{4G^{10}_{N}} \\sqrt{|\\det G_{ind}|}e^{-2\\phi},\n\\end{eqnarray}\nwhere $\\phi$ is the dilaton.\nThe holographic entanglement entropy for configurations on D2 and NS five-branes was calculated in the original work  \\cite{Ryu:2006ef},\non D3 and  D4 branes in \\cite{Klebanov:2007km,Pakman:2008ui,Arean:2008az}, on D1-D5 brane intersection in \\cite{Asplund:2011cq}.\n The dilaton destroys the scale symmetry, \nbut  we still can detect  a certain field theory on the boundaries of the branes and discuss a holographic picture.\nFor example, \nfor NS5 brane in the long distances of the theory is governed by  the (2, 0) SCFT for IIA theory and the IR free SYM with sixteen supercharges for IIB, while\nthe short distance behavior leads to a linear dilaton geometry   \\cite{Aharony:1998ub} that can be described through the so called Little String Theory  $\\mathcal{N}=(2,0)$ and $\\mathcal{N}=(1,1)$ on the Type IIA and Type IIB NS5 branes respectively.\n\n\n\n\nIn this work we aim at studying T-duality aspects of entanglement entropy for field theories living on NS five branes, including the exotic branes $5_2^r$ with $r=0,1,2,3,4$. For the NS5 brane the decoupling limit is known to be LST, which is a 6-dimensional theory describing dynamics of string-like degrees of freedom which do not have gravitational modes in their spectrum. In all other respects they exhibit essentially stringy behaviour, such as Hagedorn temperature and T-duality of spectrum \\cite{Losev:1997hx, Kutasov:2001uf, Aharony:1999ks}. This is due to the fact that in contrast to D-branes the decoupling limit for the NS branes does not involve taking $\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon'\\to0$. This preserves stringy properties on the world-volume. The origin of T-duality in LST is the simple observation that a compactified NS5 brane transform into itself under T-duality along a world-volume direction. One the language of LST this transform into T-duality symmetry of the 6d theory with one compact direction, the direct analogue of that of the 10d string theory.\n\nIn addition however one may wonder what are the properties of the theory under T-duality transformations in the transverse directions, i.e. those which change the brane, say from NS5 brane to the KK5-monopole. By simple counting of degrees of freedom one concludes that the theory does not change under that. I.e. the Type IIA\/B NS5 brane carries the same world-volume theory as the Type IIB\/A Kaluza-Klein monopole \\cite{Sen:1997js}. Continuing this logic one concludes that the theory should not change along the whole T-duality orbit.\n\\begin{equation}\n\\begin{aligned}\n5_2^0(A\/B) && \\longleftrightarrow && 5_2^1(B\/A)\\longleftrightarrow && 5_2^2(A\/B)\\longleftrightarrow && 5_2^3(B\/A)\\longleftrightarrow && 5_2^4(A\/B)\n\\end{aligned}\n\\end{equation}\nHere we use the notations for the branes of \\cite{Obers:1998fb} (see also \\cite{deBoer:2012ma} for more on that), and the last three are exotic. The fact that the corresponding world-volume field theories do not change under T-duality trivially follows from the T-duality invariant world-volume effective action for these branes presented in \\cite{Blair:2017hhy}. This is a single action for the whole orbit, which drops into actions for a representative upon removing half of the scalar fields living on the brane (geometric or dual coordinates). Since from the world-volume point of view these are just scalar fields moving in a dynamical background, replacement one by its dual does not change anything for it. \n\n\nHowever, applying the Ryu-Takayanagi prescription for geometric entropy to the background of say Kaluza-Klein monopole one gets the answer which is different from the one for the NS5 brane, which clearly breaks the T-duality invariance. In this paper we show that the reason for that is that in its geometric and straightforward form this prescription does not take into account dependence on the winding direction of the localized Kaluza-Klein monopole. Indeed, in \\cite{Tong:2002rq, Harvey:2005ab,Kimura:2013fda,Kimura:2018hph} it has been shown that instanton corrections coming from the 2d sigma-model describing the KK5 background, change the geometry such that field start depending on a winding mode. This correct the throat behaviour of the KK5-monopole to make it the same as that of the NS5 brane. In \\cite{Jensen:2011jna, Berman:2014jsa, Bakhmatov:2016kfn}  it has been shown that this has simple explanation in terms of Double Field Theory, that is to do a T-duality transformation in a direction $z$ one replaces $z$ by it dual $\\tilde{z}$ in all expressions. The same is true for producing exotic backgrounds, and the corresponding instanton interpretation has been presented in \\cite{Kimura:2013zva}. In this work we consider the invariant action of \\cite{Blair:2017hhy} and propose an algorithm to calculate entanglement entropy for theories living on branes with non-trivial dynamics in doubled space.\n\nThis paper is structures as follows. In Section \\ref{geom} we present a short technical review of how the geometric entanglement entropy is calculated and explicitly show that the RT formula gives different results when applying to NS five-brane backgrounds belonging to the same T-duality orbit. In Section \\ref{dft} we turn to invariant dynamics governed by the action of \\cite{Blair:2017hhy}, shortly review how one obtains different action from the invariant one, and describe the algorithm which produces an invariant answer for entanglement entropy. In addition we comment on the geometric meaning of the expression, which is an important and subtle point due to lack of the notions of integration, distance and area in doubled geometry.\n\n\n\\section{Geometric entanglement entropy}\n\\label{geom}\n\n\n\n\nThe usual choice of areas which carry entangled states, which significantly simplifies calculations, is the infinite strip set-up. For that one considers a surface in the space transverse to a brane one which the field theory lives (shaded on Fig. \\ref{embed}). This surface is the boundary for the minimal surface, which tends to curve closer to the brane due to the transverse geometry. For D-branes this surface is identified with the AdS conformal boundary. The geometric formula of Ryu and Takayanagi gives entanglement entropy of states in the region A and B on the picture.\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=10cm]{embed.png}\n\\caption{Configuration of the embedding}\n\\label{embed}\n\\begin{tikzpicture}[overlay]\n\\node at (-0.1,9.5) (r) {$r$};\n\\node at (-1.8,3.8) (ra) {$r_a$};\n\\node at (-4,2) (X25) {$X^{2,\\dots,5}$};\n\\node at (5,3.8) (X1) {$X^1$};\n\\node at (1.5,8.3)  (A) {$A$};\n\\node at (3.7,8)  (B1) {$B$};\n\\node at (-0.8,8.5)  (B2) {$B$};\n\\node[rotate=-11] at (1,3) (br) {brane};\n\\end{tikzpicture}\n\\end{figure}\n\n\nIn this section the standard formula for calculation of geometric entanglement entropy is applied to the standard NS5 brane and to the KK5-monopole and exotic $5_2^2$ brane. Due to the special circle already for the KK5 background one gets expressions very different from that for the NS5 background. From this we conclude that one must develop a different algorithm and a different understanding of the RT expression to properly capture transformations along the NS five-brane T-duality orbit.\n\n\n\\subsection{Non-conformal theories: geometric five-branes}\n\\label{direct_KK}\n\nWhen turning to NS five branes one encounters 6d theories which describe string-like degrees of freedom which do not have gravitational excitation in their spectrum, the so-called Little String Theory. In the field theory limit these drop to non-conformal field theories since the corresponding brane backgrounds contain non-trivial dilaton and are not asymptotically AdS. However, for these one also can define entanglement entropy using the Ruy-Takayanagi conjecture (\\ref{RTncbr}) and write\n\\begin{equation}\nS=\\int_{\\S} d^5 \\s e^{-2\\f} \\sqrt{\\det G_{\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon\\b}},\n\\end{equation}\nwhere $\\{\\s^\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon\\}$ wuth $\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon=1,\\dots,5$ are coordinates on the space-like surface $\\S$  and $G_{\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon\\b}=\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon X^\\m \\dt_\\b X^\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho G_{\\m\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho}$ is the induced metric on the surface. For our purposes we choose the simplest shape for the surface $\\S$  generated by an infinite stripe. Theory for which the entanglement entropy is calculated lives on a surface parallel to the NS5 brane placed at some $r=r_b$. Entanglement is assumed for the states defined in the interior $A$ and exterior $B$ regions of the grey surface on the Fig.\\ref{embed}. According to the conjecture this is equal to area of the minimal surface $\\S$ whose boundary satisfies $\\dt \\S=\\dt A$.\n\nEmbedding of the brane and of the surface is given by\n\\begin{equation}\n\\begin{aligned}\n&      &&  0 && 1 && 2 && 3 && 4 && 5 && r && \\q_1 && \\q_2 && \\f \\\\\n\\hline\n& NS5  &&\\times &&\\times&&\\times&&\\times&&\\times&&\\times&&  \\\\\n& \\S   &&    &&\\bullet&& L && L && L && L && \\bullet\n\\end{aligned}\n\\end{equation}\nwhere $\\times$ denote the world-volume directions. The surface $\\S$ extends from $-L$ to $L$ in the directions denoted by $L$ above, while it is somehow curved in the directions denoted by bullets. I.e. one can choose coordinates and embedding functions for the surface as follows\n\\begin{equation}\n\\begin{aligned}\nX^1&=X(r),\\\\\nX^{2,\\dots,5}&=\\s^{2,\\dots,5},\\\\\nr&=\\s^1.\n\\end{aligned}\n\\end{equation}\n\nBackground for the NS5 brane is given by\n\\begin{equation}\n\\begin{aligned}\nds^2&=\\h_{rs}dx^rdx^s+H(dr^2+r^2\\d\\W_{3}^2),\\\\\n\\mathcal{H}&=dB,\\\\\ne^{-2(\\f-\\f_0)}&=H(r)^{-1},\n\\end{aligned}\n\\end{equation}\nwith the harmonic function $H(r)=1+h\/r^2$. Hence one writes for the entropy\n\\begin{equation}\n\\begin{aligned}\\label{RT-ns5}\nS_{NS5}\n16L^4 \\int dr H(r)^{-1}\\sqrt{H(r)+X'(r)^2}.\n\\end{aligned}\n\\end{equation}\nThe usual minimisation procedure implies that the embedding function $X(r)$ should satisfy\n\\begin{equation}\n\\label{Xemb}\nX_{NS5}'(r)=\\pm\\frac} \\def\\dfr{\\dfrac} \\def\\dt{\\partial{H(r_a)^{1\/2}H(r)}{\\sqrt{H(r_a)^2-H(r)^2}},\n\\end{equation}\nwhere we used the condition that $X'(r_a)=0$, which basically means that $r_a$ is the turning point for the surface $\\S$. Note, that one has to set $r>r_a$ to keep the expression in the square root positive, which means that the turning point is closer to the brane than the surface $r=r_b$ on which the  field theory is defined. This is the usual configuration for the AdS\/CFT correspondence and hence the initial setup and the Fig.\\ref{embed}.\n\nOne can apply the same procedure to the worldvolume theory of the KK-monopole which for the Type IIA(B) monopole is the same as for the Type IIB(A) NS5 brane. Background geometry is given by the following configuration\n\\begin{equation}\n\\begin{aligned}\nds^2&=\\h_{rs}dx^rdx^s+H^{-1}(d\\tilde{z}+A_idy^i)^2+H\\d_{ij}dy^i dy^j,\\\\\nB&=0,\\\\\ne^{-2(\\f-\\f_0)}&=1.\n\\end{aligned}\n\\end{equation}\nHere $\\tilde{z}$ is the normal geometric coordinate used to measure distances in space-time, however it is dual to the coordinate $z$ of the corresponding NS5 background. Note that the harmonic function is smeared $H=1+h\/r$.\n\nRepeating the same calculation as above one gets for the entropy and for the embedding function $X_{KK5}(r)$\n\\begin{equation}\n\\begin{aligned}\nS_{KK5}&=16L^4 \\int dr \\sqrt{H(r)+X'(r)^2},\\\\\nX_{KK5}'(r)&=\\pm C\\sqrt{H(r)}, \\quad C=\\mbox{const}.\n\\end{aligned}\n\\end{equation}\nOne first notices that the crucial difference with the previous case, that is $dr\/dX=0$ at $r=0$, i.e. on the brane itself, while for the NS5 brane background the turning point is at some $r_a\\neq 0$. This can be understood in terms of the short distance behaviour of NS5 branes and KK5 monopole. As it has been shown in \\cite{Tong:2002rq,Gauntlett:1992nn} the former is the  version of the H-monopole (which is the proper T-dual of the KK monopole) localized due to instanton corrections. However, the localization breaks isometry along the compact circle of H-monopole and one observes a throat behaviour at short distances. \n\nTo cure the near-brane behaviour of the KK5-monopole background one also considers instanton corrections \\cite{Harvey:2005ab}. Only in this case one may expect result for the entropy which reproduce those for the NS5 brane. Such corrections however deform the background by introducing a non-trivial dependence on string winding coordinates, which requires double field theory to consistently address the issue, as in \\cite{Jensen:2011jna,Bakhmatov:2016kfn}. \n\n\n\n\n\n\n\\subsection{Exotic five-branes}\n\\label{direct_exotic}\n\nHence, the answer for the entropy which one obtains for the theory living on the KK monopole is different from that for the NS5 brane. The important point here is that although T-duality exchanges IIA and IIB branes the theories living on the NS5A(B) and the KK5B(A) are the same and the entropy should not change. When going further along the T-duality orbit towards exotic branes the situation does not get better. Smearing the KK5 background along $y_3$ and T-dualizing one arrives at the exotic $5_2^2$-brane with background given by \\cite{deBoer:2012ma}\n\\begin{eqnarray}\nds^{2} &=& \\eta_{rs}dx^{r}dx^{s} + HK^{-1}\\left(d\\tilde{z}^{2} + d\\tilde{y}^{2}_{3}\\right) + H \\delta_{\\alpha\\beta}dy^{\\alpha}dy^{\\beta}, \\\\\nB& = & h\\theta K^{-1} d\\tilde{z}\\wedge d\\tilde{y}_{3}, \\\\\ne^{-2(\\phi - \\phi_{0})} &=& HK^{-1},\\\\\nK &= & H^{2} + (h\\theta)^2.\n\\end{eqnarray}\nHere the harmonic function is further smeared $H(r)=1+h \\log r$ and does not behave well at space infinity\n\nThis background is globally well-defined only up to a monodromy around the brane, hence the non-geometric properties of the background. Naively applying the above procedure one obtains\n\\begin{equation}\n\\begin{aligned}\nS_{5_2^2}&=16L^4\\int d r \\frac} \\def\\dfr{\\dfrac} \\def\\dt{\\partial{H(r)\\sqrt{H(r)+X'(r)^2}}{H(r)^2+h^2 \\q^2},\\\\\nX_{5_2^2}'&=\\pm\\frac} \\def\\dfr{\\dfrac} \\def\\dt{\\partial{C \\sqrt{H(r)}(H(r)^2+h^2 \\q^2)}{\\sqrt{H(r)^2-C^2 (H(r)^2+h^2 \\q^2)^2}}.\n\\end{aligned}\n\\end{equation} \nThe embedding function $X_{5_2^2}(r)$ delivering extremum to $S_{5_2^2}$ is apparently not well-defined and moreover it explicitly depends on $\\q$. Hence, the entropy also depends explicitly on the coordinate $\\q$ respecting the monodromy property of the background.\n\nOn the other hand, the worldvolume theory on $5_2^2$-brane should not differ from that of the KK-monopole or NS5 brane (with proper replacement of Type IIA with Type IIB). To perform calculation of entanglement entropy for such theories which respect T-duality we use the T-duality covariant action of \\cite{Blair:2017hhy} for the 5-brane orbit. It suggests that the worldvolume theory is the same irrespective of the choice of the brane (equivalently, the section constraint or orientation in the doubled space) upon the proper exchange of the worldvolume scalars $X^\\m$ with their duals $\\tilde{X}_{\\m}$. \n\n\n\n\\section{Entanglement entropy in  DFT}\n\\label{dft}\n\nDouble Field Theory being a T-duality covariant formulation of supergavity (string theory) allows to consider the whole T-duality orbit instead of a single representative. In this section we propose a deformation of the geometric prescription for entanglement entropy and embed the expression for entropy itself into the DFT framework. Let us start with brief description of how NS five-branes are embedded into doubled space.\n\n\\subsection{Embedding of NS five-branes in doubled space}\n\nIn \\cite{Blair:2017hhy} it was shown in details how one can construct a T-duality covariant action for NS five-branes. The covariancy here is understood in the following way: one has a single expression which is written in terms of DFT (covariant) fields and which reproduces the effective action for the NS5B-brane, KK5A monopole and exotic branes $5_2^2$B, $5_2^3$A, $5_2^4$B. The full action smartly chooses these frames depending on which symmetries of the doubled spaces are eventually realized on the world-volume. Let us briefly describe the process focusing only on the NS-NS sector and only on the DBI part of the action, which is given by\n\\begin{equation}\n\\label{full_5}\nS_{NS,DBI}[Y(\\x)]=\\int_V d^6 \\x e^{-2d}\\sqrt{\\det h_{ab}}\\sqrt{\\displaystyle -\\det\\Big(g_{\\m\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon{X^\\m} \\dt_\\b X^\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho+ \\mathcal{H}_{MN}\\hat{D}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^M \\hat{D}_\\b Y^N\\Big)},\n\\end{equation}\nwhere we introduce\n\\begin{equation}\n\\begin{aligned}\nh_{ab}&=k_a^M k_b^N \\mathcal{H}_{MN},\\\\\n\\hat{D}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^M&=\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^M+\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon X^\\m A_\\m{}^M,\\\\\n\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^M&=\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^M-(h^{-1})^{ab}k_a^M k_b^N\\mathcal{H}_{NP}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^P,\\\\\n\\mathcal{H}_{MN}&=\n\\begin{bmatrix}\nG_{mn}-B_{m}{}^k B_{kn} & B_{n}{}^q\\\\\nB_m{}^p & G^{pq}\n\\end{bmatrix}.\n\\end{aligned}\n\\end{equation}\nHere the full space-time is split into the part parallel to the five-brane, labelled by the indices $\\m,\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho=\\{0,5\\}$, and the part transverse to the branes, which is doubled and labelled by $M,N,P,Q=\\{6,7,8,9,\\tilde{6},\\tilde{7},\\tilde{8},\\tilde{9}\\}$. The vector fields $A_\\m{}^M$ result from the Kaluza-Klein decomposition of the full 10D theory \n\\begin{equation}\nA_\\m{}^M=\n\\begin{bmatrix}\nA_\\m{}^m \\\\\n-B_{\\m m}\n\\end{bmatrix}.\n\\end{equation}\nThe integration is performed over world-volume of the brane which is parametrized by six coordinates $\\{\\x^\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon\\}$. The hatted derivative $\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon$  contains a projector part and is designed in such a way as to always remove half of the fields $Y^M$ from the action. Upon adding the action for DFT fields this results in field configurations which do not depend on half of DFT coordinates and hence is a worldvolume realization of the section constraint. Finally, the choice of the section frame and hence a representative of the T-duality orbit is done by choosing the particular form of the vectors $k_a{}^M$, which must satisfy the following algebraic section constrain\n\\begin{equation}\nk_a^Mk_b^N\\h_{MN}=0,\n\\end{equation}\nwhere $\\h_{MN}$ is the usual O(4,4) invariant tensor \n\\begin{equation}\n\\h_{MN}=\\begin{bmatrix}\n0 & 1 \\\\\n1 & 0\n\\end{bmatrix}\n\\end{equation}\nand the indices $a,b={1,4}$ enumerate the Killing vectors. The reason why we call these vectors Killing will be clear in a moment.\n\nFor the O(4,4) configuration there exist five inequivalent solutions of the algebraic section constrain, each of which corresponds to the branes $5_2^r$ with $r=0,1,2,3,4$ showing the number of quadratic direction in the mass of the corresponding 3D BPS state (see \\cite{deBoer:2012ma} for more detailed description of these notations). Here we list five representative solutions\n\\begin{equation}\n\\label{Kill}\n\\begin{aligned}\nNS5=5_2^0: && k_a^M&=(0,0,0,0;\\tilde{k}_{a 1}, \\tilde{k}_{a 2},\\tilde{k}_{a 3},\\tilde{k}_{a 4}),\\\\\nKK5=5_2^1: && k_a^M&=(0,0,0,k_{a}^{4}; \\tilde{k}_{a 1},\\tilde{k}_{a 2},\\tilde{k}_{a 3}, 0),\\\\\nQ=5_2^2: && k_a^M&=(0,0,k_{a}^{3},k_{a}^{4};\\tilde{k}_{a 1},\\tilde{k}_{a 2},0,0),\\\\\nR=5_2^3: && k_a^M&=(0,k_{a}^{2},k_{a}^{3},k_{a}^{4};\\tilde{k}_{a 1},0,0,0),\\\\\nR'=5_2^4: && k_a^M&=(k_{a}^1,k_{a}^{2},k_{a}^{3},k_{a}^{4};0,0,0,0).\n\\end{aligned}\n\\end{equation}\nFor example, for the NS5 brane case, which is the first line above, one chooses all vectors $k_a^M$ to be along the dual coordinates. Substituting this back into the action one checks that all fields $Y_m$ drop from the expression rendering field configurations independent on the corresponding DFT coordinates. This is due to\n\\begin{equation}\n\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y_m=B_{mn}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^n,\n\\end{equation}\nwhere $B_{mn}$ is the usual Kalb-Ramond two-form gauge field. The same is true for all other configurations up to the R'-brane which is a co-dimension-0 object from the point of view of the conventional supergravity.\n\nTo obtain explicit expression for the background fields for a fixed choice of the Killing vectors, one considers the full action with the embedding given by Dirac delta functions $\\d^{(8)}(\\mathbb{X}^M-Y^M(\\x))$, where $\\mathbb{X}^M=(x^m,\\tilde{x}_m)$ are the coordinates of DFT. The reparametrization invariance of the world-volume is fixed as usual as\n\\begin{equation}\n\\label{gauge_fix0}\n\\begin{aligned}\nX^\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon&=\\x^\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon.\n\\end{aligned}\n\\end{equation}\nConsider for example the KK-monopole, which is the second line above, where the fields $Y_{1,2,3}$ and $Y^4$ drop from the action meaning that the field configurations as functions are of the form $H=H(x^1,x^2,x^3,\\tilde{x}_4)$. This is interpreted as a functional dependence of the background on three geometric coordinates $x^{1,2,3}$ and one non-geometric (dual or winding) coordinate $\\tilde{x}_4$. This is due to an additional piece of information fixed in the DFT action, where one always understands $\\mathbb{X}^m=x^m$ as geometric coordinates, i.e. those used to measure space distances, and $\\mathbb{X}_m=\\tilde{x}_m$ as their non-geometric duals. Without this fixing one will just count each brane four more times obtaining the same backgrounds but with different names for the same physical coordinates.\n\nSuch dependence of exotic backgrounds (starting from the KK monopole) on dual coordinates has been shown for the DFT monopole in \\cite{Bakhmatov:2016kfn} and will be important for our further discussion.\n\n\n\n\\subsection{Invariant entropy and minimal surface}\n\nThe main feature of the effective action \\eqref{full_5} is that it does not depend on the choice of the T-duality frame and describes dynamics of all five-branes dual to NS5 brane. Since the world-volume theory does not change when switching from (Type IIB) NS5 brane to (Type IIA) KK5-monopole, the corresponding entanglement entropy should not change as well. One can conjectures the following deformation of the Ryu-Takayanagi formula which provides such invariant description:\n\\begin{equation}\n\\label{inv_entr}\nS_5=\\int_\\S d^5 \\s e^{-2d}\\sqrt{\\det h_{ab}}\\sqrt{\\displaystyle -\\det\\Big(g_{\\m\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon{X^\\m} \\dt_\\b X^\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho+ \\mathcal{H}_{MN}\\hat{D}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^M \\hat{D}_\\b Y^N\\Big)},\n\\end{equation}\nwhere the notations are the same as before. In a moment we will explicitly show that this expression gives the usual RT formula whose minimization gives the geometric entanglement entropy for the NS5 brane case. For other representatives of the orbit one gets a deformation of the formula, however the integral itself does not distinguish between the allowed choices of the duality frame.\n\nBefore that it is important to discuss the meaning of the integration and of the surface $\\S$ here. Going back to the effective action \\eqref{full_5} one notes that the integration there is performed over the world-volume $V$ parametrized by $\\s^\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon$, which is a usual geometric manifold with properly defined integration measure. On this manifold one defines $6+(4+4)$ fields $\\{X^\\m(\\x),Y^M(\\x)\\}$, which are identified with coordinates in the space-time and the doubled coordinates of the O(4,4) DFT. The crucial point here is that without such identification, these fields do not carry the meaning of coordinates on a doubled space and hence one is not actually doing doubled geometry, and rather works with a number of fields. For more discussion on this see \\cite{Blair:2017hhy}.\n\nAlthough the expressions \\eqref{full_5} and \\eqref{inv_entr} look almost the same, there is fundamental difference between them. While in action one varies with respect to the background fields keeping the embedding fixes, for the entropy the background is fixed by our choice of the brane and variation goes with respect to the embedding. The latter is defined by identification of the surface coordinates $\\{\\s^\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon\\}$ with the fields $X^\\m,Y^M$, which define the (doubled) space-time dependence of the background. Since, the harmonic function depends only on a singlet combination $r$, a natural choice of the embedding is\n\\begin{equation}\n\\label{gauge_fix}\n\\begin{aligned}\nX^{2,3,4,5}&=\\s^{2,3,4,5},\\\\\n\\s^1&=r,\n\\end{aligned}\n\\end{equation}\nand the remaining field $X^1=X(\\s^1)$ is a function delivering minimum to the expression. Here the particular form of the field $r$ depends on the choice of the background and reads\n\\begin{equation}\n\\begin{aligned}\nNS5=5_2^0: &&  r^2&=(Y^1)^2+(Y^2)^2+(Y^3)^2+(Y^4)^2,\\\\\nKK5=5_2^1: && r^2&=(Y^1)^2+(Y^2)^2+(Y^3)^2+(\\tilde{Y}_4)^2,\\\\\nQ=5_2^2: && r^2&=(Y^1)^2+(Y^2)^2+(\\tilde{Y}_3)^2+(\\tilde{Y}_4)^2,\\\\\nR=5_2^3: && r^2&=(Y^1)^2+(\\tilde{Y}_2)^2+(\\tilde{Y}_3)^2+(\\tilde{Y}_4)^2,\\\\\nR'=5_2^4: && r^2&=(\\tilde{Y}_1)^2+(\\tilde{Y}_2)^2+(\\tilde{Y}_3)^2+(\\tilde{Y}_4)^2.\n\\end{aligned}\n\\end{equation}\nThese follow from solutions of the equations of motion for the full action $S_{DFT}+S_{brane}$ which boil down to Poisson equation with delta source whose solution is the harmonic function $H=H(r)$ with $r$ given by the above expression. The number of dual coordinates entering the dependence of the fields is equal to the number of special circles.\n\nThe gauge fixing conditions  \\eqref{gauge_fix} can be understood as a proper embedding of the surface $\\S$ in the doubled 5+(4+4)-dimensional space. This is similar to the way how the magnetic charge for these branes has been calculated in \\cite{Bakhmatov:2016kfn}, however now the integration remains proper integration over a conventional manifold with conventional measure. The structure of the doubled space shows up only at the level of the Killing vectors and of the interaction between the effective action and the full DFT action. Before that, the integration does not distinguish between $Y^m$ and $\\tilde{Y}_m$, as it should be since the corresponding world-volume theories do not feel this as well. The integration is then performed in $\\s^{2,3,4,5}\\in [-L,L]$ for some large $L$ and from the points $X'=0$ in the $\\s^1$ direction. This is what is usually called the rectangular strip area, which is the simplest to perform calculations. In principle, one may choose a different embedding which will correspond to a different area inside the world-volume theory.\n\nLet us postpone the discussion, of how this process is seen from the point of view of the world-volume theory, to the Discussion section and now move to explicit examples to show invariance of the expression.\n\n\n\\subsection{Explicit examples}\n\n\nLet us start with the T-duality frame which corresponds to NS5 brane, which fixes the Killing vectors to be\n\\begin{equation}\nk_a^M=(0;\\tilde{k}_{am}).\n\\end{equation}\nThen the matrix $h_{ab}$ becomes $h_{ab}=\\tilde{k}_{am}\\tilde{k}_{bn}g^{mn}$ and one has \n\\begin{equation}\n\\det{h_{ab}}=|\\tilde{k}|^2 g^{-1},\n\\end{equation}\nwhere $|\\tilde{k}|=\\det\\tilde{k}_{am}$ and $g=\\det G_{mn}$. The inverse of the matrix $h_{ab}$ is then\n\\begin{equation}\n(h^{-1})^{ab}=(\\tilde{k}^{-1})^{am}(\\tilde{k}^{-1})^{bn}G_{mn},\n\\end{equation}\nwhere $(\\tilde{k}^{-1})$ is the inverse of $\\tilde{k}_{am}$ understood simply as a $4\\times 4$ matrix. Hence, for derivatives of the fields $Y^M$ we have\n\\begin{equation}\n\\begin{aligned}\n\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^m&=\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^m,\\\\\n\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon \\tilde{Y}_m&=\\dt \\tilde{Y}_m-(h^{-1})^{ab}\\tilde{k}_{am}\\tilde{k}_{an}\\mathcal{H}^n{}_P\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^p=B_{mn}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^n.\n\\end{aligned}\n\\end{equation}\nWith this in hands it is easy to show that\n\\begin{equation}\n\\mathcal{H}_{MN}\\hat{D}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^M \\hat{D}_\\b Y^N= G_{mn}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^m \\dt_\\b Y^n,\n\\end{equation}\nwhere we used the fact that $A_{\\m}{}^M=0$ for the chosen embedding of the brane. \n\nFinally, substituting all this into the expression for the entropy \\eqref{inv_entr} one obtains\n\\begin{equation}\n\\label{Einv_NS5}\n\\begin{aligned}\nS_{NS5}&=\\int_\\S d^5 \\s e^{-2\\f}\\sqrt{G}|\\tilde{k}|\\frac} \\def\\dfr{\\dfrac} \\def\\dt{\\partial{1}{\\sqrt{G}}\\sqrt{-\\displaystyle \\Big(g_{\\m\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon X^\\m \\dt_\\b X^\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho+G_{mn}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^m \\dt_\\b Y^n\\Big)}\\\\\n&=|\\tilde{k}|\\int_\\S d^5 \\s e^{-2\\f}\\sqrt{-\\displaystyle \\Big(g_{\\m\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon X^\\m \\dt_\\b X^\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho+G_{mn}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^m \\dt_\\b Y^n\\Big)},\n\\end{aligned}\n\\end{equation}\nwhich is the conventional expression for the geometric entanglement entropy of Ryu and Takayanagi (for the chosen embedding, i.e. $g_{\\m m}=0$).\n\nThe same calculation can be repeated for the KK5-monopole. One starts with the following Killing vectors\n\\begin{equation}\nk_a{}^M=(0,k_4^m;\\tilde{k}_{e m}),\n\\end{equation}\nwhere $e,f,g,h=1,2,3$. And the direction $4$ is identified with the Taub-NUT direction (the special circle of the monopole). For further convenience it is natural to choose such basis for the vectors $k_a^M$ where $\\tilde{k}_{e 4}=0$ and $k_4^i$=0. Then the matrix $h_{ab}=k_a{}^Mk_b{}^N\\mathcal{H}_{MN}$ becomes\n\\begin{equation}\n\\begin{aligned}\nh_{ef}&=\\tilde{k}_{e m}\\tilde{k}_{fn}g^{mn}=\\tilde{k}_{e i}\\tilde{k}_{f j}G^{ij},\\\\\nh_{e4}&=0,\\\\\nh_{44}&=k_4^4k_4^4 G_{44},\n\\end{aligned}\n\\end{equation}\nand $\\det h_{ab}=|\\tilde{k}|^2 (k_4^4)^2 g^{-1} G_{44}$, where $g=\\det g_{ij}$ is determinant of the 3-dimensional part of the metric $G_{mn}$ defined as\n\\begin{equation}\n\\begin{aligned}\nG_{ij}&=g_{ij}+A_iA_j G_{44}, && & G_{i4}&=A_i G_{44},\\\\\nG^{ij}&=g^{ij}, && & G^{i4}&=-A_{i4}G_{44},\\\\\nG_{44}&=H^{-1},  && & G^{44}&=\\frac} \\def\\dfr{\\dfrac} \\def\\dt{\\partial{1}{G_{44}}+A_i A_i G_{44}.\n\\end{aligned}\n\\end{equation}\nFollowing the same procedure as before it is straightforward to obtain the following expression for derivatives of the fields $Y^M$\n\\begin{equation}\n\\begin{aligned}\n\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^i&= \\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^i, && &\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^4&=-A_i \\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^i\\\\\n\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon \\tilde{Y}_i&=A_i \\dt_a \\tilde{Y}_4, && &\\hat{\\dt} \\tilde{Y}_4&=\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon \\tilde{Y}_4.\n\\end{aligned}\n\\end{equation}\nThe crucial difference between NS5 brane and KK5-monopole here is that for the former one is left only with the fields $Y^i$, which upon embedding into the full DFT action are identified with the usual geometric coordinates. In contrast, for KK5-monopole after projection one has the fields $\\{Y^i,\\tilde{Y}_4\\}$ which results in dependence of the background fields on the corresponding dual (winding) coordinate $\\tilde{x}_4$. This behaviour has been observed in \\cite{Harvey:2005ab,Jensen:2011jna,Bakhmatov:2016kfn} for KK5 and in \\cite{Kimura:2013zva} for the exotic $5_2^2$-brane.\n\nFinally, collecting all these pieces together one arrives at the following expression for entanglement entropy of the world-volume theory on (localized) Kaluza-Klein monopole\n\\begin{equation}\n\\label{Einv_KK55}\n\\begin{aligned}\nS_{KK5}&=|\\tilde{k}||k_4^4|\\int_\\S d^5 \\s e^{-2\\f} G_{44}\\times\\\\\n&\\sqrt{-\\displaystyle \\det\\Big[g_{\\m\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon X^\\m \\dt_\\b X^\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho + \\big(G_{ij}-G_{44}A_iA_j\\big)\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^i \\dt_\\b Y^j +\\big(G^{44}-G^{ij}A_iA_j\\big)\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon \\tilde{Y}_4 \\dt_\\b \\tilde{Y}_4\\Big]}\\\\\n&=|\\tilde{k}||k_4^4|\\int_\\S d^5 \\s e^{-2\\f}G_{44}\\sqrt{-\\displaystyle \\det\\Big[g_{\\m\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon X^\\m \\dt_\\b X^\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho + H\\big(\\d_{ij}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^i \\dt_\\b Y^j +\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon \\tilde{Y}_4 \\dt_\\b \\tilde{Y}_4\\big)\\Big]}.\n\\end{aligned}\n\\end{equation}\nWhere the last line is obtained by substituting the explicit background of KK5-monopole inside the square root. Taking into account that for the monopole one has $e^{-2\\f}=1$ and $G_{44}=H^{-1}$ the second line reproduces precisely the expression \\eqref{Einv_NS5} up to replacement $Y^4 \\to \\tilde{Y}_4$. Note however, that talking about world-volume dynamics and field theories on the branes one does not distinguish between fields $Y^m$ and their duals $\\tilde{Y}_m$. The only difference is that the latter see the background T-dual to the background seen by the former. This is a trivial consequence of the above considerations.\n\nNow, for exotic branes $5_2^r$ with $r=2,3,4$ the story is precisely the same and the algorithm is the following: fix the Killing vectors as in \\eqref{Kill}, calculate $h_{ab}$ and hatted derivatives $\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon$, substitute everything in \\eqref{inv_entr}. The result will always be \\eqref{Einv_NS5} with the corresponding replacement of the fields $Y^m$ by their duals. Hence the name ``invariant entropy''. We postpone speculations on the physical and geometrical meaning of this procedure to the next section.\n\n\n\n\n\\section{Discussion}\n\nIn this letter we propose a T-duality invariant generalization of the Ryu-Takayanagi formula for geometric entanglement entropy for the case of NS  five-branes $5_2^r$ ($r=0,\\dots,4$), with tension proportional to $g_s^{-2}$. The result is the expression \\eqref{inv_entr} which is based on the same ideas as the effective action \\eqref{full_5} for these branes. In particular, to choose a representative brane from the orbit one must specify Killing vectors, which satisfy the so-called algebraic section constraint. The choice which gives the effective action of NS5B-brane also reproduces the RT-formula for entanglement entropy of $\\mathcal{N}=(1,1)$ Little String Theory living on this brane.\n\nWe check, that the same expression gives always the same result irrespective of which representative is chosen. This is in consistency with the fact, that e.g. the world-volume theory for the KK5A-brane is also $\\mathcal{N}=(1,1)$ LST and hence the entropy should be the same. As we show in Section \\ref{direct_KK} this is in contrast with the direct application of the Ryu-Takayanagi formula, which gives different results. \n\nOn the level of world-volume scalar fields $Y^M$ transition between orbit representatives (say NS5B and KK5A) is just replacement of a field $Y^m$ by its duality partner $\\tilde{Y}_m$. Although this has crucial impact on DFT and supergravity solutions changing the background, the world-volume theory has no way to see that, and hence it is always the same. For this reason, as the carrier of the $\\mathcal{N}=(1,1)$ LST in Type IIA string theory one should consider the localized Kaluza-Klein monopole rather than the smeared one \\cite{Harvey:2005ab}. The former is a deformation of the latter by instanton corrections, and is already exotic since its harmonic function depends on one dual coordinate \\cite{Jensen:2011jna,Bakhmatov:2016kfn}. The same is true for other exotic branes, which should also be localized.\n\nThe apparent issue that needs clarification is the following. For a theory on a Dp-brane one has apparent geometric picture, where the theory lives on a timelike surface at some $r\\neq 0$ in the transverse space. For AdS\/CFT correspondence one literally takes the conformal boundary of the anti-de-Sitter space. To calculate entanglement entropy geometrically one chooses a region $A$ on this surface and a surface $\\S$ in the transverse space of the brane such that $\\dt \\S =\\dt A$, and calculates its area in the given background. \n\nIn the case in question one cannot develop such simple geometric picture, and moreover one cannot do this already for the NS5 brane. Indeed, the corresponding geometry does not drop into AdS and the corresponding field theory is not conformal. However, the $6D$ field theory associated with the brane can be as well put at any $\\r$ in the transverse space, and the choice corresponds to the RG flow and one can still calculate area properly. The procedure described here suggests the following:\n\\begin{itemize}\n\\item start with a $5_2^r$-brane with any $r\\in\\{0,1,2,3,4\\}$ and its world-volume theory described by the doubled amount of scalar fields $\\Phi} \\def\\C{\\Chi} \\def\\Y{\\Psi} \\def\\W{\\Omega^M=(\\Phi} \\def\\C{\\Chi} \\def\\Y{\\Psi} \\def\\W{\\Omega^m,\\tilde{\\Phi} \\def\\C{\\Chi} \\def\\Y{\\Psi} \\def\\W{\\Omega}_m)$ half of which is projected out by the algebraic section constraint;\n\\item choose a region $A$ in the space of the theory with boundary $\\dt A$;\n\\item consider a surface $\\S$ with boundary $\\dt\\S$ parametrized by some coordinates $\\s^\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon$;\n\\item this surface carries a doubled amount of scalar fields $\\{Y^M\\}$ with boundary conditions $Y^M\\big|_{\\dt\\S}=\\Phi} \\def\\C{\\Chi} \\def\\Y{\\Psi} \\def\\W{\\Omega^a\\big|_{\\dt A}$;\n\\item minimize the functional \\eqref{inv_entr}.\n\\end{itemize}\nThe theory in the first item here just descents from the full invariant effective action \\eqref{full_5}. The boundary condition is needed to identify the scalar fields living on the artificial surface $\\S$ with the actual fields of the theory. For the conventional geometric picture this is done automatically by the embedding functions, where both the theory and the surface live in a single geometric background. Apparently, this procedure trivially reproduces the conventional geometric calculation, and the only messages here are the following:\n\\begin{itemize}\n\\item to calculate entanglement entropy for the $\\mathcal{N}=(1,1)$ and $\\mathcal{N}=(2,0)$ $6D$ theories one may use equivalently any of the representative of the T-duality orbit;\n\\item to get the correct result one must take into account proper localization of the backgrounds in the dual space.\n\\end{itemize}\n\n\n\nAn interesting  further direction of research is to generalize the expression to the case of M5-brane which belongs to the same orbit as the $5^3$-brane under U-duality group. One then still works with Little String Theory and 6D, however the invariant expression will be different. One can also consider D-branes in DFT, which can also be non-geometric, i.e. localized in the dual space. The corresponding effective action will be presented in the forthcoming paper \\cite{axel-eric-fabio} and investigation of the corresponding world-volume theories and their entanglement entropy we reserve for future work.\n\n\n\n\n\\section*{Acknowledgements}  The authors are grateful to the Istanbul center of mathematical sciences and Bogazici University for hospitality during initial stages of this project. ETM would like to thank for hospitality Bogolyubov laboratory,  JINR, Dubna. The work of ETM was supported by the Russian state grant Goszadanie 3.9904.2017\/8.9 and by the Alexander von Humboldt return fellowship and partially by the program of competitive growth of Kazan Federal University. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Abstract}\n\nWe present the discovery of only the second radio-selected, z $\\sim$ 6 quasar.  We identified SDSS J222843.54+011032.2 (z=5.95) by matching the optical detections of the deep Sloan Digital Sky Survey (SDSS) Stripe 82 with their radio counterparts in the Stripe82 VLA Survey.  We also matched the Canadian-France-Hawaiian Telescope Legacy Survey Wide (CFHTLS Wide) with the Faint Images of the Radio Sky at Twenty-cm (FIRST) survey but have yet to find any z $\\sim$ 6 quasars in this survey area.  The discovered quasar is optically-faint, $z = 22.3$ and M$_{1450}$ $\\sim$ -24.5, but radio-bright, with a flux density of f$_{1.4GHz, peak}$ = 0.31mJy and a radio-loudness of R $\\sim$ 1100 (where R $\\equiv$ $f_{5GHz}\/f_{2500}$).  The $i-z$ color of the discovered quasar places it outside the color selection criteria for existing optical surveys.  We conclude by discussing the need for deeper wide-area radio surveys in the context of high-redshift quasars.    \n \n\\section{Introduction}\n\\label{sec:intro}\n\nHigh-redshift quasars (z $\\sim$ 6) which are powered by supermassive black holes (SMBH's) provide a window into the early universe, and, unsurprisingly, the study of these rare objects has grown in the decade since their discovery (\\citealp{2000AJ....120.1167F}; \\citealp{2001AJ....122.2833F}).  These distant SMBH's are important in the study of galaxy evolution and the epoch of re-ionization.   The strong correlation observed between the velocity dispersion of the stars in a galaxy and the mass of its central SMBH indicates a linkage between the mass of the SMBH and the evolution of the galaxy \\citep{2000ApJ...539L...9F}.  This relation may be explained by a ``quasar phase\" of the SMBH, where significant accretion is coupled with feedback into the stellar formation processes of the host galaxy \\citep{ 2005ApJ...625L..71H}.  Since nearly every galaxy is thought to have a central SMBH, active or not, high-redshift quasars are an important piece of the puzzle in understanding early galaxy evolution.  \n\nThe spectra of distant quasars probe the intergalactic medium (IGM) and can be used to study the re-ionization epoch along the line of sight (\\citealp{2001AJ....122.2850B}; \\citealp{2003AJ....126....1W}; \\citealp{2006AJ....132..117F}).  The presence of the Gunn-Peterson trough \\citep{1965ApJ...142.1633G} can indicate the neutral hydrogen fraction and act as a flag marking the re-ionization epoch.  There is evidence that the fraction of neutral hydrogen by volume increases an order of magnitude from z $=$ 5.7 to z $=$ 6.4 (\\citealp{2006AJ....132..117F}; \\citealp{2010ApJ...714..834C}).  This increase may mark the end of the re-ionization epoch, placing a high importance on any new discoveries of z $>$ 5.7 quasars, especially those which are radio-loud.  Radio-loud quasars are valuable as probes of the early IGM because the coming generation of radio telescopes (EVLA, ALMA, SKA, LOFAR, etc.) will use them to measure the properties of neutral hydrogen through 21 cm absorption studies (\\citealp{2004NewAR..48.1029C}; \\citealp{2010HiA....15..312K}).\n\nDespite large area surveys designed to find high-redshift quasars, only about 60 ($z > 5.7$) quasars have been found so far (\\citealp{2000AJ....120.1167F}; \\citealp{2007AJ....134.2435W}; \\citealp{2008AJ....135.1057J}).  These surveys all find quasars using similar methods that rely on red optical colors ($(i - z)_{AB}$ $>$ 2) and blue near-IR colors ($(z - J)_{AB}$ $<$ 1).  The red optical color of high-z quasars is a consequence of the Ly$\\alpha$ break moving through and out of the blueward optical bands.  The blue near-IR color cut is used to distinguish high-redshift quasars from the largest contaminant in these surveys: cool dwarf stars.  An alternative method to separate high-redshift quasars from cool dwarf stars is to require a radio detection, as very few stars are radio-bright (\\citealp{kimball}).  This method only finds $\\sim$5$\\%$ of quasars (3 of $\\sim$60 z $\\sim$ 6 quasars have been detected at $>$1mJy in the radio), but it can be used to select quasars with colors that fail the usual optical\/near-IR selection.  Previously, one radio-selected z $\\sim$ 6 quasar was found by \\citet{2006ApJ...652..157M} in a mere 4 ${deg}^2$ search area in the NOAO Deep Wide-Field Survey (NDWFS).  This quasar was bright enough to be discovered by other surveys; however, its red near-IR colors placed it in the color space occupied by cool dwarf stars, and hence it was missed by typical color-color selection methods.  \n\nThis work presents the discovery of the second radio-selected, z $\\sim$ 6 quasar.  We used two different survey combinations to search for radio-selected z $>$ 5.7 quasars: the Canadian-France-Hawaiian Telescope Legacy Survey Wide (CFHTLS Wide \\footnote[6]{http:\/\/www.cfht.hawaii.edu\/Science\/CFHTLS\/}) matched to the Faint Images of the Radio Sky at Twenty-cm (FIRST - \\citealp{1995ApJ...450..559B}) survey, and the deep Sloan Digital Sky Survey (SDSS Stripe 82 - \\citealp{2009ApJS..182..543A}) matched with the Stripe82 VLA Survey (Hodge et al. 2011).  We discovered the quasar in Stripe 82 with the deeper radio data of the Stripe82 VLA Survey.  \n\n In \\S 3 we describe our candidate selection methods and expected number of discoveries.  In \\S 4, we discuss the observations of SDSS J222843.53+011032.0 (SDSS J2228+0110; z $=$ 5.95). In \\S 5 we discuss the implications of further studies of high redshift radio-selected quasars.  All magnitudes are AB unless stated otherwise. This paper assumes a flat cosmological model with $\\Omega_{m} = 0.28$, $\\Omega_{\\Lambda} = 0.72$, and $H_{0} = 70$ km s$^{-1}$  Mpc$^{-1}$. \n\n\\section{Candidate Selection}\n\n\\subsection{CFHTLS Wide}\n\nCFHTLS Wide is an intermediate depth and area survey covering 171 deg$^2$ with $\\sim$130 deg$^2$ publicly available at the time of this work (release T0005).  The typical integration times are 4300s in $i$ and 3600s in $z$, reaching an average depth of 24.5 in i and 23.8 in z at the 80$\\%$ completeness limit determined through simulation\\footnote[7]{http:\/\/terapix.iap.fr\/cplt\/oldSite\/Descart\/CFHTLS-T0005-Release.pdf}. \n\n\\subsection{FIRST}\n\nFIRST is a 20 cm survey over $\\sim$10,000 deg$^2$.  With a sensitivity threshold of 1 mJy, it achieves a source density of $\\sim$90 deg$^{-2}$.  The FIRST survey covers the entirety of the CFHTLS Wide survey, and the typical astrometric accuracy between the two surveys is $<$ 0.5''.       \n\n\\subsection{Stripe 82}\n\nDuring the months when the primary SDSS area was not observable, SDSS repeatedly observed a strip of sky along the Galactic Equator known as Stripe 82.  This patch of sky is 300 deg$^2$ and spans roughly 20$^{h}$ $<$ RA $<$ 4$^{h}$ and -1.5$^{\\circ}$ $<$ Dec $<$ 1.5$^{\\circ}$.  The resulting co-additions (\\citealp{2009ApJS..182..543A}) go two magnitudes deeper than the typical SDSS images and reach 23.3 in $i$ and 22.5 in $z$ at the 95$\\%$ repeatable detection limit.  \n\n\\subsection{Stripe 82 VLA Survey}\n\nThis 1.4 GHz survey was conducted with the Very Large Array (VLA) in A-configuration and has an angular resolution of 1.8'' (Program ID AR646 and AR685).  It achieves a median rms noise of 52 $\\mu$Jy beam$^{-1}$ over 92 deg$^{2}$ (\\citealp{hodge2011}), making it the deepest 1.4 GHz survey to cover that much sky.  A catalog of 17,969 isolated radio components, for an overall source density of $\\sim$195 sources deg$^{-2}$, is publicly available.  The astrometric accuracy of the data is excellent, with an rms scatter of 0.25'' in both right ascension and declination when matched to SDSS's Stripe 82.\n\n\\subsection{Selection Method}\n\nHigh-redshift quasars are typically selected based on their very red optical colors: $(i - z)_{AB} > 1.5$.  Cool dwarf stars also have red optical colors, so in an effort to reduce contamination, surveys such as SDSS require a blue near-IR color (see Figure 1).  The optical color probes the Ly$\\alpha$ break while the near-IR color probes the quasar continuum as well as strong emission features.  Dust reddening can affect the color of the quasar continuum and displace a fraction of the high-z quasars outside of the typical selection criterion.  An alternative to a blue near-IR color cut is radio-selection, which reduces the contamination from cool dwarf stars to nearly zero but will only be able to recover $\\sim$5\\% of high-z quasars.   \n\nWe combined optical and radio data through catalog matching.  Counterparts between catalogs were defined using a matching radius of 0.6'' for Stripe 82 and 1'' for CFHTLS Wide, which was a  compromise between completeness and reliability.  We used a method similar to that of \\citet{2002ApJS..143....1M} to estimate completeness and reliability of the matches as a function of radius.  A matching radius of 1'' for FIRST-CFHTLS wide results in a completeness of 83\\% and a reliability of 94\\%.  A matching radius of 0.6'' for Stripe 82-Stripe 82 VLA results in a completeness of 91\\% and a reliability of 99\\%.   If there were multiple optical sources for a single radio source, then only the closest match was used.   After matching the two catalogs, two photometric cuts were applied to select the initial $z > 6$ quasar candidates:  $(i -z)_{AB} > 1.5$ and $z_{AB} <  23.8$ for CFHTLS Wide, and $(i -z)_{AB} > 1.7$ \\footnote[8]{The color conversion between SDSS and CFHT for typical z $\\sim$ 6 quasars is $(i-z)_{SDSS} > (i-z)_{CFHT} + 0.20$;  http:\/\/www.cadc.hia.nrc.gc.ca\/megapipe\/docs\/filters.html} and $z_{AB} <  22.5$ for Stripe 82.  The candidates went through another series of cuts which required that the \\textit{u}, \\textit{g}, and \\textit{r} band fluxes be below the 3$\\sigma$ detection limit.  Also, a visual inspection of the \\textit{i} and \\textit{z} band images was conducted to ensure there were no cosmic rays or bad pixels contaminating the photometry.  The remaining candidates were then checked against known sources as to not repeat observations.  One of the candidates was a previously found z$=$6.21 quasar by \\citet{2010AJ....139..906W}, CFHQS J1429+5447, and is the third radio-loud, z $\\sim$ 6 quasar to be identified.  CFHQS J1429+5447 is the radio-brightest z $\\sim$ 6 quasar yet to be found with f$_{1.4GHz, peak}$ = 2.93 mJy and has the highest radio-loudness value of R $\\sim$ 3200, where R $\\equiv$ $f_{5 GHz} \/ f_{2500\\AA}$.  After all of the cuts, there were 29 remaining candidates in CFHTLS Wide, i.e. $\\sim$ 0.2 per deg$^2$, and 27 in Stripe 82, giving $\\sim$ 0.3 per deg$^2$.  \n\nOur search area was previously mined by other high-z quasar searches and thus our candidate list has some overlap with those selection methods.  In the case of CFHTLS Wide, 3 of our 29 candidates satisfy the criteria set forth by \\citet{2009AJ....137.3541W}, $(i -z)_{AB} > 2.0$ and $z_{AB} < 23.0$ or 10$\\sigma$ $z_{AB}$ limit for the field (private communication).  Only 3 of the 27 candidates in Stripe 82 would have been selected by \\citet{2009AJ....138..305J} while the other 24 candidates were either too blue, $(i -z)_{AB} < 2.2$, or too faint, $z_{AB} > 21.8$ for their completeness-limited selection.      \n\nTo properly estimate how many quasars our radio-selection should find, we had to first estimate the completeness of our method as a function of redshift and rest-frame absolute magnitude, $M_{1450}$\\footnote[9]{$M_{1450}$ was calculated using the measured z-band magnitudes and assuming the \\citet{2001AJ....122..549V} spectrum corrected for the effective Gunn-Peterson optical depth due to Ly$\\alpha$ and Ly$\\beta$ absorption (\\citealp{2006AJ....132..117F})}.  The completeness of our optical selection was estimated through simulation. A relation between redshift and $i-z$ color was calculated using the median color track of z $\\sim$ 3 quasars from SDSS redshifted from z$=$5.5 to z$=$6.7 and corrected for the effective Gunn-Peterson optical depth due to Ly$\\alpha$ and Ly$\\beta$ absorption (\\citealp{2006AJ....132..117F}).  This track is very similar to that calculated by \\citet{2009AJ....137.3541W} as shown in Figure 1.  A ``measured'' $i-z$ color was drawn randomly from a gaussian distribution with a mean $i-z$ color taken from the median quasar color track and a sigma calculated using both the standard deviation of the median quasar color track and the median error as a function of the $i$ and $z$ magnitudes.  This was done hundreds of times for a grid of (z, $M_{1450}$), and the completeness was estimated by the fraction recovered by our selection method.  An example of our completeness as a function of redshift and absolute magnitude is shown in Figure 2.  The simulation of completeness only took into account the optical selection method and not the radio.  The radio-loud ($>$1mJy) fraction of quasars at 0 $<$ z $<$ 5 in SDSS is $\\sim$10\\%, but is strongly dependent on redshift and optical luminosity (\\citealp{jiang2007}).  We adopt a radio-loud fraction of 5\\% for z $\\sim$ 6 and apply this fraction to estimate the number of radio-selected quasars expected.                \n\nUsing the luminosity function for z $=$ 6 quasars calculated by \\citet{2010AJ....139..906W}, we are able to estimate the expected number of quasars in our search, N, \\begin{equation}\nN = A * \\iint \\ \\Phi(z,M_{1450}) \\, V_{c}(z)  \\,\\, p(z,M_{1450})   \\, \\mathrm{d}z \\, \\mathrm{d}M_{1450}.  \n\\end{equation}\nThe comoving volume, $V_{c}(z)$,  is corrected for the completeness, $p(z,M_{1450})$, to form an effective volume.  The luminosity function, $\\Phi(z,M_{1450})$, is in the form of a double power law, and we used the best fit parameters from \\citet{2010AJ....139..906W}.  Our search area is represented by A, in units of steradians.  From the calculation,  we expect 0.7 radio-selected quasars in Stripe 82 and 3.2 in CFHTLS Wide.  We find more candidates than the expected number of quasars for several reasons.  Firstly, some of our candidates are expected to be lower redshift, radio-loud AGN that share a red $i-z$ color with high-z quasars due to a strong Balmer break or dust extincted continuum.  Secondly, a few of our candidates may be spurious matches where the optical source is not really associated with the radio emission.  Thirdly, the use of a constant radio-loud fraction may not be appropriate as it has been observed to be a function of optical luminosity which could change our number estimates as much as a factor of 2 (\\citealp{jiang2007}).    Lastly, the best fit parameters of the luminosity function from \\citet{2010AJ....139..906W} have large errors and can change the expected number of quasars by a factor of 3 or 4.           \n\n\\section{Observations}\n\n\\subsection{Optical and Radio}\n\nThe quasar, SDSS J2228+0110, was selected for followup using the point spread function (PSF) magnitudes from the  co-added imaging catalog of Stripe 82 (\\citealp{2009ApJS..182..543A}).  It was found near our $z$-band magnitude limit at $z$ $=$ 22.28 and near our $i-z$ color limit at $i-z$ $=$ 1.81 which would have been too faint and too blue to be selected by \\citet{2009AJ....138..305J}.  It is one of the faintest quasars found to date, with M$_{1450}$ $=$ -24.53.  The quasar overlaps the UKIRT Infrared Deep Sky Survey (UKIDSS) sky coverage, but is undetected in the $J$-band placing an upper limit at $z-J \\le 1.4$\\footnote[10]{UKIDSS Large Area Survey detection limit is $J_{AB} = 20.9$, http:\/\/www.ukidss.org\/surveys\/surveys.html}.\n\nThe quasar was detected in the radio in the Stripe82 VLA Survey, which has a detection limit of 0.30 mJy.  SDSS J2228+0110 has a measured peak flux density of 0.31 mJy, just above the detection limit, and it is only the fourth z $\\sim$ 6 quasar discovered with a flux density f$_{1.4GHz} >$  0.3 mJy.  The faint optical luminosity and relatively high radio luminosity, $L_{5 GHz} = $ 2.55 $\\times$ 10$^{32}$, makes this one of the most radio-loud z $\\sim$ 6 quasars ever found, with a radio-loudness of R $\\sim$ 1100.  There is no discernible morphology from the optical or radio images given that the discovered quasar was at the limits of detection in both wavelengths (see Figure 3).\n\n\\subsection{Quasar Spectrum}\n\nFrom thirty-five candidates observed in June and December 2010 (sixteen from CFHTLS Wide and nineteen from Stripe 82 of which most remain unidentified with featureless continua), we discovered one quasar at z=5.95, SDSS J2228+0110 (see Table 1).  The discovery spectrum of SDSS J2228+0110 was taken using the Keck I telescope with the Low Resolution Imaging Spectrometer (LRIS; \\citealp{1995PASP..107..375O}).  The spectrum included four exposures of 900 seconds each and a 1\" long slit.  It was taken with a 600\/10000 grating on the red camera, resulting in a dispersion of $\\sim$0.8$\\AA$ per pixel.  The conditions were fair when the spectrum was taken, with $\\sim$1'' seeing at a high air mass of 1.5.  \n\nThe reduction of the spectrum was done in a standard way.  Bias frames were obtained and combined with the overscan bias, then subtracted off the science image.  A flat field correction was applied using flat field frames taken during the observing run.  A specialized routine was used for cosmic ray rejection which recognized and rejected cosmic rays based on their shape.  The spectrum of our quasar was extracted using optimal variance weighting through the IRAF task apall.  The wavelength was calibrated using arc line lamps.  The standard star used for flux calibration was Feige 34\\footnote[11]{The calibration flux was obtained from the Space Telescope standard star flux catalog} in combination with a custom Mauna Kea extinction curve as a function of wavelength.  A lower resolution spectrum, $\\sim$8$\\AA$ per pixel, was created using inverse sky-variance weighting.  The spectrum clearly identifies the source as a z=5.95 quasar with a large continuum break blueward of a strong emission line marked as Ly$\\alpha$ (see Figure 4).\n\nThe only strong emission feature in the discovery spectrum is Ly$\\alpha$.  The noise from sky emission lines resulted in a low signal to noise continuum, making it difficult to claim the detection of other, weaker emission lines.  However, the redshift calculated from Ly$\\alpha$ does seem consistent with possible detections of other common emission features such as NV and Si IV.  The wavelength coverage of the spectrum was not large enough to detect Ly$\\beta$ or OVI.    \n\nWe measured the rest-frame equivalent width (EW) and full width at half maximum (FWHM) of Ly$\\alpha$ for SDSS J2228+0110.  The  wavelength coverage for this spectrum was too small to fit the continuum slope; instead, we assumed it to be a power law with slope $\\alpha$ = -0.5 (f$_{\\nu} \\propto {\\nu}^{\\alpha}$).  The normalization of the power law was fit through a chi-squared minimization to the continuum redward of 1280$\\AA$.  We fit the Ly$\\alpha$ profile with a Gaussian on its red side, and we assumed the line to be symmetric to account for the Ly$\\alpha$ forest on the blue side.  We find a rest-frame FWHM of 7.68 $\\AA$, which is equivalent to a velocity of 1,890 km\/s, making it a narrow Ly$\\alpha$ but not abnormal.  The rest-frame EW is 21.9 $\\AA$ and is at the low end, but well within the range of Ly$\\alpha$ strengths of other z $\\sim$ 6 quasars.\n\n\\section{Discussion}\n\nHigh-redshift quasar searches have been popular over the last decade.  Large scale efforts have revealed that these objects are quite rare and quite difficult to find (\\citealp{2000AJ....120.1167F}; \\citealp{2007AJ....134.2435W}; \\citealp{2008AJ....135.1057J}).  Many of these surveys use optical data to select candidates and rely on very red optical colors.  Unfortunately, high-z quasars share the same red optical color space with cool dwarf stars.  In an effort to reduce contamination, a blue near-IR cut is used in the selection method to help separate the two populations.  This technique successfully increases efficiency; however, it reduces the completeness of the quasar sample by an unknown amount.   Our incompleteness was estimated by redshifting low-z quasar spectra to high-z and tracking their colors.  This track indicates that a large ``blue'' quasar population exists at high-z, and that optical surveys are selecting a significant sample of the population.  However, it is an open question as to whether low-z quasars are truly like their high-z counterparts.  \n\n\\citet{2006ApJ...652..157M} discovered, at the time, the highest redshift radio-loud quasar (FIRST J1427+3312) using radio selection in a search area of only 4 deg$^2$.  The quasar was bright enough in the optical to be detected in other surveys, but its red near-IR color would have prevented its selection.  This rare discovery in such a small area hints that there may be a larger population of ``red'' quasars than predicted.  If this is true, current estimates for the quasar number density at z $\\sim$ 6 are too low.  \n \nRadio-selection offers an alternative method for the selection of high-z quasars.  Requiring a radio detection is just as efficient in removing contaminants as a blue near-IR color cut; however, radio-selection also allows for the detection of ``red'' quasars, possibly from dust reddening.  A likely cause of dust reddening is the circumnuclear cocoon that is thought to encase relatively young quasars (\\citealp{ 2005ApJ...625L..71H}).  As the quasar ages, the dust cocoon may be blown away by energy released from accretion at near-Eddington luminosities.  The universe is less than one billion years old at z $\\sim$ 6, and it would not be unreasonable that the fraction of quasars in this dust cocoon phase is higher at higher redshift.  \n \nOf the seven z $\\sim$ 6 quasars with $z-J >$ 0.8, three of them are radio-loud.  It is not unusual that radio-loud quasars are redder than their radio-quiet counterparts, as this effect is also seen at lower redshift (\\citealp{2003AJ....126..706W}, \\citealp{2009AJ....138.1925M}).  However, when quasars at lower redshift, z $\\sim$ 3, are redshifted to z $=$ 6 and placed in the same colorspace as high-z quasars, at higher redshift radio-loud quasars tend to have redder $z-J$ colors than at lower redshift (see Figure 5).  This is a very intriguing result albeit in the small number regime, as it might suggest that radio-loud quasars are intrinsically redder at higher redshift or have higher quantities of dust.  The result should also be taken with caution as there is a substantial selection effect for z $\\sim$ 3 quasars in SDSS (\\citealp{richards}).  The selection efficiency at z $\\sim$ 3 is $\\sim$50$\\%$ and biased towards redder $u-g$ colors; however, radio-selected z $\\sim$ 3 quasars don't seem to show the same color bias (\\citealp{worseck}).  This comparison of radio-loud\/radio-selected z $\\sim$ 3 quasars redshifted to z $=$ 6 with observed radio-loud z $\\sim$ 6 quasars should be absent of a color selection-effect and serve as a viable comparison in cosmic time.    \n\nAlthough dust reddening of the quasar continuum may contribute to a significant ``red'' quasar population, it would also cause high levels of extinction in the detection bands.  This extinction can be greater than three magnitudes for E(B-V) = 0.1 using a typical SMC  reddening law (\\citealp{prevot}), placing even some of the brightest z $\\sim$ 6 quasars below the detection limit of wide-area surveys like CFHT and SDSS.  A more likely cause of a significant ``red'' quasar population is the strength of Ly$\\alpha$.  A strong Ly$\\alpha$ line leads to a ``bluer'' population of high-z quasars in the $z-J$ color while a weak Ly$\\alpha$ line leads to a ``redder'' population of z $\\sim$ 6 quasars.  A second factor that can lead to a ``red'' quasar population is the absorption blueward of Ly$\\alpha$ from neutral hydrogen in the intergalactic medium, which can lead to a ``red'' $i-z$ color for stronger absorption (see Figure 6).\n\nThis is still an ongoing search for radio-selected quasars.  We plan to observe more candidates in future observations and obtain a deeper optical spectrum of SDSS J2228+0110.  In an effort to measure the $z-J$ color of radio-selected z $\\sim$ 6 quasars, we also plan to follow up on our current discovery and any new discoveries with near-IR imaging.  It would be interesting to use the discovered z $\\sim$ 6 quasar to constrain the luminosity function of high-z quasars outside of the typical optical selection criteria; however, our survey is not complete.   \n \nThe greatest limitation to radio-selection is the depth of the radio data.  Only $\\sim$5\\% of z $\\sim$ 6 quasars are detected in FIRST.  Even a medium depth radio survey like the Stripe82 VLA Survey does not reach the median quasar radio luminosity.  With a deep and wide-area radio survey, questions about the existence of a large ``red'' high-z quasar population, as well as the mechanism that might cause the reddening, could finally be answered.                  \n\n{\\bf Acknowledgements}:  Some of the data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation.\nBased on observations obtained with MegaPrime\/MegaCam, a joint project of CFHT and CEA\/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. \nThe authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community.  We are most fortunate to have the opportunity to conduct observations from this mountain.  The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.\nGRZ acknowledges NRAO Grant GSSP 09-0010.  The work by RHB was partly performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1} \\vspace{0mm}}\n\\newcommand{\\SubSection}[1]{\\vspace{-1mm} \\subsection{#1} \\vspace{-0mm}}\n\\newcommand{\\SubSubSection}[1]{\\vspace{-1mm} \\subsubsection{#1} \\vspace{-1mm}}\n\n\\newcommand\\Mark[1]{\\textsuperscript#1}\n\n\\iccvfinalcopy\n\n\\def\\iccvPaperID{10568}\n\\def\\mbox{\\tt\\raisebox{-.5ex}{\\symbol{126}}}{\\mbox{\\tt\\raisebox{-.5ex}{\\symbol{126}}}}\n\n\n\\begin{document}\n\t\n\\begin{textblock*}{\\textwidth}(0cm,0cm)\n\t\\large\\noindent{\\copyright~2021 IEEE.  Personal use of this material is permitted.  Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting\/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.}\n\\end{textblock*}\n\\thispagestyle{empty}\t\n\n\n\\title{\\textbf{Full-Velocity Radar Returns by Radar-Camera Fusion}}\n\n\\author{\nYunfei Long\\Mark{1}, Daniel Morris\\Mark{1},  Xiaoming Liu\\Mark{1}, \\\\\nMarcos Castro\\Mark{2},\nPunarjay Chakravarty\\Mark{2},\nand Praveen Narayanan\\Mark{2} \\\\ \n\\Mark{1}Michigan State University, \\Mark{2}Ford Motor Company \\\\\n{\\tt\\small \\{longyunf,dmorris,liuxm\\}@msu.edu},\n{\\tt\\small \\{mgerard8,pchakra5,pnaray11\\}@ford.com}\n}\n\\date{}\n\n\n\\maketitle\n\\pagenumbering{arabic}\n\n\\begin{abstract}\nA distinctive feature of Doppler radar is the measurement of velocity in the radial direction for radar points. \nHowever, the missing tangential velocity component hampers object velocity estimation as well as temporal integration of radar sweeps in dynamic scenes.  \nRecognizing that fusing camera with radar provides complementary information to radar, in this paper we present a closed-form solution for the point-wise, full-velocity estimate of Doppler returns using the corresponding optical flow from camera images.  \nAdditionally, we address the association problem between radar returns and camera images with a neural network that is trained to estimate radar-camera correspondences. \nExperimental results on the nuScenes dataset verify the validity of the method and show significant improvements over the state-of-the-art in velocity estimation and accumulation of radar points.   \n\\end{abstract}\n\n\\section{Introduction}\nRadar is a mainstream automotive 3D sensor, and along with LiDAR and camera, is used in perception systems for driving assistance and autonomous driving~\\cite{stanislas2015characterisation, li2020lidar,m3d-rpn-monocular-3d-region-proposal-network-for-object-detection}. \nUnlike LiDAR, radar has been widely installed on existing vehicles due to its relatively low cost and small sensor size, which makes it an easy fit into various vehicles without changing their appearance. Thus, advances in radar vision systems have potential to make immediate impact on vehicle safety. Recently, with the release of a couple of autonomous driving datasets with radar data included,~\\emph{e.g.}, Oxford Radar RobotCar~\\cite{barnes2020oxford} and nuScenes~\\cite{caesar2020nuscenes}, there is great interest in the community to explore how to leverage radar data in various vision tasks such as object detection~\\cite{nabati2021centerfusion, yang2020radarnet}.\n\n\n\\begin{figure}[t!]\n    \\captionsetup{font=small}\n\t\\centering\n\t\\begin{subfigure}[b]{\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.9\\textwidth]{.\/figures\/indeterminacy.pdf}\n\t\t\\vspace{-2mm}\n\t\t\\caption{$ $}\n\t\t\\label{fig:y equals x}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[b]{0.2\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{.\/figures\/flow1}\n\t\t\\caption{$ $}\n\t\t\\label{fig:three sin x}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[b]{0.22\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{.\/figures\/ex_motion.pdf}\n\t\t\\caption{$ $}\n\t\t\\label{fig:five over x}\n\t\\end{subfigure}\n\t\\vspace{-2mm}\n\t\\caption{\\small (a) Full motion cannot be determined with a single sensor: all motions ending on the blue dashed line (\\emph{i.e.},~blue dashed arrows) map to the same optical flow and all motions terminated on the red dashed line (\\emph{i.e.},~red dashed arrows) fit the same radial motion. However, with a radar-camera pair, the full motion can be uniquely decided: only the motion drawn in black satisfies both optical flow and radial motion. (b) Optical flow in the camera-image and (c) a bird's-eye view of the observed vehicle. This shows measured radar points with radial velocity (red), our predicted point-wise, full velocity (black), and ground truth full velocity of the vehicle (green).}\n\t\\vspace{-3mm}\n\t\\label{Figure:indeterminacy}\n\\end{figure}\n\nIn addition to measuring 3D positions, radar has the special capability of obtaining radial velocity of returned points based on the Doppler effect. \nThis extra capability is a significant advantage over other 3D sensors like LiDAR, enabling, for instance, instantaneous moving object detection. \nHowever, due to the inherently ambiguous mapping from radial velocity to full velocity, using radial velocity directly to account for the real movement of radar points is inadequate and sometimes misleading. \nHere, the full velocity denotes the actual velocity of radar points in 2D or 3D space. \nWhile radial velocity can well approximate full velocity when a point is moving away from or towards the radar, these two can be very different when the point is moving in the non-radial directions. \nAn extreme case occurs for objects moving tangentially as these will have zero radial velocity regardless of target speed. Therefore, acquiring point-wise full velocity instead of radial velocity is crucial to reliably sense the motion of surrounding objects.\n\nApart from measuring the velocity of objects, another important application of point-wise velocity is the accumulation of radar points. \nRadar returns from a single frame are much sparser than LiDAR in both azimuth and elevation,~\\emph{e.g.}, typically LiDAR has an azimuth resolution $10\\times$ higher than radar~\\cite{yang2020radarnet}.\nThus, it is often essential to accumulate multiple prior radar frames to acquire sufficiently dense point clouds for downstream tasks,~\\emph{e.g.}, object detection~\\cite{nobis2019deep, chadwick2019distant, chang2020spatial}. \nTo align radar frames, in addition to compensating egomotion, we shall consider the motion of moving points in consecutive frames, which can be estimated by point-wise velocity and time of movement. \nAs the radial velocity does not reflect the true motion, it is desirable to have point-wise full velocity for point accumulation.\n\n\nTo solve the aforementioned dilemma of radial velocity, we propose to estimate point-wise full velocity of radar returns by fusing radar with a RGB camera. \nSpecifically, we derive a closed-form solution to infer point-wise full velocity from radial velocity as well as associated projected image motion obtained from optical flow. As shown in Fig.~\\ref{Figure:indeterminacy}, constraints imposed by optical flow resolve the ambiguities of radial-full velocity mapping and lead to a unique and closed-form solution for full velocity. \nOur method can be considered as a way to enhance raw radar measurement by upgrading point-wise radial velocity to full velocity, laying the groundwork for improving radar-related tasks, ~\\emph{e.g.}, velocity estimation, point accumulation and object detection. \n\nMoreover, a prerequisite for our closed-form solution is the association between moving radar points and image pixels. \nTo enable a reliable association, we train a neural network to predict radar-camera correspondences as well as discerning occluded radar points. \nExperimental results demonstrate that the proposed method improves point-wise velocity estimates and their use for object velocity estimation and radar point accumulation.\n\nIn summary, the main contributions of this work are:\n\\begin{itemize}\n    \\item We define a novel research task for radar-camera perception systems,~\\emph{i.e.}, estimating point-wise full velocity of radar returns by fusing radar and camera.\n    \n    \\item We propose a novel closed-form solution to infer full radar-return velocity by leveraging the radial velocity of radar points, optical flow of images, and the learned association between radar points and image pixels.\n    \n    \\item We demonstrate state-of-the-art (SoTA) performance in object velocity estimation, radar point accumulation, and 3D object localization.\n    \n\\end{itemize}\n\n\n\\section{Related Works}\n\n\\Paragraph{Application of Radar in Vision}\nRadar data differs from LiDAR data in various aspects~\\cite{brodeski2019deep}. \nIn addition to the popular point representation (also named radar target~\\cite{palffy2020cnn}), an analogy to LiDAR points, there are other radar data representations containing more raw measurements, {\\it e.g.}, range-azimuth image and spectrograms, which have been applied in tasks such as activity classification~\\cite{seyfiouglu2018deep}, detection~\\cite{lim2019radar}, and pose estimation~\\cite{roos2016reliable}.  \nOur method is based on radar points, with the format available in the  nuScenes dataset~\\cite{caesar2020nuscenes}. \n\nThe characteristics of radar have been explored to complement other sensors. \nThe Doppler velocity of radar points is used to distinguish moving targets. \nFor example, RSS-Net~\\cite{kaul2020rss} uses radial velocity as a motion cue for image semantic segmentation. Chadwick~\\emph{et al.}~\\cite{chadwick2019distant} use radial velocity to detect distant moving vehicles---difficult to detect with only images. \nFritsche~\\emph{et al.}~\\cite{fritsche2017fusion} combine radar with LiDAR for measurement under poor visibility. \nWith a longer detection range than LiDAR, radar is also deployed with LiDAR to better detect far objects~\\cite{yang2020radarnet}.\n\nThe {\\it sparsity} of radar makes it difficult to directly apply well-developed techniques for LiDAR on radar~\\cite{lim2019radar, nabati2021centerfusion}. \nFor example, Danzer~\\emph{et al.}~\\cite{danzer20192d} adopt PointNets~\\cite{qi2017pointnet} on radar points for 2D car detection, while sparsity limits it to large objects like cars. \nSimilar to LiDAR-camera depth completion~\\cite{depth-completion-with-twin-surface-extrapolation-at-occlusion-boundaries,depth-coefficients-for-depth-completion}, Long~\\emph{et al.}~\\cite{long2021radar} develop radar-camera depth completion by learning a probabilistic mapping from radar returns to images.\nTo obtain denser radar points, Lombacher~\\emph{et al.}~\\cite{lombacher2016potential} use occupancy grid~\\cite{elfes1989using} to accumulate radar frames.\nYet, the method assumes a static scene and cannot cope with moving objects. \nRadar points are projected on images and represented as regions near projected points, such as vertical bars~\\cite{nobis2019deep} and circles~\\cite{chadwick2019distant, chang2020spatial}, to account for uncertainty of projection due to measurement error. \nWhile accumulating radar frames is desirable, without reliably compensating object motion, these methods need to carefully decide the number of frames to trade off between the gain in accumulation and loss in accuracy due to delay~\\cite{nobis2019deep}. \nOur estimated point-wise velocity can compensate object motion and realize more accurate accumulation.\n\n\n\\Paragraph{Velocity Estimation in Perception Systems}\nResearchers have used monocular videos~\\cite{kinematic-3d-object-detection-in-monocular-video} or radial velocity of radar points to estimate {\\it object-wise} velocity. With only radar data of a single frame, Kellner~\\emph{et al.}~\\cite{kellner2013instantaneous, kellner2014instantaneous} compute full velocity of moving vehicles from radial velocities and azimuth angles of at least two radar hits. \nHowever, for a robust solution, the method requires that 1) radar captures more radar hits on each object, 2) radar points have significantly different azimuth angles and 3) object points are clustered before velocity estimation~\\cite{kellner2013instantaneous, schlichenmaier2019clustering, scheiner2019multi}. \nObviously due to sparsity of radar in a single frame, it is difficult to obtain at least two radar hits on distant vehicles, let alone objects of smaller sizes. \nAlso, it is common that radar points on the same object, {\\it e.g.}, a distant or small object, have similar azimuth. \n\n\\begin{figure*}[t!]\n    \\captionsetup{font=small}\n\t\\begin{center}\n\t\t\\includegraphics[width=\\linewidth]{figures\/diagram.pdf}\n\t\\end{center}\n\t\\vspace{-6mm}\n\t\\caption{\\small \\textbf{Full velocity estimation and learning to associate radar points to camera pixels}. (a) A 3D point, $\\bm{p}$, is observed by a camera at $B$.  A short interval, $\\Delta t$, later, the point has moved by $\\dot{\\bm{m}}\\Delta t$ to $\\bm{q}$ while the camera has moved by $\\dot{\\bm{c}}\\Delta t$ to $A$. At the same time, the radar measures both the position of $\\bm{q}$ and the radial speed $\\dot{r}$, which is the radial component of $\\dot{\\bm{m}}$. Using radial speed $\\dot{r}$ and the associated  optical flow of $\\bm{q}$ in images, we derive a closed-form equation (denoted as $\\bm{f}()$) to estimate $\\bm{q}$'s full velocity $\\dot{\\bm{m}}$. (b) As the closed-form solution requires point-wise association of two sensors, we train a Radar-2-Pixel (R2P) network to take a multi-channel input and predict the association probabilities for pixels within a neighborhood of the raw projection (white dot) obtained via known pose $\\prescript{A}{R}\\T$. A pixel with the highest probability (yellow arrow) is deemed as the associated pixel of a radar point. To obtain labels for training R2P, our label generation module uses $\\bm{f}()$ to compute velocities of all neighboring pixels, then calculates velocity error $E_m$ by using the ground truth velocity $\\dot{\\bm{m}}_{GT}$, and finally obtains association probabilities of these neighbors based on $E_m$.}\n\t\\label{fig:diagram_association}\n\t\t\\vspace{-3mm}\n\\end{figure*}\n\n\nRecognizing the density and accuracy limitation of radar, researchers fuse radar with other sensors, {\\it e.g.}, LiDAR and camera, for object-wise velocity estimation. \nSpecifically, existing techniques~\\cite{zhao2019object,wu2020deep,li2020deep} for images or LiDAR are employed to obtain preliminary detections. \nRadar data, including radial velocity, once associated with the initial detections, are used as additional cues to predict full velocities of objects. \nFor instance, in RadarNet~\\cite{yang2020radarnet} \ntemporal point clouds of radar and LiDAR, modeled as voxels, are used to acquire initial detections and their motions. \nObject motion direction is used to resolve the ambiguities in radar-point association by\nback-projecting their radial velocities on the motion direction. \nYet, a sequence of LiDAR frames is required to obtain the initial detection and motion estimation.\n\nCenterFusion~\\cite{nabati2021centerfusion} integrates radar with camera for object-wise velocity estimation. \nWell-developed image-based detector is applied to extract preliminary boxes.\nAfter associating radar points with detections, the method combines radar data, radial velocity and depth, with image features within detected regions to regress a full velocity per detection. \nHowever, without a closed-form solution, the mapping from radial to full velocity needs to be learned from a great number of labeled data. \nIn contrast, we present a point-wise {\\it closed-form solution} for full-velocity estimation of radar points, without performing object detection. \nTo our knowledge, there is no prior method able to perform point-wise full-velocity estimation for radar returns.\n\n\n\\section{Proposed Method}\n\nWe consider the case of a camera and radar rigidly attached to a moving platform, {\\it e.g.}, a vehicle, observing moving objects in the environment.  In this section we develop equations relating optical flow measurements in the camera to position and velocity measurements made by the radar.  \n\n\\subsection{Physical Configuration and Notation}\n\n\nThe physical configuration of our camera and radar measurements is illustrated in Fig.~\\ref{fig:diagram_association}(a). Three coordinate systems are shown: $A$ and $B$ specifying camera poses and $R$ specifying a radar pose.  The camera at $B$ observes a 3D point $\\bm{p}$.  A short interval later, $\\Delta t$, the point has moved to $\\bm{q}$, the camera to $A$ and the radar to $R$, and both the camera and radar observe the target point $\\bm{q}$.  These 3D points are specified by $4$-dim homogeneous vectors, and when needed, a left-superscript specifies the coordinate system in which it is specified, {\\it e.g.}, $\\prescript{A}{}\\bm{q}$ indicates a point relative to a coordinate system $A$.  The target velocity, $\\dot{\\bm{m}}$, and camera velocity $\\dot{\\bm{c}}$ are specified by $3$-dim vectors, again optionally with a left superscript to specify a coordinate system.    \n\nCoordinate transformations, containing both a rotation and translation, are specified by $4\\times 4$ matrices, such as $\\prescript{B}{A}{\\bm{T}}$, which transforms points from the left-subscript coordinate system to the left-superscript coordinate system.  In this case we transform a point from $A$ to $B$ with:\n\\begin{equation}\n    \\prescript{B}{}\\bm{q} = \\prescript{B}{A}{\\bm{T}} \\: \\prescript{A}{}\\bm{q}.\n\\end{equation}\nOnly the rotational component of these transformations is needed to transform velocities.  For example, $\\prescript{A}{}\\dot{\\bm{m}}$ is transformed to $\\prescript{B}{}\\dot{\\bm{m}}$ by the $3\\times 3$ rotation matrix $\\prescript{B}{A}{\\bm{R}}$:\n\\begin{equation}\n    \\prescript{B}{}\\dot{\\bm{m}} = \\prescript{B}{A}{\\bm{R}} \\: \\prescript{A}{}\\dot{\\bm{m}}.\n\\end{equation}\n\nA vector with a right subscript, {\\it e.g.}, ${\\bm{p}_i}$, indicates the $i$'th element of ${\\bm{p}}$, while a right subscript of ``1:3'' puts the first $3$ elements in a $3$-dim vector.  \nFor a matrix, the right subscript indicates the row. Thus $\\prescript{B}{A}{\\bm{R}_i}$ is a $1\\times 3$ row vector containing its $i$-th row.  A right superscript ``${}^\\mathsf{T}$'' is a matrix transpose.\n\nThe projections of points $\\bm{p}$ and $\\bm{q}$ are specified in either undistorted raw pixel coordinates, {\\it e.g.}, $(x_q,y_q)$ or their normalized image coordinates $(u_q,v_q)$ given by:\n\\begin{equation}\n    u_q = (x_q-c_x) \/ f_x, \\hspace{0.5cm} v_q = (y_q-c_y) \/ f_y.\n\\end{equation}\nHere $c_x,c_y,f_x,f_y$ are intrinsic camera parameters, while the right subscript of the pixel refers to the point being projected.  Vectors for 3D points can be expressed in terms of the normalized image coordinates:\n\\begin{align}\n\\prescript{A}{}\\qv = \\begin{pmatrix} u_qd_q\\\\ v_qd_q\\\\ d_q\\\\ 1 \\end{pmatrix} \n\\;\\; \\text{and} \\;\\;\n\\prescript{B}{}\\pv = \\begin{pmatrix} u_pd_p\\\\ v_pd_p\\\\ d_p\\\\ 1 \\end{pmatrix}.\n\\label{eq:qa}\n\\end{align}\nHere $d_q$ and $d_p$ are depths of points $\\prescript{A}{}\\qv$ and $\\prescript{B}{}\\pv$ respectively. \n\nWe assume dense optical flow is available that maps target pixel coordinates observed in $A$ to $B$ as follows:\n\\begin{equation}\n    \\text{Flow}\\left( (u_q,v_q) \\right) \\rightarrow (u_p,v_p).\n    \\label{eq:flow}\n\\end{equation}\nFurther, we assume the following are known: camera motion, $\\prescript{B}{A}\\T$, relative radar pose, $\\prescript{A}{R}\\T$, and intrinsic parameters.\n\n\\subsection{Full-Velocity Radar Returns}\n\nThe Doppler velocity measured by a radar is just one component of the three-component, full-velocity vector of an object point.  Here our goal is to leverage optical flow from a synchronized camera to augment radar and estimate this full-velocity vector for each radar return.  \n\n\\subsubsection{Relationship of Full Velocity to Radial Velocity}\n\nThe target motion from $\\bm{p}$ to $\\bm{q}$ is modeled as constant velocity, $\\dot{\\bm{m}}$, over time $\\Delta t$, such that \n\\begin{equation}\n    \\dot{\\bm{m}}=\\frac{\\bm{q}_{1:3}-\\bm{p}_{1:3}}{\\Delta t}.  \n    \\label{eq:velocity}\n\\end{equation}\nOur goal is to estimate the full target velocity, $\\dot{\\bm{m}}$.  Radar provides an estimate of the target position, $\\bm{q}$, but not the previous target location $\\bm{p}$.  Radar also provides the signed radial speed, $\\dot{r}$, which is one component of $\\dot{\\bm{m}}$.  In the nuScenes dataset $\\dot{r}$ is given by:\n\\begin{equation}\n    \\dot{r} = \\hat{\\rv}^\\mathsf{T}\\dot{\\bm{m}}.\n    \\label{eq:radial}\n\\end{equation}\nHere $\\hat{\\rv}$ is the unit-norm vector along the direction to the target $\\prescript{R}{}\\qv$.   Note that this equation is coordinate-invariant, and could  be equally written in $A$ using $\\prescript{A}{}\\hat{\\rv}$ and $\\prescript{A}{}\\mvdot$.  Now Eq.~\\eqref{eq:radial} is actually the egomotion-corrected Doppler speed.  The raw Doppler speed, $\\dot{r}_{raw}$, is the radial component of the \\emph{relative} velocity between target and sensor, $\\dot{\\bm{m}}-\\dot{\\bm{c}}$, and this constraint is given by:\n\\begin{equation}\n    \\dot{r}_{raw} = \\hat{\\rv}^\\mathsf{T}(\\dot{\\bm{m}}-\\dot{\\bm{c}}),\n    \\label{eq:radial_relative}\n\\end{equation}\nwhere $\\dot{\\bm{c}}$ is the known ego-velocity. Either Eq.~\\eqref{eq:radial} or \\eqref{eq:radial_relative} can be used in our formulation, depending on whether $\\dot{r}$ or $\\dot{r}_{raw}$ is available from the radar.\n\n\n\\subsubsection{Relationship of Full Velocity to Optical Flow}\n\nIn solving the velocity constraints, we first identify the known variables.  The radar measures $\\prescript{R}{}\\qv$, and transforming this we obtain $\\prescript{A}{}\\qv=\\prescript{A}{R}\\bm{T}\\:\\prescript{R}{}\\qv$ which contains $d_q$ as the third component.  Image coordinates $(u_q,v_q)$ are obtained by projection, and using optical flow in Eq.~\\eqref{eq:flow}, we can also obtain the $(u_p,v_p)$ components of $\\prescript{B}{}\\pv$.  The key parameter we do not know from this is the depth, $d_p$, in $B$.\n\nNext we eliminate this unknown depth from our constraints. Eq.~\\eqref{eq:velocity} can be rearranged and each component expressed in frame $B$:\n\\begin{equation}\n    \\prescript{B}{}\\pv_{1:3} = \\prescript{B}{}\\qv_{1:3} - \\prescript{B}{A}\\R\\: \\prescript{A}{}\\mvdot\\Delta t,\n    \\label{eq:pvB2}\n\\end{equation}\nwhere the second term on the right is the transformation of the target motion into $B$ coordinates.  The third row of this equation is an expression for $d_p$:\n\\begin{equation}\n    d_p = \\prescript{B}{}\\qv_3 - \\prescript{B}{A}\\R_3\\: \\prescript{A}{}\\mvdot\\Delta t.\n    \\label{eq:dp1}\n\\end{equation}\nSubstituting this for $d_p$, and the components of $\\prescript{B}{}\\pv$ from  Eq.~\\eqref{eq:qa}, into the first two rows of Eq.~\\eqref{eq:pvB2}, we obtain\n\\begin{eqnarray}\n\\begin{bmatrix}\nu_p(\\prescript{B}{}\\qv_3 - \\prescript{B}{A}\\R_3\\: \\prescript{A}{}\\mvdot\\Delta t) \\\\\nv_p(\\prescript{B}{}\\qv_3 - \\prescript{B}{A}\\R_3\\: \\prescript{A}{}\\mvdot\\Delta t) \\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\prescript{B}{}\\qv_1 - \\prescript{B}{A}\\R_1\\: \\prescript{A}{}\\mvdot\\Delta t \\\\\n\\prescript{B}{}\\qv_2 - \\prescript{B}{A}\\R_2\\: \\prescript{A}{}\\mvdot\\Delta t \\\\\n\\end{bmatrix},\n\\label{eq:dp2}\n\\end{eqnarray}\nand rearrange to give two constraints on the full velocity:\n\\begin{eqnarray}\n\\begin{bmatrix}\n\\prescript{B}{A}\\R_1 - u_p\\prescript{B}{A}\\R_3 \\\\\n\\prescript{B}{A}\\R_2 - v_p\\prescript{B}{A}\\R_3 \\\\\n\\end{bmatrix}\n\\prescript{A}{}\\mvdot\n=\n\\begin{bmatrix}\n\\left(\\prescript{B}{}\\qv_1 - u_p\\prescript{B}{}\\qv_3\\right)\/\\Delta t  \\\\\n\\left(\\prescript{B}{}\\qv_2 - v_p\\prescript{B}{}\\qv_3\\right)\/\\Delta t \\\\\n\\end{bmatrix}.\n\\label{eq:dp3}\n\\end{eqnarray}\n\n\\subsubsection{Full-Velocity Solution}\n\nWe obtain three constraints on the full velocity, $\\prescript{A}{}\\mvdot$, from Eq.~\\eqref{eq:dp3} and by converting Eq.~\\eqref{eq:radial}  to $A$ coordinates. Combining these we obtain: \n\\begin{eqnarray}\n\\begin{bmatrix}\n\\prescript{B}{A}\\R_1 - u_p\\prescript{B}{A}\\R_3 \\\\\n\\prescript{B}{A}\\R_2 - v_p\\prescript{B}{A}\\R_3 \\\\\n\\prescript{A}{}\\hat{\\rv}^\\mathsf{T} \\\\\n\\end{bmatrix}\n\\prescript{A}{}\\mvdot\n=\n\\begin{bmatrix}\n\\left(\\prescript{B}{}\\qv_1 - u_p\\prescript{B}{}\\qv_3\\right)\/\\Delta t  \\\\\n\\left(\\prescript{B}{}\\qv_2 - v_p\\prescript{B}{}\\qv_3\\right)\/\\Delta t \\\\\n\\dot{r} \\\\\n\\end{bmatrix}.\n\\label{eq:dp4}\n\\end{eqnarray}\nThen inverting the $3\\times 3$ coefficient of $\\prescript{A}{}\\mvdot$ gives a closed form solution for the full velocity:\n\\begin{eqnarray}\n\\prescript{A}{}\\mvdot =\n\\begin{bmatrix}\n\\prescript{B}{A}{\\bm{R}_1} - u_p \\prescript{B}{A}{\\bm{R}_3}\\\\\n\\prescript{B}{A}{\\bm{R}_2} - v_p \\prescript{B}{A}{\\bm{R}_3}\\\\\n\\prescript{A}{}\\hat{\\rv}^\\mathsf{T}\\\\ \n\\end{bmatrix}^{-1}\n\\begin{bmatrix}\n\\left(\\prescript{B}{}{\\bm{q}}_1 - u_p \\prescript{B}{}{\\bm{q}_3} \\right) \/ \\Delta t\\\\\n\\left(\\prescript{B}{}{\\bm{q}}_2 - v_p \\prescript{B}{}{\\bm{q}_3} \\right) \/ \\Delta t\\\\\n\\dot{r} \\\\ \n\\end{bmatrix}\n\\label{eq:full_v}.\n\\end{eqnarray}\n\n\n\\begin{figure}[t!]\n    \\captionsetup{font=small}\n\t\\centering\n\t\\scalebox{1}{\n\t\t\\begin{tabular}{@{}c@{}c@{}c@{}c@{}}\n\t\t\t\\includegraphics[width=1.4 in]{.\/figures\/flow2}  &\n\t\t\t\\includegraphics[width=1.4 in]{.\/figures\/bev1.pdf} \\vspace{-1mm}\\\\\n\t\t\t\\footnotesize{ (a) } &  \\footnotesize{ (b) } \\\\\n\t\t\t\\includegraphics[width=1.4 in]{.\/figures\/04_ev}  &\n\t\t\t\\includegraphics[width=1.4 in]{.\/figures\/02_ev} \\vspace{-1mm}\\\\\n\t\t\t \\footnotesize{ (c) } & \\footnotesize{ (d) } \\\\\n\t\\end{tabular}   }\n\t\\vspace{-2mm}\n\t\\caption{\\small (a) Optical flow; (b) Bird's-eye view of GT bounding box, radial velocity (red) and GT velocity (green); (c) and (d) show $E_m$, computed by using Eq.~(\\ref{eq:e_v}), for two radar projections (white square) over $41\\times41$ pixel regions, respectively. For radar hits reflected from the vehicle, $E_m$ is small for neighboring pixels on the car and large on the background.}\n\t\\label{Figure:label}\\vspace{-3mm}\n\\end{figure}\n\n\n\nRecall in Fig.~\\ref{Figure:indeterminacy}(a) the red\/blue dashed lines show the velocity constraints from radar\/flow.  The solution of Eq.~\\eqref{eq:full_v} is the full velocity that is consistent with both constraints. \nWe note that this can handle moving sensors, although Fig.~\\ref{Figure:indeterminacy}(a) shows the case of a stationary camera for simplicity.  Further, if we set $\\Delta t<0$, Eq.~\\eqref{eq:full_v} also applies to the case that the point shifts from $\\bm{q}$ to $\\bm{p}$ as the camera moves from $A$ to $B$.  And one limitation is that Eq.~\\eqref{eq:full_v} cannot estimate full velocity for radar points occluded in the camera view, although we can typically identify those occlusions.\n\n\n\\SubSection{Image Pixels and Radar Points Association}\n\nOur solution for point-wise velocity in Eq.~(\\ref{eq:full_v}) assumes that we know the pixel coordinates $(u_q,v_q)$ of the radar-detected point, $\\prescript{R}{}\\bm{q}$.\nIt appears straightforward to obtain this pixel correspondence by projecting a radar point onto the image using the known radar-image coordinate transformation, $\\prescript{A}{R}\\bm{T}$. \nWe refer to this corresponding pixel as ``raw projection''. \nHowever, there are a number of reasons why raw projection of radar points into an image is inaccurate. Radar beam-width typically subtends a few degrees and is large relative to a pixel, resulting in low resolution target location in both azimuth and elevation.  Also, a radar displaced from a camera can often see behind an object, as viewed by the camera, and when these returns are projected onto an image they incorrectly appear to correspond to the foreground occluding object. Using flow from an occluder or an incorrectly associated object pixel may result in incorrect full-velocity estimation.  To address these issues with raw projection, we train a neural network model, termed Radar-2-Pixel (R2P) network, to estimate associated radar pixels in the neighborhood of raw projection and identify occluded radar points.\nSimilar models have been applied to image segmentation~\\cite{kampffmeyer2018connnet} and radar depth enhancement~\\cite{long2021radar}.\n\n\n\\SubSubSection{Model Structure}\nOur method estimates association probabilities (ranging from $0$ to $1$) between a moving radar point and a set of pixels in the neighborhood of its raw projection. \nThe R2P network is an encoder-decoder structure with inputs and outputs of image resolution. \nStored in $8$ channels, the input data include image, radar depth map (with depth on raw projections) and optical flow. \nThe output has $N$ channels, representing predicted association probability for $N$ pixel neighbors. \nThe association between the radar point, $\\prescript{A}{}\\qv$, and the $k$-th neighbor of raw projection $(x,y)$ is stored in $A(x,y,k)$, where $k=1,2,...,N$.\n\n\n\\SubSubSection{Ground Truth Velocity of Moving Radar Points}\n\\label{sec:GT_velocity}\nThe nuScenes~\\cite{caesar2020nuscenes} provides the GT (ground truth) velocity of object bounding boxes. \nWe associate radar hits on an object to its labeled bounding box, and assign the velocity of the box to its associated radar points.\nThe association is determined based on two criteria:\n1) in radar coordinates, the distance between radar points and associated box is smaller than a threshold $T_d$; \nand 2) the percentage error between the radial velocity of a radar point and the radial component of the velocity of associated box is smaller than a threshold $T_p$.\n\n\\SubSubSection{Generating Association Labels}\nWe can project a radar point expressed in corresponding camera coordinates, $\\prescript{A}{}\\qv$, to pixel coordinates $(u_q,v_q)$, but as mentioned before, often this image pixel does not correspond to the radar return.  Our proposed solution is to search in a neighboring region around $(u_q,v_q)$ for a pixel whose motion is consistent with the radar return.  This neighborhood search is shown in Fig.~\\ref{fig:diagram_association}.   If a pixel is found, then we correct the 3D radar location $\\prescript{A}{}\\qv$ to be consistent with this pixel, otherwise we mark this radar return as occluded.\n\nWe learn this radar-to-pixel association and correction by training the R2P network.  \nWe generate true association score between a radar point and a pixel according to the compatibility between the true velocity and the optical flow at that pixel: high compatibility indicates high association. \nTo quantify the compatibility, assuming a pixel is associated with a radar point, we compute a hypothetical full velocity for the radar point by using the optical flow of that pixel according to Eq.~\\eqref{eq:full_v}. \nThe flow is considered compatible\nif the hypothetical velocity is close to the GT velocity. \nSpecifically, the  hypothetical velocity can be computed as\n\\begin{equation}\n\\resizebox{.88\\hsize}{!}{\n$\\prescript{A}{}\\mvdot_{est}(x,y,k) = \\bm{f}\\left(\\breve{u}_q,\\breve{v}_q,\\breve{u}_p,\\breve{v}_p, d_q, \\dot{r}, \\prescript{B}{A}\\T, \\prescript{A}{R}\\T \\right),$\n}\n\\label{eq:fvel}\n\\end{equation}\nwhere $k=1,\\cdots,N$, $\\bm{f}(\\cdot)$ is the function to solve full velocity via Eq.~\\eqref{eq:full_v}, and $(x,y)$ is the raw projection of the radar point.\nNote that  $\\breve{u}_q=u_q\\left[x+\\Delta x(k), y+\\Delta y(k)\\right]$, $\\breve{v}_q$ is defined similarly, \nand $[\\Delta x(k),\\Delta y(k)]$ is the coordinate offset from raw projection to the $k$-th neighbor.  Using flow, Eq.~\\eqref{eq:flow}, we obtain $(\\breve{u}_p,\\breve{v}_p)$ from $(\\breve{u}_q,\\breve{v}_q)$.\n\nSecond, we calculate the $L_2$ norm of errors between $\\prescript{A}{}\\mvdot_{est}(x,y,k)$ and ground truth velocity $\\prescript{A}{}\\mvdot_{GT}(x,y)$ by\n\\begin{equation}\nE_m(x,y,k)= \\lVert \\prescript{A}{}\\mvdot_{est}(x,y,k) - \\prescript{A}{}\\mvdot_{GT}(x,y) \\rVert_2\n\\label{eq:e_v}.\n\\end{equation}\nFig.~\\ref{Figure:label} shows examples of $E_m$ for two radar hits on a car.\n\nFinally, we transform $E_m$ to an association score with\n\\begin{equation}\nL(x,y,k)= e^{-\\frac{E^2_m(x,y,k)}{c}}\n\\label{eq:prob},\n\\end{equation}\nwhere $L$ is used as a label for association probability between a radar and its $k$-th neighbor. Note that $L$ increases with decreasing $E_v$, and $c$ is a parameter adjusting the tolerance of velocity errors when converting errors to association. \nWe use the cross entropy loss to train the model.\n\n\n\n\n\\SubSubSection{Estimate Association and Identify Occlusion}\nWith a trained model, we can estimate association probability between radar points and $N$ pixels around their raw projections $(x,y)$, {\\it i.e.}, $A(x,y,k)$.\nAmong the $N$ neighbors, the radar return velocity may be compatible with a number of pixels, and we select the pixel with the maximum association, $A_{max}$, as the neighbor ID $k_{max}$:\n\\begin{equation}\nk_{max}= \\underset{k}{\\arg\\max}[A(x,y,k)]\n\\label{eq:k_max}.\n\\end{equation}\n If $A_{max}$ is equal or larger than a threshold $T_a$, we estimate the associated pixel as $\\left[x+\\Delta x(k_{max}), y+\\Delta y(k_{max})\\right]$.  \nOtherwise there is no associated pixels in the neighborhood, and an occlusion is identified.\n\n\\section{Experimental Results}\n\\subsection{Comparison of Point-wise Full Velocity}\n\nTo the best of our knowledge, there is no existing method estimating {\\it point-wise} full velocity for radar returns. \nThus, we use point-wise radial velocity from raw radar returns as the baseline to compare with our estimation. \nWe extract data from the nuScenes Object Detection Dataset~\\cite{caesar2020nuscenes}, with $6432$, $632$, and $2041$ samples in training, validation and testing set, respectively. \nEach sample consists of a radar scan and two images for optical flow computation, {\\it i.e.}, one image synchronizing with the radar and the other is a neighboring image frame. \nThe optical flow is computed by the RAFT model~\\cite{teed2020raft} pre-trained on KITTI~\\cite{geiger2013vision}. \nThe R2P network is an U-Net~\\cite{ronneberger2015u, morris2018pyramid} with five levels of resolutions and $64$ channels for intermediate filters. \nThe neighborhood skips every other pixel, and its size (in pixels) is (left: $4$, right: $4$, top: $10$, bottom: $4$) and an example of the neighborhood is illustrated in Fig.~\\ref{fig:diagram_association}(b). \nThe threshold of association scores $T_a$ is $0.3$. Parameters associating radar points with GT bounding box are set as $T_d=0.5$m and $T_p=20\\%$. \nParameter $c$ in Eq.~\\eqref{eq:prob} is  $0.36$.\nTo obtain GT point-wise velocity, based on the criteria in Sec.~\\ref{sec:GT_velocity}, we first associate moving radar points to GT detection boxes, whose GT velocity is assigned to associated points as their GT velocity. The GT velocity of bounding boxes is estimated from GT center positions in neighboring frames with timestamps. \n\nTab.~\\ref{tab:baseline} shows the average velocity error for moving points. The proposed method achieves substantially more accurate velocity estimation than the baseline. \nFor instance, the error of our tangential component is only $21\\%$ of that of the baseline.\nWe also have much smaller standard deviation, indicating more stable estimates. In addition, we list in Tab.~\\ref{tab:baseline} velocity error of our method using raw radar projection for radar-camera association. Results show that, compared with using raw projection, using R2P network achieves higher estimation accuracy.\nFig.~\\ref{Figure:pv} illustrates qualitative results of our point-wise velocity estimation.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\vspace{-2mm} \n\t\\scalebox{1.02}{\n\t\t\\begin{tabular}{@{}c@{}c@{}c@{}c@{}}\n\t\t\t\\includegraphics[height=1.09in]{.\/figures\/fig_v\/Figure_1}     &\n\t\t\t\\includegraphics[height=1.16in]{.\/figures\/fig_v\/01_v_flow} &\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/01_v_aff}   &\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/01_v_bev}  \\vspace{-2mm} \\\\\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/Figure_2}     &\n\t\t\t\\includegraphics[height=1.12in]{.\/figures\/fig_v\/02_v_flow} &\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/02_v_aff}   &\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/02_v_bev}  \\vspace{-3mm} \\\\\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/Figure_3}     &\n\t\t\t\\includegraphics[height=1.15in]{.\/figures\/fig_v\/03_v_flow} &\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/03_v_aff}   &\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/03_v_bev}  \\vspace{-2mm} \\\\\n\t\t\t\\footnotesize{(a)} &  \\footnotesize{ (b)} & \\footnotesize{ (c)} & \\footnotesize{ (d)} \\vspace{-2mm}\\\\\n\t    \\end{tabular} }\n\t\\caption{\\small Visualization of point-wise velocity estimation: (a) depth of all measured radar returns as well as flow, (b) optical flow in the white box region, (c) association scores around the selected radar projections as well as predicted mapping from raw radar projections to image pixels (yellow arrow) and (d) radial velocity (red), estimated full velocity (black) and GT velocity (green) in bird's-eye view.}\n\t\\label{Figure:pv}\n\\end{figure*}\n\n\n\\begin{table}[t!]\n\t\\captionsetup{font=small}\n\t\\begin{center}\t\n\n\t\t\\scalebox{0.75}{\n\t\t\t\\begin{tabular}{|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\tMean Error (STD)  & Ours & Ours & Baseline \\\\ \n\t\t\t\t(m\/s) & (R2P Network) & (Raw Projection) & \\\\\n\t\t\t\t\\hline\n\t\t\t\tFull Velocity  \t      \t& $\\mathbf{0.433}\\; (\\mathbf{0.608})$ & $0.577\\; (1.010)$ & $1.599\\; (2.054)$ \\\\\n\t\t\t\t\\hline\n\t\t\t\tTangential Comp.    & $\\mathbf{0.322}\\; (\\mathbf{0.610})$ & $0.472\\; (1.024)$ & $1.536\\; (2.083)$ \\\\\n\t\t\t\t\\hline\n\t\t\t\tRadial Comp.    \t& $0.205\\; (0.196)$ & $0.205\\; (0.196)$ & $0.205\\; (0.196)$ \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{center}\n\t\\vspace{-5mm}\n\t\\caption{\\small Comparison of point-wise velocity error of our methods and the baseline (raw radial velocity).}\n\t\\label{tab:baseline}\n\t\\vspace{-2mm}\n\\end{table}\n\n\n\\subsection{Comparison of Object-wise Velocity}\nAlthough there are no existing methods for point-wise velocity estimation for radar, a related work, CenterFusion~\\cite{nabati2021centerfusion}, estimates {\\it object-wise} full velocity  via object detection with image and radar inputs. \nTo fairly compare with CenterFusion, we convert our point-wise velocity to object-wise velocity. \nSpecifically, we use the average velocity of radar points associated with the same detected box as our estimate of object velocity. \nPoints are associated with detected boxes according to distance. \nNote the point-wise velocity to object-wise velocity conversion is straightforward for comparison purposes, and there would be more advanced approaches to integrate point-wise full velocities in a detection network, which is beyond the scope of this work. \nTab.~\\ref{tab:det} shows that with our estimated full velocity, the velocity estimation for objects is significantly improved.\n\n\\begin{table}[t!]\n\t\\captionsetup{font=small}\n\t\\begin{center}\t\n\t\t\\scalebox{0.9}{\n\t\t\t\\begin{tabular}{|c|c|}\n\t\t\t\t\\hline\n\t\t\t\tMethods & Error (m\/s) \\\\ \n\t\t\t\t\\hline\n\t\t\t\tOurs \t        & $\\mathbf{0.451}$ \\\\\n\t\t\t\tCenterFusion~\\cite{nabati2021centerfusion}    & $0.826$  \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{center}\n\t\\vspace{-5mm}\n\t\\caption{\\small Comparison of object-wise velocity errors. For a fair comparison we inherit the same set of detected objects from~\\cite{nabati2021centerfusion}.}\n\t\\label{tab:det}\t\\vspace{-3mm}\n\\end{table}\n\n\n\\subsection{Radar Point Accumulation}\nAccumulating radar points over time can overcome the sparsity of radar hits acquired in a single sweep, achieving dense point cloud for objects and thus allowing techniques designed for processing LiDAR points to be applicable for radar. \nThe point-wise velocity estimate makes it possible to compensate the motion of dynamic objects appearing in a temporal sequence of measurements for accumulation. \nSpecifically, for a moving radar point (with estimated velocity $\\dot{\\bm{m}}$) in a previous frame $i$ captured at time $t_i$, its motion from $t_i$ to the time at the current frame, $t_0$, can be compensated by,\n\\begin{equation}\n\\bm{p_0}= \\bm{p_i} + \\dot{\\bm{m}}(t_0 - t_i),\n\\label{eq:motion_compensate}\n\\end{equation}\nwhere $\\bm{p_i}$ and $\\bm{p_0}$ are the radar point coordinates at $t_i$ and $t_0$ in radar coordinates of $t_i$. \nThen $\\bm{p_0}$ is transformed to current radar coordinates by known egomotion from $t_i$ to $t_0$.\n\n\\begin{figure}[t!]\n    \\captionsetup{font=small}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.8\\linewidth]{figures\/error_curve}\n\t\\end{center}\n\t\\vspace{-5mm}\n\t\\caption{\\small Error comparison when accumulating radar points from increasing number of frames. The lines represent mean error and shaded area $\\pm0.1\\times$ STD. Our  full velocity based accumulation outperforms the ones with radial velocity, or no compensation.}\n\t\\label{fig:curve}\\vspace{-3mm}\n\\end{figure}\n\n\\Paragraph{Qualitative results}\nFig.~\\ref{Figure:acc} shows accumulated points of moving vehicles in radar coordinates. \nFor comparison, we show accumulated radar points compensated by our estimated full velocity, compensated with radial velocity (baseline) and without motion compensation. Compared with the baseline and no motion compensation, our accumulated points are more consistent with the GT bounding boxes.\n\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\scalebox{1}{\n\t\t\\begin{tabular}{@{}c@{}c@{}c@{}c@{}c@{}}\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/02_acc_im} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/02_acc_one}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/02_acc_no} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/02_acc_old}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/02_acc_our}\\vspace{-2mm}\\\\\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/03_acc_im} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/03_acc_one}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/03_acc_no} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/03_acc_old}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/03_acc_our}\\vspace{-2mm}\\\\\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/04_acc_im} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/04_acc_one}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/04_acc_no} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/04_acc_old}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/04_acc_our}\\vspace{-2mm}\\\\\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/05_acc_im} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/05_acc_one}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/05_acc_no} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/05_acc_old}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/05_acc_our}\\vspace{-2mm}\\\\\n\t\t\t\\footnotesize{(a)} &  \\footnotesize{ (b)} & \\footnotesize{ (c)} & \\footnotesize{ (d)} & \\footnotesize{ (e)}\\\\\n\t\\end{tabular} }\n\t\\vspace{-2mm}\n\t\\caption{\\small Moving radar points are plotted with point-wise radial (red) and full (black) velocity, including image with bounding box (a), single-frame radar points in bird's-eye view (b), accumulated radar points from $20$ frames without motion compensation (c), with radial velocity based compensation (d), and with our full-velocity based compensation (e).\n\t Our accumulated points are tightly surrounding the bounding box, which will benefit downstream tasks such as pose estimation and object detection.}\n\t\\label{Figure:acc}\t\\vspace{-1mm}\n\\end{figure*}\n\n\n\n\\Paragraph{Quantitative results}\nTo quantitatively evaluate the accuracy of radar point accumulation, we use the mean distance from accumulated points (of up to $25$ frames) to their corresponding GT boxes as the accumulation error. \nThis distance for points inside the box is zero, and outside it is the distance from the radar point to the closest point on the box's boundary. \nIn Fig.~\\ref{fig:curve}, we compare the accumulation for our method, the baseline and accumulation without motion compensation. \nWhile error increases with the number of frames for all methods, our method has the lowest rate of error escalation. \n\n\n\\begin{table}[t!]\n    \\captionsetup{font=small}\n\t\\begin{center}\t\n\t\t\\scalebox{0.85}{\n\t\t\t\\begin{tabular}{|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t  \t\t                    Metric\t& Ours &  Baseline \\\\ \n\t\t\t\t\\hline\t\t\t\t\n\t\t\t\t\tCenter Error (m) $\\downarrow$            & $\\mathbf{0.834}$ & $0.997$  \\\\\t\t\t\n\t\t\t\t\tOrientation Error (degree) $\\downarrow$  & $\\mathbf{6.873}$ & $7.517$ \\\\\n\t\t\t\t\tIoU $\\uparrow$ \t\t\t\t\t\t& $\\mathbf{0.546}$ & $0.462$ \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{center}\n\t\\vspace{-4mm}\n\t\\caption{\\small Comparison of pose estimation performance: average error in center and orientation as well as Intersection over Union (IoU), by using BoxNet~\\cite{nezhadarya2019boxnet} on radar points accumulated using our velocity and the radial velocity as a baseline.}\n\t\\label{tab:iou}\t\\vspace{-3mm}\n\\end{table}\n\n\\Paragraph{Application of pose estimation}\nTo demonstrate the utility of accumulated radar points for downstream applications, we apply a pose estimation method, {\\it i.e.}, BoxNet~\\cite{nezhadarya2019boxnet}, on the accumulated 2D radar points via our full velocity and radial velocity (baseline), respectively. BoxNet takes pre-segmented 2D point clouds of an object as input and predicts a 2D bounding box with parameters as center position, length, width and orientation. \nWe use accumulated radar points of $5702$, $559$ and $2001$ moving vehicles with corresponding GT bounding boxes as training, validation and testing data, respectively. Tab.~\\ref{tab:iou} shows our accumulated radar achieves higher accuracy than the baseline.\n\n\n\\Section{Conclusion}\nA drawback of Doppler radar has been that it provides only the radial component of velocity, which limits its utility in object velocity estimation, motion prediction and radar return accumulation. This paper addresses this drawback by presenting a closed-form solution to the full velocity of radar returns.  It leverages optical flow constraints to upgrade radial velocity into full velocity.  As part of this work, we use GT bounding-box velocities to supervise a  network that predicts association corrections for the raw radar projections.  We experimentally verify the effectiveness of our method and demonstrate its application on motion compensation for integrating radar sweeps over time.\n\nThis method developed here may apply to additional modalities such as full-velocity estimation from Doppler LiDAR and cameras.  \n\n\\vspace{1mm}\n\\noindent\\textbf{Acknowledgement}\nThis work was supported by the Ford-MSU Alliance.\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTopology freezing or fixing are important issues in quantum field theory, in particular in QCD. For example, when simulating chirally symmetric overlap quarks, the corresponding algorithms do not allow transitions between different topological sectors, i.e.\\ topological charge is fixed (cf.\\ e.g.\\ \\cite{Aoki:2008tq,Aoki:2012pma}). Also when using other quark discretizations, e.g.\\ Wilson fermions, topology freezing is expected at lattice spacings $a \\lesssim 0.05 \\, \\textrm{fm}$, which are nowadays still fine, but realistic \\cite{Luscher:2011kk,Schaefer:2012tq}. There are also applications, where one might fix topology on purpose. For example, when using a mixed action setup with light overlap valence and Wilson sea quarks, approximate zero modes in the valence sector are not compensated by the sea. The consequence is an ill-behaved continuum limit \\cite{Cichy:2010ta,Cichy:2012vg}. A possible solution to overcome this problem is to restrict computations to a single topological sector, either by sorting the generated gauge link configurations with respect to their topological charge or by directly employing so-called topology fixing actions (cf.\\ e.g.\\ \\cite{Fukaya:2005cw,Bietenholz:2005rd,Bruckmann:2009cv}).\n\nIn view of these issues it is important to develop methods, which allow to obtain physically meaningful results (i.e.\\ results corresponding to unfixed topology) from fixed topology simulations. The starting point for our work are calculations from the seminal papers \\cite{Brower:2003yx,Aoki:2007ka}. We extend these calculations by including all terms proportional to $1\/V^2$ and $1\/V^3$. We apply the resulting equations to a quantum mechanical particle on a circle, to the Schwinger model and to SU(2) Yang-Mills theory and determine ``hadron masses'' at unfixed topology from fixed topology computations and simulations (for related exploratory studies in the Schwinger model and the $O(2)$ and $O(3)$ non-linear Sigma model cf.\\ \\cite{Bietenholz:2011ey,Bietenholz:2012sh,Bautista:2014tba}).\n\nPart of this work has already been published \\cite{Dromard:2013wja,Dromard:2014wja,Czaban:2013haa}.\n\n\n\n\n\\section{\\label{SEC1}Hadron masses from fixed topology simulations}\n\n\n\\subsection{\\label{SEC11}Two-point correlation functions at fixed topology}\n\nThe partition function and the two-point correlation function of a hadron creation operator $O$ at fixed topological charge $Q$ and finite spacetime volume $V$ are given by\n{\\small\n\\begin{equation}\n\\begin{aligned}\n & Z_{Q,V} \\equiv \\int DA \\, D\\psi \\, D\\bar{\\psi}\\, \\delta_{Q,Q[A]} e^{-S_E[A,\\bar{\\psi},\\psi]} \\\\\n & C_{Q,V}(t) \\equiv \\frac{1}{Z_{Q,V}} \\int DA \\, D\\psi \\, D\\bar{\\psi} \\, \\delta_{Q,Q[A]} O^\\dagger(t) O(0) e^{-S_E[A,\\bar{\\psi},\\psi]} .\n\\end{aligned}\n\\end{equation}\n}Using a saddle point approximation the correlation function has been expanded in \\cite{Brower:2003yx} according to\n{\\small\n\\begin{equation}\n\\label{EQN673} C_{Q,V}(t) = \\alpha(0) \\exp\\bigg(-M_H(0) t - \\frac{M^{(2)}_H(0) t}{2\\mathcal{E}_2 V} \\bigg(1- \\frac{ Q^2}{\\mathcal{E}_2 V}\\bigg)\\bigg) + \\mathcal{O}\\bigg(\\frac{1}{ V^2} \\bigg) ,\n\\end{equation}\n}where $\\alpha(0)$ is a constant, $M_H(\\theta)$ the hadron mass at vacuum angle $\\theta$, $\\mathcal{E}_k \\equiv e_0^{(k)}(\\theta)|_{\\theta=0}$ ($\\mathcal{E}_2 = \\chi_t$, the topological susceptibility) and $e_0$ is the vacuum energy density. In \\cite{Dromard:2014wja} we have extended this calculation by including all terms proportional to $1\/V^2$ and $1\/V^3$,\n{\\small{\n\\begin{equation}\n\\label{EQN674} \\begin{aligned}\n & C_{Q,V}(t) = \\alpha(0) \\exp\\bigg(-M_H(0) t - \\frac{x_2}{2\\mathcal{E}_2 V} - \\bigg(\\frac{x_4 - 2 (\\mathcal{E}_4\/\\mathcal{E}_2) x_2 - 2 x_2^2-4x_2Q^2}{8(\\mathcal{E}_2 V)^2} \\bigg) \\\\\n & \\hspace{0.6cm} - \\bigg(\\frac{16 (\\mathcal{E}_4\/\\mathcal{E}_2)^2 x_2 + x_6 - 3 (\\mathcal{E}_6\/\\mathcal{E}_2) x_2 - 8 (\\mathcal{E}_4\/\\mathcal{E}_2) x_4 - 12 x_2 x_4 + 18 (\\mathcal{E}_4\/\\mathcal{E}_2) x_2^2 + 8 x_2^3}{48(\\mathcal{E}_2 V)^3} \\\\\n & \\hspace{1.2cm} - \\frac{x_4 - 3 (\\mathcal{E}_4\/\\mathcal{E}_2) x_2 - 2 x_2^2}{4(\\mathcal{E}_2 V)^3} Q^2\\bigg)\\bigg) + \\mathcal{O}\\bigg(\\frac{1}{(\\mathcal{E}_2 V)^4} \\, , \\, \\frac{1}{(\\mathcal{E}_2 V)^4} Q^2 \\, , \\, \\frac{1}{(\\mathcal{E}_2 V)^4} Q^4\\bigg) ,\n\\end{aligned}\n\\end{equation}\n}where $x_n \\equiv M^{(n)}_H (0) t+ \\beta^{(n)}(0)$ (for the definition of $\\beta^{(n)}$ cf.\\ \\cite{Dromard:2014wja}). The expansions (\\ref{EQN673}) and (\\ref{EQN674}) are rather accurate approximations, if the following conditions are fulfilled: \\vspace{0.1cm}\n\\\\\\textbf{(C1)} $\\phantom{xxx} 1 \/ \\mathcal{E}_2 V \\ll 1 \\quad , \\quad |Q| \/ \\mathcal{E}_2 V \\ll 1$. \\vspace{0.1cm}\n\\\\\\textbf{(C2)} $\\phantom{xxx} |x_2| = |M_H^{(2)}(0) t + \\beta^{(2)}(0)| \\lesssim 1$. \\vspace{0.1cm}\n\\\\\\textbf{(C3)} $\\phantom{xxx} m_\\pi(\\theta) L \\gtrsim 3 \\ldots 5 \\gg 1$ $\\ \\ $ ($m_\\pi$: pion mass, $L$: periodic spatial extension). \\vspace{0.1cm}\n\\\\\\textbf{(C4)} $\\phantom{xxx} (M_H^\\ast(\\theta) - M_H(\\theta)) t \\gg 1 \\quad , \\quad M_H(\\theta) (T-2 t) \\gg 1$. \\vspace{0.1cm}\n\nNote that the effective mass at fixed topology, defined in the usual way,\n{\\small\n\\begin{equation}\n\\label{eq:MQ} M^\\textrm{eff}_{Q,V}(t) \\equiv -\\frac{1}{C_{Q,V}(t)}\\frac{d C_{Q,V}(t)}{dt} ,\n\\end{equation}\n}exhibits severe deviations from a constant behavior at large temporal separations $t$ \\cite{Dromard:2014wja}, which is in contrast to ordinary quantum field theory at unfixed topology.\n\n\n\\subsection{\\label{SEC13}Extracting hadron masses}\n\nA straightforward method to determine physical hadron masses (i.e.\\ hadron masses at unfixed topology) from fixed topology simulations is to fit either (\\ref{EQN673}) or (\\ref{EQN674}) to two-point correlation functions computed at fixed topology. Among the results of the fit are then the hadron mass at unfixed topology $M_H(0)$ and the topological susceptibility $\\mathcal{E}_2 = \\chi_t$. A similar method is to first determine hadron masses $M_{Q,V}$ at fixed topological charge $Q$ and spacetime volume $V$ and then use equations based on (\\ref{EQN673}) or (\\ref{EQN674}) to determine $M_H(0)$ and $\\mathcal{E}_2 = \\chi_t$. For a detailed discussion cf.\\ \\cite{Dromard:2014wja}}.\n\n\n\n\n\\section{\\label{SEC2}A quantum mechanical particle on a circle at fixed topology}\n\nFor a first test of the methods mentioned in section~\\ref{SEC13} we decided for a simple toy model, a quantum mechanical particle on a circle in a square well potential. This model shares some important features with QCD, e.g.\\ the existence of topological charge and the symmetry $+\\theta \\leftrightarrow -\\theta$. Moreover, it can be solved numerically up to arbitrary precision. We determine $M_H(0)$ (which is the energy difference between the ground state and the first excitation) and $\\chi_t$ from fixed topology two-point correlation functions as outlined in section~\\ref{SEC13}. We compare the $1\/V$ expansion from \\cite{Brower:2003yx} (eq.\\ (\\ref{EQN673})) and our $1\/V^3$ version (eq.\\ (\\ref{EQN674})). We find rather accurate results for $M_H(0)$ and $\\chi_t$ (cf.\\ Table~\\ref{TAB001}). Note that the relative errors for both $M_H(0)$ and $\\chi_t$ are smaller, when using the $1\/V^3$ version (\\ref{EQN674}). For details cf.\\ \\cite{Dromard:2013wja,Dromard:2014wja}.\n\n\\begin{table}[htb]\n\\begin{center}\n\\small{\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline \n & expansion  & $\\hat{M}_H(0)$ & error& $\\hat{\\chi}_t$ & error \\tabularnewline\n\\hline \n\\hline \n\\multirow{2}{*}{$\\frac{|Q|}{\\chi_{t}V} \\leq 0.5$} & (\\ref{EQN674}), hep-lat\/0302005 & $0.40702$ & $0.029\\%$& $0.00629$ & $2.5\\%$\\tabularnewline\n\\cline{2-6} \n & (\\ref{EQN674}) &  $0.40706$ & $0.019\\%$ & $0.00633$ & $1.9\\%$ \\tabularnewline\n\\hline \n\\end{tabular}\n}\n\n\\caption{\\label{TAB001}$M_H(0)$ and $\\chi_t$ from fixed topology two-point correlation functions; ``error'' denotes relative differences to the exact results $\\hat{M}_{H} = 0.40714$ and $\\hat{\\chi}_t = 0.00645$ at unfixed topology.}\n\\end{center}\n\\end{table}\n\n\n\n\n\\section{The Schwinger model at fixed topology}\n\n\nThe Schwinger model, defined by the Lagrangian\n{\\small\n\\begin{equation}\n\\mathcal{L}(\\psi,\\bar{\\psi},A_{\\mu}) \\equiv \\bar{\\psi} (\\gamma_\\mu (\\partial_\\mu + i g A_\\mu) + m) \\psi + \\frac{1}{2} F_{\\mu \\nu} F_{\\mu \\nu} ,\n\\end{equation}\n}also shares certain features with QCD, most prominently confinement. Furthermore, simulations are computationally inexpensive, because there are only $2$ spacetime dimensions.\n\nWe have studied the ``pion'' mass $m_\\pi$ and the static quark-antiquark potential ${\\mathcal V}_{q\\bar{q}}$ for various separations. Results are summarized in Table~\\ref{TAB002}. In the first line (``fixed top.'') results obtained from two-point correlation functions at fixed topology (as outlined in section~\\ref{SEC13}) are listed. In the second line (``unfixed top.'') they are compared to results from standard lattice simulations, where gauge link configurations from all topological sectors are taken into account. One can observe agreement demonstrating that one can obtain correct and accurate physical results from fixed topology simulations. For details cf.\\ \\cite{Czaban:2013haa}.\n\n\\begin{table}[htb]\n\\begin{center}\n{\\small\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n & $m_\\pi a$ & ${\\mathcal V}_{q\\bar{q}}(1a) a$ & ${\\mathcal V}_{q\\bar{q}}(2a) a$ & ${\\mathcal V}_{q\\bar{q}}(3a) a$ & ${\\mathcal V}_{q\\bar{q}}(4a) a$ \\\\ \\hline\\hline\nfixed top.\\ &  0.2747(2) & 0.12551(4) & 0.2247(2) & 0.3005(3) & 0.3581(7)  \\\\ \\hline\nunfixed top.\\ & 0.2743(3) & 0.12551(4) & 0.2247(2) & 0.3008(4) & 0.3577(9) \\\\ \\hline\n\\end{tabular}\n}\n\n\\caption{\\label{TAB002}Comparison of results obtained from computations at fixed and at unfixed topology.}\n\\end{center}\n\\end{table}\n\n\n\n\n\\section{SU(2) Yang-Mills theory at fixed topology}\n\nCurrently we perform fixed topology studies of SU(2) Yang-Mills theory,\n{\\small\n\\begin{equation}\n\\mathcal{L}(A_\\mu) \\equiv \\frac{1}{4} F_{\\mu \\nu}^a F_{\\mu \\nu}^a ,\n\\end{equation}\n}which is expected to be rather similar to QCD. Again we explore the static quark-antiquark potential for various separations.\n\nThe left plot in Fig.~\\ref{FIG342} shows that there is a significant discrepancy between the potential from computations restricted to a single topological sector and corresponding results obtained at unfixed topology. The plot, therefore, underlines the necessity of a method to extract physical results from fixed topology computations.\n\nIn the right plot of Fig.~\\ref{FIG342} we compare the static potential obtained from Wilson loops at fixed topology (as outlined in section~\\ref{SEC13}) and from standard lattice simulations, where gauge link configurations from all topological sectors are taken into account. As for the Schwinger model, one can observe excellent agreement demonstrating again that one can obtain correct and accurate physical results from fixed topology simulations.\n\nDetails regarding our study of Yang-Mills theory at fixed topology will be published in the near future.\n\n\\begin{figure}[htb]\n\\input{FIGSU001.pstex_t}\n\\caption{\\label{FIG342} \\textbf{(left)} ${\\mathcal V}_{q\\bar{q}}(6a)$ for different topological sectors $Q = 0, 1, 2, 3$ for spacetime volume $V\/a^4 = 16^4$. \\textbf{(right)} Comparison of potential results obtained from computations at fixed and at unfixed topology.}\n\n\\end{figure}\n\n\n\n\n\\section{Conclusions and outlook}\n\nWe have extended relations from the literature \\cite{Brower:2003yx,Aoki:2007ka} relating two-point correlation functions at fixed topology to physical hadron masses (i.e.\\ hadron masses at unfixed topology). We have successfully applied our resulting equations to various models. We plan to test the same methods for QCD in the near future, where hadron masses obtained from different topological sectors also exhibit clear differences (for an example cf.\\ \\cite{Galletly:2006hq}, where the pion mass has been computed in various topological charge sectors).\n\n\n\n\n\\section*{Acknowledgments}\n\nWe thank Wolfgang Bietenholz, Krzysztof Cichy, Dennis Dietrich, Gregorio Herdoiza, Karl Jansen and Andreas Wipf for discussions. We acknowledge support by the Emmy Noether Programme of the DFG (German Research Foundation), grant WA 3000\/1-1. This work was supported in part by the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the State of Hesse.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}