00footnotetext:  Mathematics Subject Classification (2020): Primary: 19E20;
Secondary: 19A31, 19B28, 20F36. The second author was supported by grant
PID2020-117971GB-C21 of the Spanish Ministery of Science and Innovation, and
grant FQM-213 of the Junta de Andalucía.

# Bredon homology of Artin groups of dihedral type

Yago Antolín<EMAIL_ADDRESS>Universidad de Sevilla, Departmento de
Geometría y Topología, c/ Tarfia, s/n 41080, Seville (Spain)  and  Ramón
Flores<EMAIL_ADDRESS>
###### Abstract.

For Artin groups of dihedral type, we compute the Bredon homology groups of
the classifying space for the family of virtually cyclic subgroups with
coefficients in the K-theory of a group ring.

##  1\. Introduction

This paper is part of a program aimed to understand equivariant and
$K$-theoretic invariants for Artin groups. These 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 [42] for a good
survey. In this context, Azzali _et al._ have recently computed explicitly
both sides of Baum-Connes for pure braid groups [4]; in joint work of the
second author with J. González-Meneses [21] 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
_et al._ ([1], [2]) and by Petrosyan and Nucinkis [40].

In the present paper we consider the case of Artin groups of dihedral type,
denoted by $A_{n}$, which are the groups defined by

$A_{n}=\left\langle
a,b\mid\mathrm{prod}(a,b;n)=\mathrm{prod}(b,a;n)\right\rangle.$

Here $\mathrm{prod}(x,y;n)$ denotes the word of length $n$ that alternates $x$
and $y$ and starts with $x$. 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 [37], systems of equations [13] or geodesics [50].

We 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 [8], 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.

Our main technical result is the following:

Theorem 4.17. 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:

  1. (1)

$H_{i}^{vc}({\underline{\underline{E}}}A_{n},K_{q}(R[-]))=\\{0\\}$ for $i\geq
4$.

  2. (2)

$H_{3}^{vc}({\underline{\underline{E}}}A_{n},K_{q}(R[-]))=\begin{cases}\bigoplus_{[H]\neq[Z(A_{n})]}K_{q}(R)&\text{$n$
odd}\\\ \ker g_{2}^{2}&\text{$n$ even.}\end{cases}$

  3. (3)

$H_{2}^{vc}({\underline{\underline{E}}}A_{n},K_{q}(R[-]))=\ker g_{2}^{1}$.

  4. (4)

$H_{1}^{vc}({\underline{\underline{E}}}A_{n},K_{q}(R[-]))=\emph{coker
}g_{2}^{1}=\begin{cases}(\bigoplus_{[H]\neq[Z(A_{n})]}N_{q}^{[H]})\oplus
T_{1}(K_{q}(R))\oplus T_{2}(K_{q}(R))&\text{$n$ odd}\\\
\bigoplus_{[H]}N_{q}^{[H]}&\text{$n$ even.}\end{cases}$

  5. (5)

$H_{0}^{vc}({\underline{\underline{E}}}A_{n},K_{q}(R[-]))=\emph{ coker
}g_{2}^{0}=\begin{cases}(\bigoplus_{[H]\neq[Z(A_{n})]}N_{q}^{[H]})\oplus
K_{q}(R)\oplus\overline{C}(K_{q}(R))&\text{$n$ odd}\\\
(\bigoplus_{[H]}N_{q}^{[H]})\oplus K_{q}(R)&\text{$n$ even.}\end{cases}$

Here $g^{j}_{i}$ stands for a homomorphism in the Degrijse-Petrosyan exact
sequence [17, Section 7], $[H]$ 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 $K_{q}(R[\mathbb{Z}])$. See Section 4 for details.

The main tool used in the proof of Theorem 4.17 is an exact sequence in Bredon
homology [17], which is in turn the Mayer-Vietoris sequence associated to the
push-out that defines the Lück-Weiermann model for
${\underline{\underline{E}}}G$ ([34], see also Section 2 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 5).

Next 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 [20] the existence of an assembly
map:

$H_{n}^{G}({\underline{\underline{E}}}G,\mathbf{K}(R))\rightarrow K_{n}(RG),$

where ${\underline{\underline{E}}}G$ is the classifying space of $G$ with
respect to the family of virtually cyclic groups, $H_{*}^{G}(-,\mathbf{K}(R))$
is the $G$-homology theory defined in Section 1 of [20] and $K_{n}(RG)$ stands
for the $n$-th group of algebraic $K$-theory of the group ring $RG$. The
Farrell-Jones conjecture predicts that the assembly map is an isomorphism. The
conjecture has been verified for a big family of groups (see [31] for an
excellent survey), and no counterexample has been found so far. The 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 [9], [15], [27] or [45],
for example.

The 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 [5] [32]. In this setting, Theorem 4.17 and the examples of
Section 5 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 [30, Lemma 16.12], using previously a splitting result of
Lück-Steimle [33] (see the end of Section 5 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.

We 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 [7]. Hence, all the computations in the left-hand side can be
read in terms of algebraic $K$-theory of $RG$.

Summary of contents. In Section 2 we recall the main definitions about
classifying spaces for families and Bredon homology, with special emphasis in
the Lück-Weiermann model and its associated Mayer-Vietoris sequence. In
Section 3 the main properties of Artin groups that will be used on the rest of
the paper are studied. Then, in Section 4, we carefully analyze the
homomorphisms in the exact sequence and prove Theorem 4.17; and in final
Section 5 we apply our results to describe different concrete examples.

##  2\. Preliminaries

In this section we state some notions of $G$-equivariant homotopy that will
frequently appear in the rest of the paper. The exposition will be sketchy and
very focused to our goals; the reader interested in a thorough treatment of
the subject is referred to [18] for the theory of $G$-$CW$-complexes and
actions on them, to [29] for the theory of classifying spaces and to the first
part of [39] for Bredon homology.

###  2.1. Classifying spaces for families

In 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 Lück-Weiermann model and the
definition of commensurator, which will be crucial in our computations.

######  Definition 2.1.

Let $G$ 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
_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.

The classifying space for the family $\mathcal{F}$ is usually denoted by
$E_{\mathcal{F}}G$. Moreover, 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.

If 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 $F\in{\mathcal{F}}$. The family ${\mathcal{F}}\cap H$ of subgroups of
$H$ is again closed under $H$-conjugation and taking subgroups. In these
conditions the action of $H$ over $E_{{\mathcal{F}}}G$ by restriction turns
$E_{{\mathcal{F}}}G$ into a model for $E_{{\mathcal{F}}\cap H}H$.

The 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. It is also a standard argument to show that torsion-free
virtually cyclic groups are cyclic (see for example Lemma 3.2 in [36]). Then,
for torsion-free groups, $\mathcal{F}_{vc}$ is the set of cyclic subgroups.

From now on we will describe the model of ${\underline{\underline{E}}}G$
developed by Lück-Weiermann in [34], 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 $K$. Observe 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. Also remark that if
$H\in\mathcal{F}_{Fin}$, then $H\sim K$ if and only if
$K\in\mathcal{F}_{Fin}$. The equivalence class of $H$ will be denoted by
$[H]$. This 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 $g\in
G$.

Now we can define the notion of _commensurator_ , central in this model and in
our paper:

######  Definition 2.2.

Given an equivalence class $[H]$ of the relation $\sim$, the _commensurator_
of $[H]$ in $G$ is defined as the subgroup

$\textrm{Comm}_{G}[H]=\\{g\in G\>|\ g^{-1}[H]g=[H]\\}.$

In [34], it is also defined a family of subgroups of $\textrm{Comm}_{G}[H]$
as:

${\mathcal{F}}[H]:=\\{K<{\textrm{Comm}_{G}[H]}\>|\ K\sim H\text{ or
}|K|<\infty\\}.$

If $H$ 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.

Now we have all the ingredients needed for building Lück-Weiermann model:

######  Theorem 2.3.

([34], 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 $\underline{E}\operatorname{Comm}_{G}[H]$ and
$E_{{\mathcal{F}}[H]}\operatorname{Comm}_{G}[H]$. We also choose a model for
$\underline{E}G$. Consider the $G$-pushout:

$\textstyle{\coprod_{[H]\in
I}G\times_{\operatorname{Comm}_{G}[H]}\underline{E}\operatorname{Comm}_{G}[H]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
56.9055pti}$$\scriptstyle{\coprod_{[H]\in
I}id_{G}\times_{\operatorname{Comm}_{G}[H]}f_{[H]}}$$\textstyle{\underline{E}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\coprod_{[H]\in
I}G\times_{\operatorname{Comm}_{G}[H]}E_{{\mathcal{F}}[H]}\operatorname{Comm}_{G}[H]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$

where $f_{[H]}$ is a cellular $\operatorname{Comm}_{G}[H]$-map for every
$[H]\in I$ and $i$ is the inclusion. In these conditions, $X$ is a model for
${\underline{\underline{E}}}G$.

In 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.

###  2.2. Bredon homology

In this subsection we will briefly review the main definitions concerning
Bredon homology. We follow the topological concise approach from [44], which
we use in our computations.

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
_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. Then
a _left Bredon module_ $N$ over $O_{\mathcal{F}}(G)$ is a covariant functor

$N:O_{\mathcal{F}}(G)\rightarrow\textbf{Ab},$

where Ab is the category of abelian groups.

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
_Bredon chain complex_ $(C_{n}^{\mathcal{F}}(X,N),\Phi_{n})$ can be defined in
the following way. For 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
$\operatorname{Stab}(e_{i}^{d})$ the stabilizer of $e_{i}^{d}$. Then we define
the _n-th group of Bredon chains_ as $C_{n}^{\mathcal{F}}(X,N)=\bigoplus_{i\in
I}N(G/\operatorname{Stab}(e_{i}^{d}))$.

Consider now a $(d-1)$-face of $e_{i}^{d}$, which can be given as $ge$ for a
certain $(d-1)$-cell $e$. Then we have an inclusion of stabilizers
$g^{-1}\operatorname{Stab}(e_{i}^{d})g\subseteq\operatorname{Stab}(e)$. As
$g^{-1}\operatorname{Stab}(e_{i}^{d})g$ and $stab(e_{i}^{d})$ are isomorphic,
the previous inclusion induces an equivariant $G$-map $f\colon
G/\operatorname{Stab}(e_{i}^{d})\rightarrow G/\operatorname{Stab}(e)$. In
turn, as $N$ is a functor, we have an induced homomorphism $N(f)\colon
N(G/\operatorname{Stab}(e_{i}^{d}))\rightarrow N(G/\operatorname{Stab}(e))$.
Taking 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 $d>0$. So we have the following
definition:

######  Definition 2.4.

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 _Bredon homology
groups_ of $X$ with coefficients in $N$ with respect to the family
$\mathcal{F}$.

We define $H_{i}^{\mathcal{F}}(G,N)$, the _B_ redon homology groups of $G$
with coefficients in $N$ with respect to the family $\mathcal{F}$ as
$H_{i}^{{\mathcal{F}}}(E_{\mathcal{F}}G,M)$.

These groups are preserved under $G$-equivariant homotopy equivalence.

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. It 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 [39, page 15]. To 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.

The cyclic group of order $n$ will be denoted by $C_{n}$. When we want to
consider its ring structure we might use $\mathbb{Z}/{\bf n}$. When $n$ is
prime, we might use $\mathbb{F}_{n}$ to emphasize its field structure.

##  3\. Artin groups of dihedral type

In 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:

######  Definition 3.1.

Let $n\geq 1$. By $\mathrm{prod}(x,y;n)$ we denote the word of length $n$ that
alternates $x$ and $y$ and starts with $x$. For example,
$\mathrm{prod}(x,y;3)=xyx$ and $\mathrm{prod}(x,y;4)=xyxy$. With this
notation, a dihedral Artin group of type $n$ is the group

$A_{n}=\left\langle
a,b\mid\mathrm{prod}(a,b;n)=\mathrm{prod}(b,a;n)\right\rangle.$

The name “dihedral” comes from the associated Coxeter group,

$\left\langle a,b\mid
a^{2}=b^{2}=1,\mathrm{prod}(a,b;n)=\mathrm{prod}(b,a;n)\right\rangle$

which is the dihedral group of order $2n$. Dihedral Artin groups are torsion-
free, even more $A_{n}\cong F_{n-1}\rtimes\mathbb{Z}$, where $F_{k}$ is a free
group of rank $k$. To see this, one can check that the kernel of
$A_{n}\to\mathbb{Z}$, $a,b\mapsto 1$ is free on rank $n-1$. In particular,
Dihedral Artin groups satisfy the Farrell-Jones conjecture [7].

We 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.

As 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

$\operatorname{Comm}_{G}[H]=\operatorname{Comm}_{G}(\left\langle
h\right\rangle)=\\{g\in G\mid gh^{m}g^{-1}=h^{n}\text{ for some
}n,m\in\mathbb{Z}-\\{0\\}\\}.$

We will be interested in computing commensurators up to isomorphism, and we
will use the fact that commensurators of conjugated subgroups are conjugated.
Let us denote by $\pi$ the natural projection map $\pi\colon
A_{n}\to\overline{{A_{n}}}\mathrel{\mathop{\mathchar
0\relax}"303A}\mkern-1.2mu=A_{n}/Z(A_{n})$. Then, for every $g\in A_{n}$,

$\pi(\operatorname{Comm}_{A_{n}}[\left\langle
g\right\rangle])\leqslant\operatorname{Comm}_{\overline{{A_{n}}}}[\left\langle\pi(g)\right\rangle],$

and

$Z(A_{n})\leqslant\operatorname{Comm}_{A_{n}}[\left\langle g\right\rangle].$

We will prove the following.

######  Lemma 3.2.

Let $A_{n}$ be a dihedral Artin group and $g\in A_{n}$ of infinite order. Then

  1. (1)

If $\langle g\rangle\cap Z(A_{n})\neq\\{1\\}$ then
$\operatorname{Comm}_{A_{n}}[\langle g\rangle]=A_{n}$ and $Z(A_{n})\in[H]$,

  2. (2)

If $\langle g\rangle\cap Z(A_{n})=\\{1\\}$ then
$\operatorname{Comm}_{A_{n}}[\langle g\rangle]\cong\mathbb{Z}^{2}$ and there
is $\langle g^{\prime}\rangle\in[\langle g\rangle]$ that is a direct factor of
$\operatorname{Comm}_{A_{n}}[\langle g\rangle]$.

###### Proof.

It 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.

Case $n$ even:  In the group

$A_{2n}=\left\langle
a,b\mid\mathrm{prod}(a,b;2n)=\mathrm{prod}(b,a;2n)\right\rangle,$

it is known that $Z(A_{2n})=\left\langle(ab)^{n}\right\rangle$. Moreover,
$A_{2n}\cong\left\langle x,y\mid x^{-1}y^{n}x=y^{n}\right\rangle$ via the
isomorphism defined by $x\to b$, $y\mapsto ba$. Therefore
$A_{2n}/Z(A_{2n})=\left\langle x,y\mid
x^{-1}y^{n}x=y^{n},y^{n}=1\right\rangle=\left\langle\overline{{x}},\overline{{y}}\mid\overline{{y}}^{n}=1\right\rangle\cong
C_{\infty}*C_{n}$.

Since $\overline{{A_{2n}}}$ is hyperbolic (even more, virtually free), the
commensurator of an infinite order element is virtually cyclic. (See for
example [3, Theorem 2] bearing in mind that infinite cyclic subgroups of
hyperbolic groups are quasi-convex).

Suppose first that $\pi(g)$ has infinite order. Then
$\operatorname{Comm}_{\overline{A_{2n}}}[\langle\pi(g)\rangle]$ is infinite
and virtually cyclic and hence $\pi(\operatorname{Comm}_{A_{2n}}[\left\langle
g\right\rangle])$ is infinite and virtually cyclic. Since all infinite
virtually cyclic subgroups of $\overline{{A_{2n}}}$ are infinite cyclic we
have that $\pi(\operatorname{Comm}_{A_{2n}}[\left\langle g\right\rangle])$ is
infinite cyclic. Therefore, $\operatorname{Comm}_{A_{2n}}[\left\langle
g\right\rangle]$ is a central extension of $\mathbb{Z}$ by $\mathbb{Z}$ and
hence $\operatorname{Comm}_{A_{n}}[\left\langle
g\right\rangle]\cong\mathbb{Z}^{2}$. We can take $g^{\prime}$ as a pre-image
of a generator of $\pi(\operatorname{Comm}_{A_{2n}}[\left\langle
g\right\rangle])$.

Suppose now that $\pi(g)$ has finite order. Thus $\langle g\rangle\cap
Z(A_{2n})$ is non-trivial and infinite, and thus $[\langle
g\rangle]=[Z(A_{2n})]$ and $\operatorname{Comm}_{A_{2n}}[\left\langle
g\right\rangle]=\operatorname{Comm}_{A_{2n}}Z(A_{n})=A_{2n}.$

Case $n$ odd: In the group

$A_{2n+1}=\left\langle
a,b\mid\mathrm{prod}(a,b;2n+1)=\mathrm{prod}(b,a;2n+1)\right\rangle$

we have that $Z(A_{2n+1})=\left\langle(ab)^{2n+1}\right\rangle$. Moreover,
$A_{2n+1}\cong\left\langle x,y\mid xy^{n}=y^{n+1}x^{-1}\right\rangle$ via the
isomorphism defined by $x\to b$, $y\mapsto ab$. Therefore

$\displaystyle A_{2n+1}/Z(A_{2n+1})$ $\displaystyle=\left\langle x,y\mid
xy^{n}x=y^{n+1},y^{2n+1}=1\right\rangle=\left\langle
x,y\mid(xy^{n})^{2}=1=y^{2n+1}\right\rangle$
$\displaystyle=\left\langle\overline{{z}},\overline{{y}}\mid\overline{{z}}^{2}=1=\overline{{y}}^{2n+1}\right\rangle\cong
C_{2}*C_{2n+1},$

where $\overline{{z}}$ denotes the class of $xy^{n}Z(A_{2n+1})$.

Since $\overline{{A_{2n+1}}}$ is hyperbolic and infinite virtually cyclic
subgroups of $\overline{{A_{2n+1}}}$ are infinite cyclic we get, arguing as
above, that if $\pi(g)$ has infinite order then
$\operatorname{Comm}_{A_{2n+1}}[\left\langle
g\right\rangle]\cong\mathbb{Z}^{2}$.

Suppose now that $\pi(g)$ has finite order. Arguing as above,
$\operatorname{Comm}_{A_{2n+1}}[\left\langle
g\right\rangle]=\operatorname{Comm}_{A_{2n+1}}[Z(A_{2n+1})]=A_{2n+1}$. ∎

We recall the ordinary homology of these groups, which will be important in
the remaining sections of the paper.

######  Proposition 3.3.

For $n\geq 2$.

If $n$ 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 $i>2$.

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 $i>1$.

###### Proof.

As 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 [35] or [12] for more general Artin groups). 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 $n$. The
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 [14]. ∎

##  4\. Bredon homology of Artin groups of dihedral type

In 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.

Let $G$ 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 [31].

Although the path to the computation is very clear, in general it is difficult
to obtain explicit formulae for the Bredon homology 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
[52].

In 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 4.17); 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 Lück-Weiermann model (Theorem 2.3), that was
explicitly stated by Degrijse-Petrosyan in Section 7 of [17], 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.

######  Proposition 4.1.

([17, Proposition 7.1], see also [34, Theorem 2.3]) Let $M$ be a left Bredon
module over $O_{{\mathcal{F}}_{vc}}(A_{n})$. There is an exact sequence:

$\ldots\stackrel{{\scriptstyle}}{{\rightarrow}}H^{vc}_{i+1}(A_{n},M)\stackrel{{\scriptstyle
g_{1}^{i+1}}}{{\rightarrow}}\bigoplus_{[H]\in
I}H_{i}^{Fin\cap\operatorname{Comm}_{A_{n}}[H]}(\operatorname{Comm}_{A_{n}}[H],M)\stackrel{{\scriptstyle
g_{2}^{i}}}{{\rightarrow}}$

$\stackrel{{\scriptstyle g_{2}^{i}}}{{\rightarrow}}(\bigoplus_{[H]\in
I}H_{i}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],M))\oplus
H_{i}^{Fin}(A_{n},M)\stackrel{{\scriptstyle
g_{3}^{i}}}{{\rightarrow}}H^{vc}_{i}(A_{n},M)\rightarrow\ldots$

Before 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.

######  Proposition 4.2.

The geometric dimension of $A_{n}$ with respect of the family of virtually
cyclic groups is 3.

###### Proof.

As seen in Lemma 3.2, $A_{n}=\langle
a,b\,|\,\textrm{prod}(a,b;n)=\textrm{prod}(b,a;n)\rangle$, contains subgroups
isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$ (for example
$\operatorname{Comm}_{A_{n}}(\langle a\rangle))$. According to Example 5.21 in
[34], this implies that ${\underline{\underline{\textrm{gd}}}\ }A_{n}\geq 3.$

On the other hand, as the Artin groups $A_{n}$ are one-relator, Corollary 3 in
[16] implies that ${\underline{\underline{\textrm{gd}}}\ }A_{n}\leq 3$, and
then ${\underline{\underline{\textrm{gd}}}\ }A_{n}=3$. ∎

As stated, the strategy to compute $H_{i}^{vc}(A_{n},M)$ goes through
describing the different elements of the exact sequence of Proposition 4.1. We
first understand the terms $H_{i}^{\mathcal{F}[H]}$ on Subsection 4.1. Then,
in Subsection 4.2, we give a more concrete description of the different
homology groups when we take coefficients in the $K$-theory. Finally, in
Subsection 4.3 we study the homomorphisms $g_{1}$ and $g_{2}$ of the exact
sequence of Proposition 4.1 and prove our main theorems.

###  4.1. Computing the homology of the commensurators

Let us start the main calculations of this section. It is clear from the
previous Mayer-Vietoris sequence that the computations of the ordinary
homology of the commensurators (which includes the homology of $A_{n}$, as the
center of $A_{n}$ is virtually cyclic) 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, so we will perform these calculations in the sequel. We start with
the ordinary homology, which is straightforward and depends on the shape of
the commensurators:

  * •

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 $i>2$.

  * •

If $\textrm{Comm}_{A_{n}}[H]\simeq A_{n}$, see Proposition 3.3.

We concentrate now in the case of $\mathcal{F}[H]$, which we recall is the
family of subgroups of $\operatorname{Comm}_{A_{n}}[H]$ 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 $A_{n}$. We 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.

######  Convention 4.3.

For $[H]$ a class of infinite cyclic subgroups, we can always take a
representative $H$ that is normal in $\operatorname{Comm}_{A_{n}}[H]$.

If $H\cap Z(A_{n})$ is trivial, then Lemma 3.2 states that
$H\unlhd\operatorname{Comm}_{A_{n}}[H]$. Moreover, by Lemma 3.2 we assume that
$H$ is a direct factor of $\operatorname{Comm}_{A_{n}}[H]$.

If $H\cap Z(A_{n})$ 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 $H\unlhd\operatorname{Comm}_{A_{n}}[H]$ again.

With this convention, the projection
$\pi\colon\textrm{Comm}_{A_{n}}[H]\rightarrow\textrm{Comm}_{A_{n}}[H]/H$ is
well-defined and $\operatorname{Comm}_{A_{n}}[H]/H$ is isomorphic to
$C_{\infty}$, $C_{\infty}*C_{n}$ or $C_{2}*C_{2n+1}$.

Let 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 $\operatorname{Comm}_{A_{n}}[H]/H$ with respect to the
family of finite groups. That is, for $K\leqslant\textrm{Comm}_{A_{n}}[H]/H$
finite it is defined

$\pi^{-1}M((\operatorname{Comm}_{An}[H]/H)/K)\mathrel{\mathop{\mathchar
0\relax}"303A}\mkern-1.2mu=M(\operatorname{Comm}_{A_{n}}[H]/\pi^{-1}(K)).$

Observe that this assignation gives rise to a natural transformation of
functors $\pi^{-1}M\rightarrow M$. In the next result, which is essentially
[17, Lemma 4.2], 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.

######  Proposition 4.4.

Let $H$ be an infinite cyclic subgroup of $A_{n}$ normal in its commensurator.
For every $n\geq 0$, and every module $M$ over the orbit category of
$\operatorname{Comm}_{A_{n}}[H]$ there is an isomorphism

$H_{i}^{Fin}(\operatorname{Comm}_{A_{n}}[H]/H,\pi^{-1}M)\simeq
H_{i}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],M).$

Moreover, every model for
$\operatorname{\underline{E}}(\operatorname{Comm}_{A_{n}}[H]/H)$ is a model
for $E_{\mathcal{F}[H]}\operatorname{Comm}_{A_{n}}H$, with the action induced
by the quotient map
$\operatorname{Comm}_{A_{n}}[H]\to\operatorname{Comm}_{A_{n}}[H]/H$.

###### Proof.

The argument here is taken from the proof of [17, Lemma 4.2]. We write it here
to make our paper more self-contained.

The projection
$\pi\colon\operatorname{Comm}_{A_{n}}[H]\to\operatorname{Comm}_{A_{n}}[H]/H$
maps the family $\mathcal{F}[H]$ onto the family $Fin$ of finite subgroups of
the quotient $\textrm{Comm}_{A_{n}}[H]/H$. Moreover, the pre-image
$\pi^{-1}(K)$ for any finite group $K$ of $\operatorname{Comm}_{A_{n}}[H]/H$
lies in $\mathcal{F}[H]$. Therefore,
$\operatorname{\underline{E}}\operatorname{Comm}_{A_{n}}[H]/H$ is a model for
$E_{\mathcal{F}[H]}\textrm{Comm}_{A_{n}}[H]$, with the action induced by the
projection $\pi$.

Consider now the spectral sequence associated to the short exact sequence
$1\to
H\to\operatorname{Comm}_{A_{n}}[H]\stackrel{{\scriptstyle\pi}}{{\to}}\operatorname{Comm}_{A_{n}}[H]/H\to
1$ for homology [38]. For every module $M$ over the orbit category of
$\operatorname{Comm}_{A_{n}}[H]$ we have

$E^{p,q}_{2}(M)=H_{p}^{{Fin}}(\operatorname{Comm}_{A_{n}}[H]/H,H_{q}^{\mathcal{F}[H]\cap\pi^{-1}(-)}(\pi^{-1}(-),M)),$

which converges to
$E^{p,q}_{\infty}(M)=H_{p+q}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],M)$.

Observe that $E^{p,q}_{2}$ is trivial for $q\geq 1$, as for every finite
subgroup $K<\operatorname{Comm}_{A_{n}}[H]/H$, $\pi^{-1}(K)$ 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
$E^{i,0}_{2}(M)=H_{i}^{Fin}(\operatorname{Comm}_{A_{n}}[H]/H,\pi^{-1}M)$ in
the 0-th row of the sequence, we have

$H_{i}^{Fin}(\operatorname{Comm}_{A_{n}}[H]/H,\pi^{-1}M)\simeq
H_{i}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],M).$

for every $i\geq 0.$ We are done. ∎

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:

  * •

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 $\mathbb{Z}$; 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.

  * •

If $\textrm{Comm}_{A_{n}}[H]\simeq A_{n}$, then $Z(A_{n})$ is a representative
of the class $[H]$. We 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 $n$. In this case
a tree model for ${\operatorname{\underline{E}}}(\textrm{Comm}_{A_{n}}[H]/H)$
can be explicitly constructed.

If $n$ is even, we have that $A_{n}/Z(A_{n})\cong C_{\infty}*C_{n}$. Denote
$C_{\infty}*C_{n}$ by $\overline{A_{n}}$. Let $s$ be a generator of
$C_{\infty}$. Then 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.

If $n$ is odd, we have that $A_{n}/Z(A_{n})\cong C_{2}*C_{n}$. Denote
$C_{2}*C_{n}$ by $\overline{A_{n}}$. Then 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.

Note 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 $\operatorname{\underline{E}}\overline{A_{n}}$.

As the classifying spaces of the commensurators that we have described are all
1-dimensional, we will use the following result of Mislin:

######  Lemma 4.5 ([39], Lemma 3.14).

Suppose 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:

$0\rightarrow
H_{1}^{\mathcal{F}}(G,N)\rightarrow\bigoplus_{[e]}N(G/S_{e})\rightarrow\bigoplus_{[v]}N(G/S_{v})\rightarrow
H_{0}^{\mathcal{F}}(G,N)\rightarrow 0,$

where $[e]$ and $[v]$ run over the $G$-orbits of edges and vertices of $T$,
respectively.

######  Remark 4.6.

It 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.

In particular we obtain:

######  Corollary 4.7.

For 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
$H_{i}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],M)=0$.

###### Proof.

Recall that by Proposition 4.4
$H_{i}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],M)\simeq
H_{i}^{Fin}(\operatorname{Comm}_{A_{n}}[H]/H,\pi^{-1}M)$. As
$\operatorname{\underline{E}}\operatorname{Comm}_{A_{n}}[H]/H$ is
1-dimensional, the result follows by Lemma 4.5. ∎

So 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 $M(A_{n}/-)=K_{q}(R[-])$, $q\in\mathbb{Z}$.

######  Proposition 4.8.

Let $A_{n}=\langle a,b\,\textrm{ }|\textrm{
}\mathrm{prod}(a,b;n)=\mathrm{prod}(b,a;n)\rangle$ be a dihedral Artin group,
let $H\leqslant A_{n}$ be an infinite cyclic group that is normal in its
commensurator, let
$\pi\colon\operatorname{Comm}_{A_{n}}[H]\to\operatorname{Comm}_{A_{n}}[H]/H$
be the natural projection. Let $M$ be a module over the orbit category of
$\operatorname{Comm}_{A_{n}}[H]$. Then

  1. (i)

If $H\cap Z(A_{n})$ is trivial, then

$H_{0}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],M)=H_{1}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],M)=M(\operatorname{Comm}_{A_{n}}[H]/H).$

  2. (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

$H_{0}^{\mathcal{F}[H]}(A_{n},M)=\mathrm{coker}f\quad\text{ and }\quad
H_{1}^{\mathcal{F}[H]}(A_{n},M)=\ker f.$

  3. (iii)

If $H=Z(A_{n})$ and $n=2k$, we have

$H_{0}^{\mathcal{F}[H]}(A_{n},M)=M(A_{n}/\langle ab\rangle)\quad\text{ and
}\quad H_{1}^{\mathcal{F}[H]}(A_{n},M)=M(A_{n}/Z(A_{n})).$

Note that in cases (ii) and (iii), $A_{n}=\operatorname{Comm}_{A_{n}}[H]$.

###### Proof.

In the case $H\cap Z(A_{n})$ is trivial, the classifying space
$\operatorname{\underline{E}}\operatorname{Comm}_{A_{n}}[H]/H$ is the real
line, there is one orbit of edges and one orbit of vertices, and the action is
free. Then by Remark 4.6, the central map in that exact sequence is trivial
and by Lemma 4.5,

$H_{1}^{Fin}(\operatorname{Comm}_{A_{n}}[H]/H,\pi^{-1}M)=\pi^{-1}M((\operatorname{Comm}_{A_{n}}[H]/H)/\\{1\\})$

and

$H_{0}^{Fin}(\operatorname{Comm}_{A_{n}}[H]/H,\pi^{-1}M)=\pi^{-1}M((\operatorname{Comm}_{A_{n}}[H]/H)/\\{1\\}).$

Using Proposition 4.4, case (i) follows.

Let us discuss the case $H=Z(A_{n})$. Now
$\operatorname{Comm}_{A_{n}}[H]/H=A_{n}/Z(A_{n})$ 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
$\operatorname{Comm}_{A_{n}}[H]/H$ by $\overline{A_{n}}$ for brevity.

We consider first the case $n=2k+1$. Here $\overline{A_{n}}=S*L$, with
$S=\langle b(ab)^{k}Z(A_{n})\rangle\cong C_{2}$ and $L=\langle
abZ(A_{n})\rangle\cong C_{n}$. The Bass-Serre tree has two orbits of vertices,
with stabilizers conjugated to $S$ or $L$, and one free orbit of edges. Then,
by Lemma 4.5

(1)
$H_{1}^{Fin}(\overline{A_{n}},\pi^{-1}M)\hookrightarrow\pi^{-1}M(\overline{A_{n}}/\\{1\\})\stackrel{{\scriptstyle
f}}{{\rightarrow}}$ $\stackrel{{\scriptstyle
f}}{{\rightarrow}}\pi^{-1}M(\overline{A_{n}}/S)\oplus\pi^{-1}M(\overline{A_{n}}/L)\twoheadrightarrow
H_{0}^{Fin}(\overline{A_{n}},\pi^{-1}M).$

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

$H_{1}^{Fin}(\overline{A_{n}},\pi^{-1}M)=\textrm{Ker }f.$

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 4.4.

Consider now the case $n=2k$. In this situation
$\overline{A_{n}}=A_{n}/Z(A_{n})$ is a free product $G=S\ast L$, with
$S=\langle bZ(A_{n})\rangle\simeq C_{\infty}$ and $L=\langle
abZ(A_{n})\rangle\simeq C_{n}$. We denote by $s$ the element $bZ(A_{n})$. The
Bass-Serre tree for this group has one orbit of vertices with stabilizers
conjugate to $L$ and one orbit of edges with trivial stabilizers.

The exact sequence of Proposition 4.5 has the following form:

(2) $H_{1}^{Fin}(\overline{A_{n}},\pi^{-1}M)\hookrightarrow
M(\overline{A_{n}}/\\{1\\})\stackrel{{\scriptstyle
f}}{{\rightarrow}}\pi^{-1}M(\overline{A_{n}}/L)\twoheadrightarrow
H_{0}^{Fin}(\overline{A_{n}},\pi^{-1}M).$

The function $f$ is induced by $g\\{1\\}\in A_{2n}/\\{1\\}\mapsto gsL-gL$ by
the functor $M$. And hence $f$ is trivial.

Hence, we deduce that

$H_{1}^{Fin}(\overline{A_{n}},\pi^{-1}M)=\pi^{-1}M(\overline{A_{n}}/\\{1\\})$

and

$H_{0}^{Fin}(\overline{A_{n}},\pi^{-1}M)=\pi^{-1}M(\overline{A_{n}}/L).$

The result now follows by applying Proposition 4.4. ∎

Observe that taking into account that $\overline{A_{2n}}$ is one-relator, this
computation agrees with the result of Corollary 3.23 in [39], in the case of
$M=R_{\mathbb{C}}$, the complex representation ring.

###  4.2.  Coefficients in $K$-theory groups

From 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 4.1 with
coefficients in this module, except the groups $H^{vc}(A_{n},K_{q}(R[-]))$
which will be studied in the next section.

We begin recalling the following definiton.

######  Definition 4.9.

A regular ring is a 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.

Examples 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 $R$.

Before stating our results, we also need to recall the _Bass-Heller-Swan
decomposition_ in $K$-theory, which permits to decompose the $K$-theory of
$R[\mathbb{Z}]$. For thorough approaches to algebraic $K$-theory the reader is
referred to [6], [54] or [30].

######  Theorem 4.10 ([47], Theorem 9.8).

Given a ring $R$ and $q\in\mathbb{Z}$, there exists an isomorphism

$K_{q}(R[\mathbb{Z}])\simeq K_{q}(R)\oplus K_{q-1}(R)\oplus NK_{q}(R)\oplus
NK_{q}(R),$

which is natural in the ring $R$.

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 [47, Section 9].

In our computations it will be important to understand the endomorphism
$ind_{n}$ of $K_{q}(R[\mathbb{Z}])$ induced by multiplication by $n$ in
$\mathbb{Z}$. The references for the sequel are [23, Section 2], [49] and
[51].

According to Bass-Heller-Swan-decomposition, $ind_{n}$ can be seen as a
homomorphism

$ind_{n}\colon K_{q}(R)\oplus K_{q-1}(R)\oplus NK_{q}(R)\oplus
NK_{q}(R)\rightarrow K_{q}(R)\oplus K_{q-1}(R)\oplus NK_{q}(R)\oplus
NK_{q}(R).$

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 [19], $ind_{n}$
admits a transfer $res_{n}$ such that $res_{n}\circ ind_{n}$ is multiplication
by $n$. This implies the following:

######  Proposition 4.11.

In 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 $NK_{q}(R)\oplus NK_{q}(R).$

It should be remarked that the restriction of $ind_{n}|_{NK_{q}(R)}$
(sometimes call the _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.

The following propositions particularize our previous results when the
coefficient module is $K$-theory.

For the sake of clarity, we maintain the notation from Proposition 4.1,
although we are aware that sometimes can be found redundant.

######  Proposition 4.12.

Let us consider an Artin group $A_{n}$ of dihedral type, $q\in\mathbb{Z}$, $R$
a ring. Then we have:

  * •

$H_{0}^{Fin}(A_{n},K_{q}(R[-]))=K_{q}(R)$.

  * •

$H_{1}^{Fin}(A_{n},K_{q}(R[-]))=K_{q}(R)\oplus K_{q}(R)$ if $n$ is even

  * •

$H_{1}^{Fin}(A_{n},K_{q}(R[-]))=K_{q}(R)$ if $n$ is odd.

  * •

$H_{2}^{Fin}(A_{n},K_{q}(R[-]))=K_{q}(R)$ if $n$ is even

  * •

$H_{2}^{Fin}(A_{n},K_{q}(R[-]))=0$ if $n$ is odd.

  * •

$H_{i}^{Fin}(A_{n},K_{q}(R[-]))=0$ if $i\geq 3$.

###### Proof.

As $A_{n}$ 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 [24, Theorem 3A.3] to the ordinary
homology groups of $A_{n}$ (Proposition 3.3), taking into account that the
latter are torsion-free and that the $K$-theory groups are abelian. ∎

We consider now the Bredon homology of the commensurators.

######  Proposition 4.13.

Let us consider an Artin group $A_{n}$ of dihedral type, $q\in\mathbb{Z}$, $R$
a ring. Let $\operatorname{Comm}_{A_{n}}[H]$ be the commensurator of a
virtually cyclic group in $A_{n}$. Then:

  * •

If $\operatorname{Comm}_{A_{n}}[H]=\mathbb{Z}\oplus\mathbb{Z}$, then

$H_{i}^{Fin\cap\operatorname{Comm}_{A_{n}}[H]}(\operatorname{Comm}_{A_{n}}[H],K_{q}(R[-]))=\begin{cases}K_{q}(R)&\text{
if $i=0,2$}\\\ K_{q}(R)\oplus K_{q}(R)&\text{ if i =1}\\\
0&\text{otherwise}.\end{cases}$

  * •

If $\operatorname{Comm}_{A_{n}}[H]=A_{n}$, then

$H_{i}^{Fin\cap\operatorname{Comm}_{A_{n}}[H]}(\operatorname{Comm}_{A_{n}}[H],K_{q}(R[-]))=H_{i}^{Fin}(A_{n},K_{q}(R))$

and this case was described in the previous proposition.

###### Proof.

Given 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. ∎

We now undertake the remaining case. In item (3) we follow the notation of
Proposition 4.11. Moreover, we use the following notation.

######  Notation 4.14.

For $n$ odd, we denote by $C(K_{q}(R))$ the cokernel of the homomorphism

$\tilde{f}\colon K_{q}(R[\mathbb{Z}])\to K_{q}(R[\mathbb{Z}])\oplus
K_{q}(R[\mathbb{Z}])$

induced by $f\colon R[\mathbb{Z}]\rightarrow R[\mathbb{Z}]\oplus
R[\mathbb{Z}]$ induced in the first component by multiplication by 2 and in
the second by multiplication by $n=2k+1$.

By 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 $K_{q}(R[\mathbb{Z}])$ to the
diagonal map to $K_{q}(R)\oplus K_{q}(R)$, a component of
$K_{q}(R[\mathbb{Z}])\oplus K_{q}(R[\mathbb{Z}])$. Thus, $C(K_{q}(R))$ can be
viewed as a quotient of
$K_{q}(R)\bigoplus(\oplus_{i=1}^{2}K_{q-1}(R))\bigoplus(\oplus_{i=1}^{4}NK_{q}(R))$
that can be identified in many cases, see Section 5.

######  Proposition 4.15.

Let $A_{n}$ be an Artin group of dihedral type, $q\in\mathbb{Z}$, $R$ a ring.
Consider for every non-trivial virtually cyclic $H\leqslant A_{n}$ and the
family $\mathcal{F}[H]$. Then:

  1. (1)

If $i\geq 2$, then

$H_{i}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],K_{q}(R[-]))=0.$

  2. (2)

If $H\cap Z(A_{n})$ is trivial, then

$H_{0}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],K_{q}(R[-]))=H_{1}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],K_{q}(R[-]))=K_{q}(R[\mathbb{Z}]).$

  3. (3)

If $n$ is odd and $\operatorname{Comm}_{A_{n}}[H]=A_{n}$, then

$H_{0}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],K_{q}(R[-]))=C(K_{q}(R))$

and

$H_{1}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],K_{q}(R[-]))=T_{1}(K_{q}(R))\oplus
T_{2}(K_{q}(R)).$

  4. (4)

If $n$ is even and $\operatorname{Comm}_{A_{n}}[H]=A_{n}$, then

$H_{0}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],K_{q}(R[-]))=H_{1}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],K_{q}(R[-]))=K_{q}(R[\mathbb{Z}]).$

###### Proof.

We will check every item separately.

  1. (1)

It follows straightforward from Corollary 4.7.

  2. (2)

The claim follows from item (i) in Proposition 4.8, taking into account that
$H$ is infinite cyclic an then $K_{q}(R[H])=K_{q}(R[\mathbb{Z}])$.

  3. (3)

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 4.8. Taking $M(A_{n}/-)=K_{q}(R[-])$, we obtain a
homomorphism $f_{K}:K_{q}(R[Z(A_{n})])\to K_{q}(R[\langle
b(ab)^{k}\rangle])\oplus K_{q}(R[\langle ab\rangle])$. Observe 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
$\mathbb{Z}\hookrightarrow\mathbb{Z}$ given respectively by multiplication by
2 and multiplication by $n$. Then, $f_{K}$ can be seen as the homomorphism
$K_{q}(R[\mathbb{Z}])\rightarrow K_{q}(R[\mathbb{Z}])\oplus
K_{q}(R[\mathbb{Z}])$ induced by each multiplication in the corresponding
component. According to Proposition 4.11, 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. The result now follows from item
(2) in Proposition 4.8.

  4. (4)

In this case $A_{n}=\operatorname{Comm}_{A_{n}}[H]$. The claim follows from
item (3) in Proposition 4.8, taking into account that $H$ is infinite cyclic
and then $K_{q}(R[H])=K_{q}(R[\mathbb{Z}])$.

∎

###  4.3. Understanding the homomorphisms of Proposition 4.1

In 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 4.1.

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. Note that the superscript of
the homomorphism $g^{k}_{i}$ specifies the degree of the homology in the
source of $g^{k}_{i}$. Moreover, when we need to refer to the $j$-th component
of $g_{i}^{k}$, we will write $g_{ij}^{k}$. For example, $g_{22}^{1}$ is the
second component of the homomorphism $g_{2}^{1}$ defined over the first
homology group.

To prove our results, we need to analyze in detail the following homomorphism

(3)
$\bigoplus_{[H]}H_{i}^{Fin\cap\operatorname{Comm}_{A_{n}}[H]}(\operatorname{Comm}_{A_{n}}[H],M)\stackrel{{\scriptstyle
g_{2}^{i}}}{{\rightarrow}}$ $\stackrel{{\scriptstyle
g_{2}^{i}}}{{\rightarrow}}\left(\bigoplus_{[H]}H_{i}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],M)\right)\oplus
H_{i}^{Fin}(A_{n},M).$

The homomorphism $g_{21}^{i}$ is induced by the vertical left arrow
$\sqcup_{[H]\in
I}\mathrm{id}_{A_{n}}\times_{\operatorname{Comm}_{A_{n}}[H]}f_{[H]}$ of Lück-
Weiermann push-out (4) below (see also Theorem 2.3). In turn, the homomorphism
$g_{22}^{i}$ is induced by the inclusion in the upper horizontal arrow of the
push-out (4)

(4) $\textstyle{\coprod_{[H]\in
I}A_{n}\times_{\operatorname{Comm}_{A_{n}}[H]}\operatorname{\underline{E}}\operatorname{Comm}_{A_{n}}[H]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
56.9055pti}$$\scriptstyle{\coprod_{[H]\in
I}id_{A_{n}}\times_{\operatorname{Comm}_{A_{n}}[H]}f_{[H]}}$$\textstyle{\operatorname{\underline{E}}A_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\coprod_{[H]\in
I}A_{n}\times_{\operatorname{Comm}_{A_{n}}[H]}E_{{\mathcal{F}}[H]}\operatorname{Comm}_{A_{n}}[H]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X.}$

We will decompose the homomorphism $g_{2}^{i}$ of Equation (3) 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

(5) $g_{2[H]}^{i}\colon
H_{i}^{Fin\cap\operatorname{Comm}_{A_{n}}[H]}(\operatorname{Comm}_{A_{n}}[H],M)\to
H_{i}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],M)\oplus
H_{i}^{Fin}(A_{n},M).$

Note 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. Moreover, 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.

As all the commensurators are torsion-free,
$H_{i}^{Fin\cap\operatorname{Comm}_{A_{n}}[H]}(\operatorname{Comm}_{A_{n}}[H],M)$
is ordinary homology with coefficients in
$M(\operatorname{Comm}_{A_{n}}[H]/\\{1\\})$.

We now analyze the homomorphism $g_{2}^{i}$ on the case $M=K_{q}(R[-])$.
Before that, we introduce the following notation.

######  Notation 4.16.

We will denote $N_{q}^{[H]}=K_{q-1}(R)\oplus NK_{q}(R)\oplus NK_{q}(R)$, where
the superindex means that this group is associated to a concrete
commensurability class $[H]$.

We now will give more detailed descriptions of the homomorphism $g_{2}^{i}$
and the cokernels introduced above.

####  4.3.1. The homomorphism $g_{2}^{2}$ when $n$ odd:

In this case we have that the codomain of $g_{2}^{2}$ is $(\bigoplus
H_{2}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],M))\oplus
H_{2}^{Fin}(A_{n},M)=\\{0\\}$, and therefore for all commensurability classes
of infinite cyclic subgroups $[H]$ the homomorphisms $g_{2[H]}^{2}$ are
trivial.

####  4.3.2. The homomorphism $g_{2}^{2}$ when $n$ even:

In this case we have that the codomain of $g_{2}^{2}$ is $(\bigoplus
H_{2}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],M))\oplus
H_{2}^{Fin}(A_{n},M)=(\bigoplus_{[H]}\\{0\\})\oplus K_{q}(R)$. Moreover, by
Proposition 4.12 and Proposition 4.13, the domain of all $g_{2[H]}^{2}$ are
the same, and we have

$g_{2[H]}^{2}\colon K_{q}(R)\to\\{0\\}\oplus K_{q}(R).$

Note that $g_{22[H]}^{2}$ is induced by the inclusion given by the upper arrow
in the push-out. When $H=Z(A_{n})$, we have that
$\operatorname{Comm}_{A_{n}}[H]=A_{n}$ and therefore this inclusion is the
identity. Since $g_{22[Z(A_{n})]}^{2}$ is surjective, we have that $g_{2}^{2}$
is surjective.

####  4.3.3. The homomorphism $g_{2}^{1}$ when $n$ is odd:

Before we describe this case, we need to make some considerations. Recall that
we assume that $H$ is normal in $\textrm{Comm}_{A_{n}}[H]$. Observe that in
the models described in Section 4.1 for
$\underline{E}(\textrm{Comm}_{A_{n}}[H]/H)$, the stabilizers of the edges are
trivial for every $H$. This means that if we consider the space
$\underline{E}(\textrm{Comm}_{A_{n}}[H]/H)$ as a model for
$E_{{\mathcal{F}}[H]}\operatorname{Comm}_{A_{n}}[H]$ (see Proposition 4.4) the
stabilizers of the edges are always isomorphic to $H$. On 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.
Then, the homomorphism induced by
$f_{[H]}:\underline{E}\textrm{Comm}_{A_{n}}[H]\rightarrow
E_{{\mathcal{F}}[H]}\operatorname{Comm}_{A_{n}}[H]$ 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.

In the following we use the notation of Proposition 4.11 when needed. We 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, $[H]\neq[Z(A_{n})]$

$g_{2[H]}^{1}\colon K_{q}(R)\oplus K_{q}(R)\to K_{q}(R[\mathbb{Z}])\oplus
K_{q}(R).$

Note 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. We now examine the cases $[H]\neq[Z(A_{n})]$ and
$[H]=[Z(A_{n})]$ separately.

If $[H]=[Z(A_{n})]$ then we have $\operatorname{Comm}_{A_{n}}[H]=A_{n}$.

In order to describe $g^{1}_{21[Z(A_{n})]}$, consider the homomorphism

$C_{1}^{Fin}(\underline{E}\textrm{Comm}_{A_{n}}[H],K_{q})\rightarrow
C_{1}^{{\mathcal{F}}[H]}(E_{{\mathcal{F}}[H]}\operatorname{Comm}_{A_{n}}[H],K_{q})$

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 $i:K_{q}(R)\hookrightarrow K_{q}(R[H])=K_{q}(R[\mathbb{Z}])$ given
by Bass-Heller-Swan decomposition. But by item (3) of Proposition 4.15, the
image of this homomorphism is trivial in
$H_{1}^{{\mathcal{F}}[H]}(E_{{\mathcal{F}}[H]}\operatorname{Comm}_{A_{n}}[H],K_{q})$,
and hence $g^{1}_{21[Z(A_{n})]}$ is also trivial. In turn,
$g^{1}_{22[Z(A_{n})]}$ is the identity.

If $[H]\neq[Z(A_{n})]$ then $\operatorname{Comm}_{A_{n}}[H]=\mathbb{Z}^{2}$,
and $\operatorname{Comm}_{A_{n}}[H]/H$ is isomorphic to $\mathbb{Z}$. In this
case $g_{21[H]}^{1}$ is given by a homomorphism $K_{q}(R)\oplus
K_{q}(R)\rightarrow K_{q}(R[\mathbb{Z}])$, 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
$K_{q}(R[\mathbb{Z}])$.

On the other hand, $g_{22[H]}^{1}\colon(\mathbb{Z}\oplus\mathbb{Z})\otimes
K_{q}(R)\rightarrow\mathbb{Z}\otimes K_{q}(R)$ is defined by the
abelianization of the inclusion
$H_{1}(\operatorname{Comm}_{A_{n}}[H])\rightarrow H_{1}(A_{n})$ in the first
component of the tensor product and by the identity in the second.

Now 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
$(\bigoplus_{[H]\neq[Z(A_{n})]}N_{q}^{[H]})\oplus T_{1}(K_{q}(R))\oplus
T_{2}(K_{q}(R))$.

####  4.3.4.  The homomorphism $g_{2}^{1}$ when $n$ is even:

This homomomorphism is defined in equation (5) and, according to Propositions
4.12, 4.13 and 4.15, its first component is given by

$g_{2[H]}^{1}\colon K_{q}(R)\oplus K_{q}(R)\to
K_{q}(R[\mathbb{Z}])\oplus(K_{q}(R)\oplus K_{q}(R))$

for every commensurability class $[H]$. Let us describe this component with
more detail.

We 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 $K_{q}(R)\hookrightarrow K_{q}(R[\mathbb{Z}])$ via Bass-
Heller-Swan decomposition, and the other component is trivial.

Also when $[H]\neq[Z(A_{n})]$ the homomorphism $g_{22[H]}^{1}$ is identified
(via Proposition 4.13 and the Universal Coefficient Theorem) with a
homomorphism $(\mathbb{Z}\oplus\mathbb{Z})\otimes
K_{q}(R)\rightarrow(\mathbb{Z}\oplus\mathbb{Z})\otimes K_{q}(R)$, which comes,
as above, from tensoring with $K_{q}(R)$ the homomomorphism
$H_{1}(\operatorname{Comm}_{A_{n}}[H])\rightarrow H_{1}(A_{n})$ given by
abelianization of the inclusion of the commensurator in $A_{n}$.

Now we consider $[H]=[Z(A_{n})]$. As in the previous case, the homomorphism
$g_{21[H]}^{1}$ can be described as:

$H_{1}^{Fin}(A_{n},K_{q}(-))\to
H_{1}^{{\mathcal{F}}[H]}(A_{n},\pi^{-1}K_{q}(-))\cong
H_{1}^{Fin}(A_{n}/Z(A_{n}),\pi^{-1}K_{q}(-)).$

Now recall from Proposition 4.8 that

$H_{1}^{Fin}(A_{n}/Z(A_{n}),\pi^{-1}K_{q}(-))=\pi^{-1}K_{q}(R[\\{1\\}])=K_{q}(R[Z(A_{n})])=K_{q}(R[\mathbb{Z}]),$

where $Z(A_{n})$ 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 4.1. On 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
$g_{21[Z(A_{n})]}^{1}\colon K_{q}(R)\oplus K_{q}(R)\to K_{q}(R[\mathbb{Z}])$
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.

As $H_{1}(A_{n})$ 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 $\mathbb{Z}$, 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 $K_{q}(R[\mathbb{Z}])$
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 $K_{q}(R[\mathbb{Z}])$, and the remaining two copies
corresponding to the ordinary homology of $A_{n}$. In particular, we have that
$\textrm{coker }g_{2}^{1}=\bigoplus_{[H]}N_{q}^{[H]}.$

####  4.3.5. The homomorphism $g_{2}^{0}$ for every $n$:

Similar considerations to those of the beginning of Section 4.3.3 hold here.
If $x$ 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 (4)), then the induced map $K_{q}(R)\rightarrow K_{q}(R[\mathbb{Z}])$
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. This will be enough to describe
$g_{2}^{0}$ to the extent we need.

We begin describing $g_{2[H]}^{0}$ according to the different prossibilities
of $[H]$ and $n$.

When $H\neq Z(A_{n})$, the results of Propositions 4.12, 4.13 and 4.15 imply
that $g_{2[H]}^{0}$ is defined in the following way:

$g_{2[H]}^{0}\colon K_{q}(R)\to K_{q}(R[\mathbb{Z}])\oplus K_{q}(R).$

By 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.

When $n$ is even and $H=Z(A_{n})$, we have that
$H_{0}^{{\mathcal{F}}[H]}(\textrm{Comm}_{A_{n}}[H],K_{q}(-))=K_{q}(R[\mathbb{Z}])$
by Proposition 4.15, and this copy of $\mathbb{Z}$ 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 $H\neq Z(A_{n})$.

In the case $n$ odd and $H=Z(A_{n})$, again by Proposition 4.15 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 4.14. Then $g_{21[H]}^{0}$ is
defined as a homomorphism

$g_{2[H]}^{0}\colon K_{q}(R)\to C(K_{q}(R))\oplus K_{q}(R).$

Remark 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
$\oplus_{i=1}^{4}NK_{q}(R)$, and 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
$K_{q}(R[\langle ab\rangle])$. These 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 $C(K_{q}(R))$. In turn, $g_{22[H]}^{0}$ is
again the identity, arguing as in the even case.

With the previous information, we can describe $\mathrm{coker}g_{2}^{0}$.

Now if $n$ is even, observe that the source of $g_{2}^{0}$ is
$\oplus_{[H]}K_{q}(R)$ and the codomain is
$(\oplus_{[H]}K_{q}(R[\mathbb{Z}]))\oplus K_{q}(R)$. Moreover, the image of
every $g_{2[H]}^{0}$ consists in a copy of $K_{q}(R)$ inside
$K_{q}(R[\mathbb{Z}])$ (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 $K_{q}(R[\mathbb{Z}])$ (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
$K_{q}(R[\mathbb{Z}])$ are “glued” by the copy of $K_{q}(R)$ that corresponds
to $H_{0}^{Fin}(A_{n},M)$, and hence we have

$\mathrm{coker}g_{2}^{0}=(\bigoplus_{[H]}N_{q}^{[H]})\oplus K_{q}(R).$

When $n$ is odd, we only need to take into account that in the case of
$H=Z(A_{n})$ the role of $K_{q}(R[\mathbb{Z}])$ in the codomain is played by
$C(K_{q}(R))$. Then, 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:

$\mathrm{coker}g_{2}^{0}=(\bigoplus_{[H]\neq[Z(A_{n})]}N_{q}^{[H]})\oplus
K_{q}(R)\oplus\overline{C}(K_{q}(R)).$

Observe that $g_{2}^{0}$ is a monomorphism for every $n$, as the inclusions of
$K_{q}(R)$ in $K_{q}(R[\mathbb{Z}])$ and $C(K_{q}(R))$ are so.

Now we can describe the Bredon homology of $A_{n}$ with respect to the family
of virtually cyclic groups. Observe that the (co)kernels of the statement have
been previously described, and that the groups $N_{q}^{[H]}$ were defined in
Section 4.3.1.

######  Theorem 4.17.

Let $A_{n}$ be an Artin group of dihedral type. In the previous notation, we
have the following:

  1. (1)

$H_{i}^{vc}(A_{n},K_{q}(R[-]))=\\{0\\}$ for $i\geq 4$.

  2. (2)

$H_{3}^{vc}(A_{n},K_{q}(R[-]))=\begin{cases}\bigoplus_{[H]\neq[Z(A_{n})]}K_{q}(R)&\text{$n$
odd}\\\ \ker g_{2}^{2}&\text{$n$ even.}\end{cases}$

  3. (3)

$H_{2}^{vc}(A_{n},K_{q}(R[-]))=\ker g_{2}^{1}$.

  4. (4)

$H_{1}^{vc}(A_{n},K_{q}(R[-]))=\emph{coker
}g_{2}^{1}=\begin{cases}(\bigoplus_{[H]\neq[Z(A_{n})]}N_{q}^{[H]})\oplus
T_{1}(K_{q}(R))\oplus T_{2}(K_{q}(R))&\text{$n$ odd}\\\
\bigoplus_{[H]}N_{q}^{[H]}&\text{$n$ even.}\end{cases}$

  5. (5)

$H_{0}^{vc}(A_{n},K_{q}(R[-]))=\emph{ coker
}g_{2}^{0}=\begin{cases}(\bigoplus_{[H]\neq[Z(A_{n})]}N_{q}^{[H]})\oplus
K_{q}(R)\oplus\overline{C}(K_{q}R)&\text{$n$ odd}\\\
(\bigoplus_{[H]}N_{q}^{[H]})\oplus K_{q}(R)&\text{$n$ even.}\end{cases}$

###### Proof.

The proof is based on the sequence of Proposition 4.1 and the previous
homological computations. We will check the five items in a separate way.

  1. (1)

By Proposition 4.12, Proposition 4.13 and Proposition 4.15 the sequence of
Proposition 4.1 is identically trivial to the left of $g_{3}^{3}$.

  2. (2)

As the exact sequence of Proposition 4.1 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.

For the case $n$ odd, we claim that $g_{1}^{3}$ is an isomorphism. Indeed,
from Proposition 4.12 and Proposition 4.15, the term $(\bigoplus
H_{2}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],M))\oplus
H_{2}^{Fin}(A_{n},M)=\\{0\\}$ and in particular
$\textrm{Im}(g_{2}^{2})=\\{0\\}$. Since $g_{1}^{3}$ is injective, the claim
follows. By Proposition 4.13, the specific description of the odd case
follows.

  3. (3)

There are two different arguments depending if $n$ is odd or even.

For $n$ odd, as the term $(\bigoplus
H_{2}^{\mathcal{F}[H]}(\operatorname{Comm}_{A_{n}}[H],M))\oplus
H_{2}^{Fin}(A_{n},M)$ is trivial, the statement is a direct consequence of the
exactness of the sequence of Proposition 4.1.

For $n$ 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. This implies that
$H_{2}^{vc}(A_{n},K_{q}(R[-]))$ is the image of $g_{1}^{2}$, or equivalently,
the kernel of $g_{2}^{1}$.

  4. (4)

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 $N_{q}^{[H]}$.

  5. (5)

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.

∎

As 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. For instance, we have

######  Corollary 4.18.

Let $A_{n}$, $n>2$ be an Artin group of dihedral type. Let $R$ be a regular
ring. Then $K_{0}(RA_{n})=K_{0}(R)$.

###### Proof.

As $R$ is regular, $K_{i}(R)$ vanishes for $i<0$ and also the Nil-Terms of the
Bass-Heller-Swan decomposition. In particular, by Theorem 4.17,
$H_{0}^{vc}(A_{n},K_{0}(R[-]))=K_{0}(R)$. Moreover, as $K_{i}(R)$ vanishes for
$i<0$ we see by Theorem 4.17, that $H_{j}^{vc}(A_{n},K_{i}(R[-]))=\\{0\\}$ for
$i<0$. Then 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 $E_{\infty}^{0,0}=E_{2}^{0,0}=K_{0}(R)$. ∎

In particular, this implies that every finitely dominated $CW$-complex whose
fundamental group is $A_{n}$ has the homotopy type of a finite $CW$-complex.

In the following section we compute these Bredon homology groups for different
choices of the ring, including some non-regular ones.

##  5\. Computations of $H_{i}^{vc}(A_{n},K_{q}(R[-]))$ for several
coefficient rings

In this section we use Theorem 4.17 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 $\mathbb{Z}[\mathbb{Z}/{\bf 4}]$). We 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 $\mathbb{Z}$ is nearly
known and the $K$-theory of $\mathbb{F}_{q}$ is known (see the corresponding
examples below). In 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.

Recall that the groups $H_{i}^{vc}(A_{n},K_{0}(R[-]))$ are trivial for $i\geq
4$ and any ring $R$.

######  Example 5.1.

First we compute the Bredon homology of $A_{n}$ with respect to the family of
virtually cyclic subgroups, taking as coefficients $K_{q}(\mathbb{Z}[-])$, for
$q=0,1,2$.

We need in our computations the lower algebraic $K$-theory groups of the
integers. The groups $K_{q}(\mathbb{Z})$ are known for all $q\leq 7$ and all
$q\geq 8$ such that $q\not\equiv 0\mod 4$, see [52, page 2]. For example, the
first values of $K_{q}(\mathbb{Z})$ are given in the following table:

$q$ | $<0$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---|---
$K_{q}(\mathbb{Z})$ | 0 | $\mathbb{Z}$ | $\mathbb{Z}/{\bf 2}$ | $\mathbb{Z}/{\bf 2}$ | $\mathbb{Z}/{\bf 48}$ | 0 | $\mathbb{Z}$ | 0 | $\mathbb{Z}/{\bf 240}$

Observe that, as $\mathbb{Z}$ is regular, the Nil-terms in the Bass-Heller-
Swan decomposition are trivial, and then for every $i\in\mathbb{N}$,
$K_{i}(\mathbb{Z}[\mathbb{Z}])=K_{i}(\mathbb{Z})\oplus K_{i-1}(\mathbb{Z})$
and $N_{q}^{[H]}=K_{q-1}(\mathbb{Z})$. Moreover, also by regularity,
$K_{i}(\mathbb{Z})=0$ if $i\leq-1$.

Taking into account of all these considerations, the previous theorem implies
the following.

For $q=0$, we have:

  1. (1)

$H_{3}^{vc}(A_{n},K_{0}(\mathbb{Z}[-]))\simeq
H_{2}^{vc}(A_{n},K_{0}(\mathbb{Z}[-]))\simeq\bigoplus_{\aleph_{0}}\mathbb{Z}$.

  2. (2)

$H_{1}^{vc}(A_{n},K_{0}(\mathbb{Z}[-]))=0$.

  3. (3)

$H_{0}^{vc}(A_{n},K_{0}(\mathbb{Z}[-]))=\mathbb{Z}$.

Now for $q=1$,

  1. (1)

$H_{3}^{vc}(A_{n},K_{1}(\mathbb{Z}[-]))\simeq
H_{2}^{vc}(A_{n},K_{1}(\mathbb{Z}[-]))\simeq\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
2}$.

  2. (2)

$H_{1}^{vc}(A_{n},K_{1}(\mathbb{Z}[-]))=\bigoplus_{\aleph_{0}}\mathbb{Z}$.

  3. (3)

$H_{0}^{vc}(A_{n},K_{1}(\mathbb{Z}[-]))=(\bigoplus_{\aleph_{0}}\mathbb{Z})\oplus\mathbb{Z}/{\bf
2}$.

And finally, for $q=2$, all these groups are isomorphic to
$\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf 2}$.

Let us briefly explain these computations. We consider first $q=0$. For
$H_{3}^{vc}(A_{n},K_{0}(\mathbb{Z}[-]))$, the odd case is immediate. In the
even case, $H_{3}^{vc}(A_{n},K_{1}(\mathbb{Z}[-]))$ is the kernel of a
homomorphism $\bigoplus_{\aleph_{0}}\mathbb{Z}\rightarrow\mathbb{Z}$, and then
isomorphic to $\bigoplus_{\aleph_{0}}\mathbb{Z}$. This easy argument will be
frequently used in the examples of this section without express mention.

We now deal with $H_{2}^{vc}(A_{n},K_{0}(\mathbb{Z}[-]))$. We only deal with
the odd case, the even one is very similar. First observe that the kernel of
$g_{2}^{1}$ is isomorphic to $\bigoplus_{\aleph_{0}}\mathbb{Z}$. More
precisely, taking account the description of Section 4.3.3 and the values of
the $K_{i}(\mathbb{Z})$, $g_{2}^{1}$ (for $n$ odd) is defined in the following
way:

$g_{2}^{1}:(\bigoplus_{[H]\neq[Z(A_{n})]}\mathbb{Z}^{2})\oplus\mathbb{Z}\rightarrow(\bigoplus_{[H]\neq[Z(A_{n})]}\mathbb{Z})\oplus\mathbb{Z}.$

Here the kernel of $g_{21[H]}^{1}$ is isomorphic to $\mathbb{Z}$ for every
$[H]\neq[Z(A_{n})]$. As there is an infinite number of such commensurators, we
conclude that the kernel should be also infinite, and then
$H_{2}^{vc}(A_{n},K_{1}(\mathbb{Z}[-]))=\bigoplus_{\aleph_{0}}\mathbb{Z}$.

Observe that, as $K_{-1}(\mathbb{Z})=0$ and $\mathbb{Z}$ is regular,
$N_{0}^{[H]}=T_{i}(K_{0}\mathbb{Z})=\overline{C}(K_{0}\mathbb{Z})=0$. Now the
values of $H_{1}^{vc}(A_{n},K_{0}(\mathbb{Z}[-]))$ and
$H_{0}^{vc}(A_{n},K_{0}(\mathbb{Z}[-]))$ are easily deduced from items (4) and
(5) of Theorem 4.17.

Now take $q=1$. For $H_{3}^{vc}(A_{n},K_{1}(\mathbb{Z}[-]))$ and
$H_{2}^{vc}(A_{n},K_{1}(\mathbb{Z}[-]))$ 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. Now observe that
$N_{1}^{[H]}=K_{0}(\mathbb{Z})=\mathbb{Z}$ by the Bass-Heller-Swan
decomposition, and regularity and the fact that $K_{0}(\mathbb{Z})=\mathbb{Z}$
is torsion-free imply that $T_{1}(K_{1}\mathbb{Z})=0$,
$T_{2}(K_{1}\mathbb{Z})=0$ and $\overline{C}(K_{1}\mathbb{Z})=\mathbb{Z}$. The
values of $H_{1}^{vc}(A_{n},K_{1}(\mathbb{Z}[-]))$ and
$H_{0}^{vc}(A_{n},K_{1}(\mathbb{Z}[-]))$ follow again from items (4) and (5)
of Theorem 4.17.

Finally, for $q=2$, the values of the homology are immediately implied by
regularity and the fact that
$K_{2}(\mathbb{Z})=K_{1}(\mathbb{Z})=\mathbb{Z}/{\bf 2}$.

######  Example 5.2.

Now 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 [43]) includes the
algebraic $K$-groups that are necessary in our computations:

$q$ | $<0$ | 0 | 1 | 2 | 3
---|---|---|---|---|---
$K_{q}(\mathbb{F}_{2})$ | 0 | $\mathbb{Z}$ | 0 | 0 | $\mathbb{Z}/{\bf 3}$

As $\mathbb{F}_{2}$ 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 4.17 we have the following results. For $q=0$ and every
$i\in\mathbb{N}$,
$H_{i}^{vc}(A_{n},K_{0}(\mathbb{F}_{2}[-]))=H_{i}^{vc}(A_{n},K_{0}(\mathbb{Z}[-]))$,
because $K_{0}(\mathbb{Z})=K_{0}(\mathbb{F}_{2})$ and then
$K_{0}(\mathbb{F}_{2}[-])=K_{0}(\mathbb{Z}[-])$ because both have the same
Bass-Heller-Swan decomposition.

For $q=1$, 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 $N_{1}^{[H]}=\mathbb{Z}$
for every $H$,
$H_{1}^{vc}(A_{n},K_{1}(\mathbb{F}_{2}[-]))=H_{0}^{vc}(A_{n},K_{1}(\mathbb{F}_{2}[-]))=\bigoplus_{\aleph_{0}}\mathbb{Z}$.

For $q=2$, 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 $i$.

Finally, for $q=3$, we have $H_{2}^{vc}(A_{n},K_{3}(\mathbb{F}_{2}[-]))\simeq
H_{3}^{vc}(A_{n},K_{3}(\mathbb{F}_{2}[-]))=\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
3}$. As the groups $N_{3}^{[H]}$ are trivial and $K_{2}(\mathbb{F}_{2})$ is
so, we obtain that $H_{1}^{vc}(A_{n},K_{3}(\mathbb{F}_{2}[-]))=\mathbb{Z}/{\bf
3}$ is trivial, and
$H_{0}^{vc}(A_{n},K_{3}(\mathbb{F}_{2}[-]))=\mathbb{Z}/{\bf 3}$.

We follow with some non-regular examples, namely the rings
$\mathbb{Z}[\mathbb{Z}/{\bf 2}]$, $\mathbb{Z}[\mathbb{Z}/{\bf
2}\times\mathbb{Z}/{\bf 2}]$ and $\mathbb{Z}[\mathbb{Z}/{\bf 4}]$, which we
will respectively denote by $R_{1}$, $R_{2}$ and $R_{3}$. We will compute the
groups $H_{i}^{vc}(A_{n},K_{q}(R_{j}[-]))$, for $0\leq i\leq 3$, $q=0,1$ and
$1\leq j\leq 3$. In order to do this we will need the values of their lower
algebraic $K$-theory groups, as well as the Nil groups. All these groups are
displayed in the following table:

| $K_{1}$ | $K_{0}$ | $K_{-1}$ | $NK_{0}$ | $NK_{1}$
---|---|---|---|---|---
$\mathbb{Z}[\mathbb{Z}/{\bf 2}]$ | $(\mathbb{Z}/{\bf 2})^{2}$ | $\mathbb{Z}$ | 0 | 0 | 0
$\mathbb{Z}[\mathbb{Z}/{\bf 2}\times\mathbb{Z}/{\bf 2}]$ | $(\mathbb{Z}/{\bf 2})^{3}$ | $\mathbb{Z}$ | $\mathbb{Z}^{r}$ | $\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf 2}$ | $\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf 2}$
$\mathbb{Z}[\mathbb{Z}/{\bf 4}]$ | $\mathbb{Z}/{\bf 2}\times\mathbb{Z}/{\bf 4}$ | $\mathbb{Z}$ | $\mathbb{Z}^{s}$ | $\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf 2}$ | $\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf 2}$

Let us briefly explain the values of the table. By work of Oliver [41, Theorem
14.1-2], the kernels $SK_{1}(R_{j})$ of the determinant maps are trivial, and
hence (see for example [48]) $K_{1}(R_{j})$ is the group of units of $R_{j}$.
Now the values of $K_{1}$ follow from a theorem of Higman ([26], see also [46,
II.4.1]). In turn, by [11, Proposition 6], the reduced $K_{0}$ of these three
rings is trivial, and then [30, Lemma 2.18] implies that
$K_{0}(R_{j})=\mathbb{Z}$ for every $j$. The values of the third column follow
from work of Carter [10, Theorem 1], 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 [53].

We 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
4.17.

######  Example 5.3.

We start with $R_{1}=\mathbb{Z}[\mathbb{Z}/{\bf 2}]$. When $q=0$,
$H^{vc}_{3}(A_{n},K_{0}(R_{1}[-]))=H^{vc}_{2}(A_{n},K_{0}(R_{1}[-]))=\bigoplus_{\aleph_{0}}\mathbb{Z}$.
Taking 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
$H^{vc}_{1}(A_{n},K_{0}(R_{1}[-]))=0$. On the other hand,
$H^{vc}_{0}(A_{n},K_{0}(R_{1}[-]))=\mathbb{Z}$ too.

If $q=1$,
$H^{vc}_{3}(A_{n},K_{1}(R_{1}[-]))=H^{vc}_{2}(A_{n},K_{1}(R_{1}[-]))=\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
2}$. Moreover, as $\mathbb{Z}$ is torsion-free and then the correspondent
$T_{1}(K_{1}(R_{1}))$ is trivial, we have
$H^{vc}_{1}(A_{n},K_{1}(R_{1}[-]))=\bigoplus_{\aleph_{0}}\mathbb{Z}$. Finally,
taking into account that $C(K_{1}(R_{1}))=\mathbb{Z}$ in this case,
$H^{vc}_{0}(A_{n},K_{1}(R_{1}[-]))=\mathbb{Z}/{\bf 2}\oplus\mathbb{Z}/{\bf
2}\oplus(\bigoplus_{\aleph_{0}}\mathbb{Z})$.

######  Example 5.4.

We continue by considering $R_{2}=\mathbb{Z}[\mathbb{Z}/{\bf
2}\times\mathbb{Z}/{\bf 2}]$. Now
$H^{vc}_{3}(A_{n},K_{0}(R_{2}[-]))=H^{vc}_{2}(A_{n},K_{0}(R_{2}[-]))=\bigoplus_{\aleph_{0}}\mathbb{Z}$,
exactly as in the previous example. We have $N_{0}^{[H]}=K_{-1}(R_{2})\oplus
NK_{0}(R_{2})\oplus
NK_{0}(R_{2})=\mathbb{Z}^{r}\oplus(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf 2})$
and then
$H^{vc}_{1}(A_{n},K_{0}(R_{2}[-]))=(\bigoplus_{\aleph_{0}}\mathbb{Z})\oplus(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
2})$. 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. Finally,
$H^{vc}_{0}(A_{n},K_{0}(R_{2}[-]))=(\bigoplus_{\aleph_{0}}\mathbb{Z})\oplus(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
2})$ 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.

When $q=1$,
$H^{vc}_{3}(A_{n},K_{1}(R_{2}[-]))=H^{vc}_{2}(A_{n},K_{0}(R_{2}[-]))=\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
2}$, because $K_{1}(R_{2})=(\mathbb{Z}/{\bf 2})^{3}$. Now
$N_{q}^{[H]}=\mathbb{Z}\oplus\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf 2}$, and
then by similar reasons to the previous case,
$H^{vc}_{1}(A_{n},K_{1}(R_{2}[-]))=(\bigoplus_{\aleph_{0}}\mathbb{Z})\oplus(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
2})$. Similarly
$H^{vc}_{0}(A_{n},K_{1}(R_{2}[-]))=(\bigoplus_{\aleph_{0}}\mathbb{Z})\oplus(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
2})$.

######  Example 5.5.

To conclude, we consider $R_{2}=\mathbb{Z}[\mathbb{Z}/{\bf 4}]$. As the groups
$K_{0}$, $NK_{0}$ and $NK_{1}$ are all isomorphic as abelian groups to their
counterparts in the previous example and $K_{-1}$ is also free abelian,
$H^{vc}_{3}(A_{n},K_{0}(R_{3}[-]))=H^{vc}_{2}(A_{n},K_{0}(R_{3}[-]))=\bigoplus_{\aleph_{0}}\mathbb{Z}$,
$H^{vc}_{1}(A_{n},K_{0}(R_{3}[-]))=(\bigoplus_{\aleph_{0}}\mathbb{Z})\oplus(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
2})$, and
$H^{vc}_{0}(A_{n},K_{0}(R_{3}[-]))=(\bigoplus_{\aleph_{0}}\mathbb{Z})\oplus(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
2})$.

For $q=1$, it is clear that
$H^{vc}_{3}(A_{n},K_{1}(R_{3}[-]))=(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
2})\oplus(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf 4})$. Now, the analysis of
$g_{2}^{1}$ in Section 4.3.3 guarantees that
$H^{vc}_{2}(A_{n},K_{1}(R_{3}[-]))$ should contain a subgroup isomorphic to
$(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
2})\oplus(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf 4})$, and then be isomorphic
to it, as the source of $g_{2}^{1}$ is also isomorphic to
$(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
2})\oplus(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf 4})$. We have
$N_{1}^{[H]}=K_{0}(R_{3})\oplus NK_{0}(R_{2})\oplus
NK_{0}(R_{2})=\mathbb{Z}\oplus(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf 2})$. By
an analogous reasoning to the case of $R_{2}$,
$H^{vc}_{1}(A_{n},K_{1}(R_{3}[-]))=(\bigoplus_{\aleph_{0}}\mathbb{Z})\oplus(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
2})$. Finally,
$H^{vc}_{0}(A_{n},K_{1}(R_{3}[-]))=(\bigoplus_{\aleph_{0}}\mathbb{Z})\oplus(\bigoplus_{\aleph_{0}}\mathbb{Z}/{\bf
2})\oplus\mathbb{Z}/{\bf 4}$, corresponding the extra term to $K_{1}(R_{3})$.

######  Remark 5.6.

Observe 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. We remark that the computation of the lower algebraic
$K$-theory groups is a hot topic nowadays (see for example [28], [25] or
[22]), 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.

When 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 [33], 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 $R$. Now if $R$ is regular and $G$ is torsion-
free, the second term of the direct sum vanishes [31, Proposition 2.6].
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 [30] 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 [43] provide a complete knowledge of the groups
$H^{(}BA_{n},\mathbf{K}(R))$ , while for $K=\mathbb{Z}$ much information is
available ([52, page 2]). 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 4.17, 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.

Acknowledgments.

We warmly thank D. Juan-Pineda, W. Lück, N. Petrosyan, L.J. Sánchez-Saldaña,
V. Srinivasan and J. Stienstra and an anonymous referee for their useful
comments.

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