# On the equivariant $K$\- and $KO$-homology of some special linear groups

Sam Hughes

###### Abstract.

We compute the equivariant $KO$-homology of the classifying space for proper
actions of $\textrm{SL}_{3}(\mathbb{Z})$ and $\textrm{GL}_{3}(\mathbb{Z})$. We
also compute the Bredon homology and equivariant $K$-homology of the
classifying spaces for proper actions of
$\textrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ and
$\textrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$ for each prime $p$. Finally, we
prove the Unstable Gromov-Lawson-Rosenberg Conjecture for a large class of
groups whose maximal finite subgroups are odd order and have periodic
cohomology.

##  1\. Introduction

There has been considerable interest in the Baum-Connes conjecture, which
states that for a group $\Gamma$ a certain ‘assembly map’, from the
equivariant $K$-homology of the classifying space for proper actions
$\underline{E}\Gamma$ to the topological $K$-theory of the reduced group
$C^{*}$-algebra, is an isomorphism [3]. The Baum-Connes conjecture is known to
hold for several families of groups, including word-hyperbolic groups, ${\rm
CAT}(0)$-cubical groups and groups with the Haagerup property. An excellent
survey can be found in [2].

######  Conjecture 1.1 (The Baum-Connes Conjecture).

Let $\Gamma$ be a discrete group, then the following _a_ ssembly map is an
isomorphism

$\mu:K^{\Gamma}_{\ast}(\underline{\emph{E}}\Gamma)\rightarrow K^{\rm
top}_{\ast}(C^{\ast}_{r}(\Gamma)).$

There is also a ‘real’ Baum-Connes conjecture which predicts that an assembly
map from the equivariant $KO$-homology of $\underline{E}\Gamma$ to the
topological $K$-theory of the real group $C^{\ast}$-algebra is an isomorphism.
It is known that the Baum-Connes Conjecture implies the Real Baum-Connes
Conjecture [4].

######  Conjecture 1.2 (The Real Baum-Connes Conjecture).

Let $\Gamma$ be a discrete group, then the following _a_ ssembly map is an
isomorphism

$\mu_{\mathbb{R}}:KO^{\Gamma}_{\ast}(\underline{\emph{E}}\Gamma)\rightarrow
KO^{\rm top}_{\ast}(C^{\ast}_{r}(\Gamma)).$

In spite of the interest, to date there have been very few computations of
$K^{\Gamma}$\- and $KO^{\Gamma}$-homology. Indeed, for $K^{\Gamma}$-homology
there are complete calculations for one relator groups [34], NEC groups [31],
some Bianchi groups and hyperbolic reflection groups [29, 37, 38], some
Coxeter groups [16, 45, 46], Hilbert modular groups [43],
$\mathrm{SL}_{3}(\mathbb{Z})$ [44], and $\mathrm{PSL}_{4}(\mathbb{Z})$ [10].
Explicit assembly maps have also been computed for solvable Baumslag-Solitar
groups [36], lamplighter groups of finite groups [18] and certain wreath
products [35, 30]. For $KO^{\Gamma}$-homology the author is aware of two
complete computations; the first, due to Davis and Lück, on a family of
Euclidean crystallographic groups [14], and the second, due to Mario Fuentes-
Rumí, on simply connected graphs of cyclic groups of odd order and of some
Coxeter groups [17].

In this paper we compute the equivariant $K$-homology of
$\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$ and the equivariant $KO$-homology
of $\mathrm{SL}_{3}(\mathbb{Z})$. We give the relevant background and the
connection to Bredon homology in Section 2.

The calculation for $KO^{\Gamma}$-homology is of particular interest because
it is (to the author’s knowledge) the first computation of
$KO^{\Gamma}_{\ast}$ for a property (T) group. For background on property (T)
the reader may consult the monograph [5]. This interest stems from the fact
that property (T) is a strong negation of the Haagerup property which implies
Baum-Connes [21]. Moreover, the (real) Baum-Connes conjecture is still open
for $\mathrm{SL}_{n}(\mathbb{Z})$ when $n\geq 3$. We note that there are
counterexamples for the Baum-Connes conjecture for groupoids constructed from
$\mathrm{SL}_{3}(\mathbb{Z})$ and more generally a discrete group with
property (T) for which the assembly map is known to be injective [22].

######  Theorem 1.3 (Theorem 3.2).

Let $\Gamma=\mathrm{SL}_{3}(\mathbb{Z})$, then for $n=0,\dots,7$ we have

$KO^{\Gamma}_{n}(\underline{\emph{E}}\Gamma)=\mathbb{Z}^{8},\quad\mathbb{Z}_{2}^{8},\quad\mathbb{Z}_{2}^{8},\quad
0,\quad\mathbb{Z}^{8},\quad 0,\quad 0,\quad 0$

and the remaining groups are given by $8$-fold Bott-periodicity.

Applying a Künneth type theorem [45, Theorem 3.6] to the isomorphism
$\mathrm{GL}_{3}(\mathbb{Z})\cong\mathrm{SL}_{3}(\mathbb{Z})\times\mathbb{Z}_{2}$
on the level of Bredon homology, we obtain the following result for
$\mathrm{GL}_{3}(\mathbb{Z})$.

######  Corollary 1.4 (Theorem 3.3).

Let $\Gamma=\mathrm{GL}_{3}(\mathbb{Z})$, then for $n=0,\dots,7$ we have

$KO^{\Gamma}_{n}(\underline{\emph{E}}\Gamma)=\mathbb{Z}^{16},\quad\mathbb{Z}_{2}^{16},\quad\mathbb{Z}_{2}^{16},\quad
0,\quad\mathbb{Z}^{16},\quad 0,\quad 0,\quad 0$

and the remaining groups are given by $8$-fold Bott-periodicity.

We also consider $\Gamma=\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ or
$\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$, for $p$ a prime, computing the
equivariant $K$-homology groups $K^{\Gamma}_{n}(\underline{E}\Gamma)$. There
has been considerable interest in determining homological properties of the
groups $\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{m}])$ and groups related to them
[1, 9, 23]. It appears, however, that even with computer based methods the
problem of determining the cohomology of
$\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{m}])$ for $m$ a product of $3$ primes is
out of reach [9]. In Lemma 5.4 we give a short proof of the Baum-Connes
conjecture for $\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$ and so we obtain the
topological $K$-theory of the reduced group $C^{*}$-algebra of
$\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$ as well.

######  Theorem 1.5 (Theorem 5.6).

Let $p$ be a prime and $\Gamma=\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$,
then $K_{n}^{\Gamma}(\underline{{\emph{E}}}\Gamma)$ is a free abelian group
with rank as given in Table 1. Moreover, since the Baum-Connes Conjecture
holds for $\Gamma$ we have
$K^{\Gamma}_{\ast}(\underline{{\emph{E}}}\Gamma)\cong
K^{\textrm{top}}_{\ast}(C^{\ast}_{r}(\Gamma))$.

| $p=2$ | $p=3$ | $p\equiv 1\pmod{12}$ | $p\equiv 5\pmod{12}$ | $p\equiv 7\pmod{12}$ | $p\equiv 11\pmod{12}$
---|---|---|---|---|---|---
$n=0$ | $7$ | $6$ | $4+\frac{1}{6}(p-7)$ | $6+\frac{1}{6}(p+1)$ | $5+\frac{1}{6}(p-1)$ |  $7+\frac{1}{6}(p+7)$
$n=1$ | $0$ | $0$ | $3$ | $1$ | $2$ | $0$
Table 1. $\mathbb{Z}$-rank of the equivariant $K$-homology of the classifying
space for proper actions of $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$.

######  Theorem 1.6 (Theorem 5.7).

Let $p$ be a prime and $\Gamma=\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$. Then
$K_{n}^{\Gamma}(\underline{\emph{E}}\Gamma)$ is additively isomorphic to the
direct sum of two copies of the corresponding equivariant $K$-homology group
of $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$.

Finally, we will give a proof of the Unstable Gromov-Lawson-Rosenberg
Conjecture for positive scalar curvature for a large class of groups whose
torsion subgroups have periodic cohomology. The statement and background
concerning this conjecture is given in Section 6. However, we will introduce
the following notation now before the theorem statement. We say a group
$\Gamma$ satisfies:

  1. (M)

If every finite subgroup is contained in a unique maximal finite subgroup.

  2. (NM)

If $M$ is a maximal finite subgroup of $\Gamma$, then the normaliser
$N_{\Gamma}(M)$ of $M$ is equal to $M$.

  3. (BC)

If $\Gamma$ satisfies the Baum-Connes conjecture.

  4. (PFS)

If all maximal finite subgroups of $\Gamma$ are odd order and have periodic
cohomology.

A large number of arithmetic groups satisfy the following theorem including
many finite index subgroups $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ for
$p\equiv 11\pmod{12}$ and Hilbert modular groups. We will detail a number of
additional examples in Section 6.

######  Theorem 1.7 (Theorem 6.3).

Let $\Gamma$ be a group satisfying (BC), (M), (NM) and (PFS). If
$\underline{\emph{B}}\Gamma$ is finite and has dimension at most $9$, then the
Unstable Gromov-Lawson-Rosenberg Conjecture holds for $\Gamma$.

In Section 2 we give the relevant background on equivariant $K$ and
$KO$-homology. In Section 3 we give the computations of the equivariant
$KO$-homology for $\mathrm{SL}_{3}(\mathbb{Z})$ and
$\mathrm{GL}_{3}(\mathbb{Z})$. In Section 4 we provide auxiliary computations
of the equivariant $K$-homology of Fuchsian groups. In Section 5 we compute
the equivariant $K$-homology of $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$
and $\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$. Finally, in Section 6 we prove
the results about the Unstable Gromov-Lawson-Rosenberg Conjecture and give a
number of examples of groups satisfying the conjecture.

### Acknowledgements

This paper contains material from the author’s PhD thesis. The author would
like to thank his PhD supervisor Professor Ian Leary for his guidance and
support. He would also like to thank Jim Davis for pointing out Remark 6.4,
Naomi Andrew, Guy Boyde, and Kevin Li for helpful conversations and the
anonymous reviewer whose feedback greatly improved the exposition of this
paper. This work was supported by the Engineering and Physical Sciences
Research Council grant number 2127970.

##  2\. Preliminaries

In this section we introduce the relevant background from Bredon homology and
its interactions with equivariant $K$\- and $KO$-homology. We follow the
treatment given in Mislin’s notes [34].

###  2.1. Classifying spaces for families

Let $\Gamma$ be a discrete group. A $\Gamma$-CW complex $X$ is a CW-complex
equipped with a cellular $\Gamma$-action. We say the $\Gamma$ action is proper
if all of the cell stabilisers are finite.

Let $\mathcal{F}$ be a family of subgroups of $\Gamma$ which is closed under
conjugation and finite intersections. A model for the classifying space
$E_{\mathcal{F}}\Gamma$ for the family $\mathcal{F}$ is a $\Gamma$-CW complex
such that all cell stabilisers are in $\mathcal{F}$ and the fixed point set of
every $H\in\mathcal{F}$ is weakly-contractible. This is equivalent to the
following universal property: For every $\Gamma$-CW complex $Y$ there is
exactly one $\Gamma$-map $Y\rightarrow E_{\mathcal{F}}\Gamma$ up to
$\Gamma$-homotopy.

In the case where $\mathcal{F}=\mathcal{FIN}$, the family of all finite
subgroups of $\Gamma$, we denote $E_{\mathcal{FIN}}(\Gamma)$ by
$\underline{E}\Gamma$. We call such a space, the classifying space for proper
actions of $\Gamma$. Note that if $\Gamma$ is torsion-free then
$\underline{E}\Gamma=E\Gamma$.

###  2.2. Bredon homology

Let $\Gamma$ be a discrete group and $\mathcal{F}$ be a family of subgroups.
We define the orbit category $\mathbf{Or}_{\mathcal{F}}(\Gamma)$ to be the
category with objects given by left cosets $\Gamma/H$ for $H\in\mathcal{F}$
and morphisms the $\Gamma$-maps $\phi:\Gamma/H\rightarrow\Gamma/K$. A morphism
in the orbit category is uniquely determined by its image $\varphi(H)=\gamma
K$ and $\gamma H\gamma^{-1}\subseteq K$; conversely, each such
$\gamma\in\Gamma$ defines a $G$-map.

A (left) Bredon module is a covariant functor
$M:\mathbf{Or}_{\mathcal{F}}(\Gamma)\rightarrow\mathbf{Ab}$, where
$\mathbf{Ab}$ is the category of Abelian groups. Consider a $\Gamma$-CW
complex $X$ and a family of subgroups $\mathcal{F}$ containing all cell
stabilisers. Let $M$ be a Bredon module and define the Bredon chain complex
with coefficients in $M$ as follows:

Let $\\{c_{\alpha}\\}$ be a set of orbit representatives of the $n$-cells in
$X$ and let $\Gamma_{\alpha}$ denote the stabiliser of the cell $\alpha$. The
$n$th chain group is then

$C_{n}:=\bigoplus_{\alpha}M(\Gamma/\Gamma_{c_{\alpha}}).$

If $\gamma c^{\prime}$ is an $(n-1)$-cell in the boundary of $c$, then
$\gamma^{-1}\Gamma_{c}\gamma\subseteq\Gamma_{c^{\prime}}$. This defines a
$\Gamma$-map $\varphi:\Gamma/\Gamma_{c}\rightarrow\Gamma/\Gamma_{c^{\prime}}$,
which in turn gives an induced homomorphism
$M(\varphi):M(\Gamma/\Gamma_{c})\rightarrow M(\Gamma/\Gamma_{c^{\prime}})$.
Therefore, we obtain a differential $\partial:C_{n}\rightarrow C_{n-1}$.
Taking homology of the chain complex $(C_{\ast},\partial)$ gives the Bredon
homology groups $H^{\mathcal{F}}_{n}(X;M)$. A right Bredon module and Bredon
cohomology are defined analogously using contravariant functors.

###  2.3. Equivariant $K$-homology

The original definition of equivariant $K$-homology used Kasparov’s
$KK$-theory [3]. There is also homotopy theoretic approach using spaces and
spectra over the orbit category due to Davis-Lück [12]. We will highlight the
details we need.

Let $\Gamma$ be a discrete group. In the context of the Baum-Connes conjecture
we are specifically interested in the case where $X=\underline{E}\Gamma$,
$\mathcal{F}=\mathcal{FIN}$ and $M=\mathcal{R}_{\mathbb{C}}$ the complex
representation ring. We consider $\mathcal{R}_{\mathbb{C}}(-)$ as a Bredon
module in the following way: For
$\Gamma/H\in\mathbf{Or}_{\mathcal{F}}(\Gamma)$ set
$\mathcal{R}_{\mathbb{C}}(\Gamma/H):=\mathcal{R}_{\mathbb{C}}(H)$, the ring of
complex representations of the finite group $H$. Morphisms are then given by
induction of representations.

We note that
$\mathcal{R}_{\mathbb{C}}(\Gamma):=H_{0}^{\mathcal{F}}(\Gamma)=\textrm{colim}_{\Gamma/H\in\mathbf{Or}(\Gamma)}R_{\mathbb{C}}(H)$.
In the case that $\Gamma$ has finitely many conjugacy classes of finite
subgroups, $\mathcal{R}_{\mathbb{C}}(\Gamma)$ is a finitely generated quotient
of $\bigoplus\mathcal{R}_{\mathbb{C}}(H)$, where $H$ runs over conjugacy
classes of finite subgroups.

We now exhibit the connection between Bredon homology and
$K^{\Gamma}_{\ast}(\underline{E}\Gamma)$, the equivariant $K$-homology of the
classifying space for proper actions. Indeed, for each subgroup $H\leq\Gamma$
equivariant $K$-homology satisfies

$K_{n}^{\Gamma}(\Gamma/H)=K_{n}^{\rm{top}}(C_{r}^{\ast}(H)).$

In the case $H$ is a finite subgroup we have $C_{r}^{\ast}(H)=\mathbb{C}H$,
$K_{0}^{\Gamma}(\Gamma/H)=K_{0}^{\rm{top}}(\mathbb{C}H)=\mathcal{R}_{\mathbb{C}}(H)$,
and $K_{1}^{\Gamma}(\Gamma/H)=K_{1}^{\rm{top}}(\mathbb{C}H)=0$. The remaining
$K^{\Gamma}$ groups are given by $2$-fold Bott periodicity. This allows us to
view $K^{\Gamma}_{n}(-)$ as a Bredon module over
$\mathbf{Orb}_{\mathcal{FIN}}(\Gamma)$.

We may use an equivariant Atiyah-Hirzebruch spectral sequence to compute the
$K^{\Gamma}$-homology of a proper $\Gamma$-CW-complex $X$ from its Bredon
homology.

######  Theorem 2.1.

_[34, Page 50]_ Let $\Gamma$ be a group and $X$ a proper $\Gamma$-CW complex,
then there is an Atiyah-Hirzebruch type spectral sequence

$E^{2}_{p,q}:=H^{\mathcal{FIN}}_{p}(X;K^{\Gamma}_{q}(-))\Rightarrow
K^{\Gamma}_{p+q}(X).$

###  2.4. Equivariant $KO$-homology

In this section we summarise the material from [14, Section 9] which we will
require for our calculations. Again fixing a discrete group $\Gamma$ and
$\mathcal{F}=\mathcal{FIN}$, we introduce two more Bredon modules, the real
representation ring $\mathcal{R}_{\mathbb{R}}(-)$, and the quaternionic
representation ring $\mathcal{R}_{\mathbb{H}}(-)$. These are defined on
$\mathbf{Or}_{\mathcal{F}}(\Gamma)$ in exactly the same way as the complex
representation ring. We have natural transformations between the functors.
Indeed for a finite subgroup $H\leq\Gamma$ we have a diagram:

${\mathcal{R}_{\mathbb{R}}(H)}$${\mathcal{R}_{\mathbb{C}}(H)}$${\mathcal{R}_{\mathbb{H}}(H).}$$\scriptstyle{\nu}$$\scriptstyle{\sigma}$$\scriptstyle{\rho}$$\scriptstyle{\eta}$

Note that the diagram does not commute. For instance let
$\mathbf{1}\in\mathcal{R}_{\mathbb{R}}(H)$ denote the trivial representation,
then $\rho\nu(\mathbf{1})=2\cdot(\mathbf{1})$.

For a real representation $\psi$, the complexification is
$\nu(\psi)=\psi\otimes\mathbb{C}$. For a complex representation $\phi$, the
symplectification is $\sigma(\phi)=\phi\otimes\mathbb{H}$. Going the other
way, for an $n$-dimensional quaternionic representation $\omega$, the
complexification is $\eta(\omega)=\eta$ considered as $2n$-dimensional complex
representation. Similarly, for an $n$-dimensional complex representation
$\phi$, the realification is $\rho(\phi)=\phi$ considered as a
$2n$-dimensional real representation. Note that any composition of the
x-ification natural transformations with the same source and target is
necessarily not the identity.

The situation for the equivariant $KO$-homology, denoted
$KO^{\Gamma}_{\ast}(-)$, is similar to the equivariant $K$-homology but more
complicated. For a subgroup $H\leq\Gamma$ we set
$KO_{n}^{\Gamma}(\Gamma/H)=KO_{n}^{\rm{top}}(C_{r}^{\ast}(H))$. By [8, Section
1.2], in the case that $H$ is a finite subgroup we have that

$KO_{n}^{\Gamma}(\Gamma/H)=KO_{n}^{\rm{top}}(C_{r}^{\ast}(H))=\begin{cases}\mathcal{R}_{\mathbb{R}}(H)&n=0,\\\
\mathcal{R}_{\mathbb{R}}(H)/\rho(\mathcal{R}_{\mathbb{C}}(H)&n=1,\\\
\mathcal{R}_{\mathbb{C}}(H)/\eta(\mathcal{R}_{\mathbb{H}}(H))&n=2,\\\
0&n=3,\\\ \mathcal{R}_{\mathbb{H}}(H)&n=4,\\\
\mathcal{R}_{\mathbb{H}}(H)/\sigma(\mathcal{R}_{\mathbb{C}}(H))&n=5,\\\
\mathcal{R}_{\mathbb{C}}(H)/\nu(\mathcal{R}_{\mathbb{R}}(H))&n=6,\\\
0&n=7,\end{cases}$

with the remaining groups given by $8$-fold Bott-periodicity. For $X$ a proper
$\Gamma$-space, the Atiyah-Hirzebruch spectral sequence from before now takes
the form

$E^{2}_{p,q}:=H^{\mathcal{FIN}}_{p}(X;KO^{\Gamma}_{q}(-))\Rightarrow
KO^{\Gamma}_{p+q}(X).$

###  2.5. Spectra and homotopy

The section gives an alternative $\Gamma$-equivariant homotopy theoretic
viewpoint. Now, we consider $\Gamma$-equivariant homology theories as functors
$\mathbf{E}\colon\mathbf{Or}_{\mathcal{F}}(\Gamma)\rightarrow\mathbf{Spectra}$.
Technically, to avoid functorial problems one must take composite functors
through the categories $C^{\ast}$-$\mathbf{Cat}$ and $\mathbf{Groupoids}$. We
do not concern ourselves with this complication and refer the reader to [12]
and [15].

Instead we will take for granted that there is a composite functor

$\mathbf{KO}\colon\mathbf{Or}_{\mathcal{F}}(\Gamma)\rightarrow\mathbf{Spectra}$

which satisfies
$\pi_{n}\mathbf{KO}(\Gamma/H)=KO_{n}^{\rm{top}}(C_{r}^{\ast}(H))$. When
$\mathcal{F}=\mathcal{FIN}$ this perspective gives a homotopy theoretic
construction of the (real) Baum-Connes assembly map. Indeed, we have maps

$B\Gamma_{+}\wedge\mathbf{KO}\simeq\underset{\mathbf{Or}_{\mathcal{TRV}}(\Gamma)}{\operatorname{hocolim}}\mathbf{KO}\rightarrow\underset{\mathbf{Or}_{\mathcal{F}}(\Gamma)}{\operatorname{hocolim}}\mathbf{KO}\rightarrow\underset{\mathbf{Or}_{\mathcal{ALL}}(\Gamma)}{\operatorname{hocolim}}\mathbf{KO}\simeq\mathbf{KO}(C^{\ast}_{r}(\Gamma;\mathbb{R})).$

The assembly map $\mu_{\mathbb{R}}$ is then $\pi_{n}$ applied to the
composite.

###  2.6. Group $C^{\ast}$-algebras and $KK$-theory

In this section we give a brief outline of Kasparov’s $KK$-theory, the
material here will not be used elsewhere in the paper. The theory was
introduced by Kasparov in [25, 26] in relation to the Novikov Conjecture. The
original formulation of the Baum-Connes Conjecture using $KK$-theory was given
in [3].

For a $C^{\ast}$-algebra $A$ define $\mathbf{M}_{\infty}(A)$ to be the direct
limit of sets of $(n\times n)$-matrices over $A$ as $n\rightarrow\infty$.
Similarly, define $\mathrm{GL}_{\infty}(A)$ to be the direct limit of groups
of invertible $(n\times n)$-matrices over $A$.

Topological or operator $K$-theory is a $2$-periodic homology theory of unital
$C^{\ast}$-algebras denoted $K^{\rm{top}}_{\ast}(-)$. The zeroth $K$-group of
a unital $C^{\ast}$-algebra $A$ is defined to be the Grothendieck group of the
set of projections in $\mathbf{M}_{\infty}(A)$ up to Murray von Neumann
equivalence. The first $K$-group is defined to be
$\mathrm{GL}_{\infty}(A)/\mathrm{GL}_{\infty}(A)_{0}$, where
$\mathrm{GL}_{\infty}(A)_{0}$ is the path component of the identity.

An alternative formulation is given by Kasparov’s bifunctor $KK(-,-)$. For any
two $C^{\ast}$-algebras $A$ and $B$ there is an abelian group $KK(A,B)$. An
element of $KK(A,B)$ is a homotopy class of $(A,B)$-Fredholm bimodules (see
[2, Section 3] for the precise definition). The zeroth $K$-group of $A$ from
before is recovered as $KK(\mathbb{C},A)$ and the first $K$-group is recovered
as $KK(C_{0}(\mathbb{R}),A)$.

Let $\Gamma$ be a discrete group. The _reduced $C^{\ast}$-algebra_ of
$\Gamma$, denoted $C^{\ast}_{r}(\Gamma)$, is the norm closure of the algebra
of bounded operators on $\ell^{2}(\Gamma)$ by the left regular representation
of $\Gamma$. The algebra and its $K$-groups are intimately related with the
theory of elliptic operators on manifolds $M$ with fundamental group $\Gamma$.
For more information the reader should consult the survey [2] and the
references therein.

##  3\. Equivariant $KO$-homology of $\mathrm{SL}_{3}(\mathbb{Z})$

###  3.1. A classifying space for proper actions

A model for $X=\underline{E}\mathrm{SL}_{3}(\mathbb{Z})$ can be constructed as
a $\mathrm{SL}_{3}(\mathbb{Z})$-equivariant deformation retract of the
symmetric space $\mathrm{SL}_{3}(\mathbb{R})/O(3)$. This construction has been
detailed several times in the literature ([49, Theorem 2], [20, Theorem 2.4]
or [44, Theorem 13]), so rather than detailing it again here, we simply
extract the relevant cell complex and cell stabilisers. Specifically, we
follow the notation of Sánchez-García [44] and collect the information in
Table 2.

$\begin{array}[]{|c|c|c|c|c|}\hline\cr\rm Dimension&\rm Cell&\rm Boundary&\rm
Stabiliser\\\ \hline\cr\hline\cr
3&T_{1}&-t_{1}+t_{2}-t_{3}+t_{4}-t_{5}&\\{1\\}\\\
\hline\cr\hbox{\multirowsetup 2}&t_{1}&e_{1}-e_{2}-e_{4}&\mathbb{Z}_{2}\\\
&t_{2}&e_{4}-e_{5}+e_{6}&\\{1\\}\\\
&t_{3}&e_{6}-e_{7}+e_{8}&\mathbb{Z}_{2}^{2}\\\
&t_{4}&e_{1}-e_{3}+e_{5}+e_{8}&\mathbb{Z}_{2}\\\
&t_{5}&e_{2}-e_{3}+e_{6}-e_{6}+e_{7}&\mathbb{Z}_{2}\\\
\hline\cr\hbox{\multirowsetup 1}&e_{1}&v_{1}-v_{2}&\mathbb{Z}_{2}^{2}\\\
&e_{2}&v_{3}-v_{1}&D_{3}\\\ &e_{3}&v_{5}-v_{1}&D_{3}\\\
&e_{4}&v_{3}-v_{2}&\mathbb{Z}_{2}\\\ &e_{5}&v_{4}-v_{2}&\mathbb{Z}_{2}\\\
&e_{6}&v_{4}-v_{3}&\mathbb{Z}_{2}^{2}\\\ &e_{7}&v_{5}-v_{3}&D_{4}\\\
&e_{8}&v_{5}-v_{4}&D_{4}\\\ \hline\cr\hbox{\multirowsetup
0}&v_{1}&\hbox{\multirowsetup-}&{\rm Sym}(4)\\\ &v_{2}&&D_{6}\\\ &v_{3}&&{\rm
Sym}(4)\\\ &v_{4}&&D_{4}\\\ &v_{5}&&{\rm Sym}(4)\\\ \hline\cr\end{array}$
Table 2. Cell structure and stabilisers of a model for
$\underline{E}\mathrm{SL}_{3}(\mathbb{Z})$.

###  3.2. Proof of Theorem 1.3

The calculation of the equivariant $KO$-groups will require the following
proposition and an analysis of the representation theory of the finite
subgroups of $\mathrm{SL}_{3}(\mathbb{Z})$. We remark that one could prove a
dozen subtle variations on the theme of the following proposition. However,
rather than do this we offer the slogan: “Computations with coefficients in
$KO_{n}^{\Gamma}(-)$ can be greatly simplified by looking for chain maps to
the Bredon chain complex with coefficients in $\mathcal{R}_{\mathbb{C}}(-)$.”

######  Proposition 3.1.

Let $\Gamma$ be a discrete group, $\mathcal{F}=\mathcal{FIN}$ and suppose $X$
is a proper $\Gamma$-CW complex with finitely many $\Gamma$ orbits of cells in
each dimension. Assume that for every cell stabiliser the real, complex and
quarternionic character tables are equal, then the Atiyah-Hirzebruch spectral
sequence converging to $KO^{\Gamma}_{\ast}(X)$ has $E^{2}$-page isomorphic to

$E^{2}_{p,q}=H_{p}^{\mathcal{FIN}}(X;K^{\Gamma}_{0})\otimes
KO_{q}(\ast)\oplus\operatorname{\mathrm{Tor}}^{\mathbb{Z}}_{1}[H_{p-1}^{\mathcal{FIN}}(X;K^{\Gamma}_{0}),KO_{q}(\ast)]$

where for $q=0,\dots,7$ we have

$KO_{q}(\ast)=\mathbb{Z},\quad\mathbb{Z}_{2},\quad\mathbb{Z}_{2},\quad
0,\quad\mathbb{Z},\quad 0,\quad 0,\quad 0$

and the remaining groups are given by $8$-fold Bott-periodicity.

Note that the $\operatorname{\mathrm{Tor}}$ terms vanish except possibly when
$q=1$ or $2$.

###### Proof.

Since the three character tables are equal, the complexification from
$\nu:\mathcal{R}_{\mathbb{R}}\to\mathcal{R}_{\mathbb{C}}$ and the
symplectification from
$\sigma:\mathcal{R}_{\mathbb{C}}\to\mathcal{R}_{\mathbb{H}}$ are isomorphisms.
In the other direction, the complexification from
$\eta:\mathcal{R}_{\mathbb{H}}\to\mathcal{R}_{\mathbb{C}}$ and the
realification from $\rho:\mathcal{R}_{\mathbb{C}}\to\mathcal{R}_{\mathbb{R}}$
correspond to multiplication by $2$. We will now compute each row of the
spectral sequence in turn.

$\mathbf{q=0:}$ We have $E^{2}_{p,0}=H^{\mathcal{FIN}}_{p}(X;KO_{0}^{\Gamma})$
which is exactly equal to $H^{\mathcal{FIN}}_{p}(X;\mathcal{R}_{\mathbb{R}})$,
the result follows from the isomorphism
$H^{\mathcal{FIN}}_{p}(X;\mathcal{R}_{\mathbb{R}})\cong
H_{p}^{\mathcal{FIN}}(X;K^{\Gamma}_{0})\otimes\mathbb{Z}$ and the vanishing of
the $\operatorname{\mathrm{Tor}}$ group.

$\mathbf{q=1:}$ The realification
$\rho:\mathcal{R}_{\mathbb{C}}\to\mathcal{R}_{\mathbb{R}}$ is multiplication
by $2$, thus the cokernel of the map

$\rho_{\ast}:C^{\mathcal{FIN}}_{\ast}(X;\mathcal{R}_{\mathbb{C}})\rightarrow
C^{\mathcal{FIN}}_{\ast}(X;\mathcal{R}_{\mathbb{R}})$

is the modulo $2$ reduction of
$C^{\mathcal{FIN}}_{\ast}(X;\mathcal{R}_{\mathbb{R}})$. Consider
$C^{\mathcal{FIN}}_{\ast}(X;\mathcal{R}_{\mathbb{R}})$ as a chain complex of
abelian groups. The result follows from the Universal Coefficient Theorem in
homology with $\mathbb{Z}_{2}$ coefficients applied to the homology of the
chain complex $C^{\mathcal{FIN}}_{\ast}(X;\mathcal{R}_{\mathbb{R}})$.

$\mathbf{q=2:}$ The complexification
$\eta:\mathcal{R}_{\mathbb{H}}\to\mathcal{R}_{\mathbb{C}}$ is multiplication
by $2$ and $C^{\mathcal{FIN}}_{\ast}(X;\mathcal{R}_{\mathbb{C}})$ is
isomorphic to $C^{\mathcal{FIN}}_{\ast}(X;\mathcal{R}_{\mathbb{R}})$. The
result now follows as in the case $q=1$.

$\mathbf{q=3:}$ Immediate since $KO^{\Gamma}_{3}(-)=0$.

$\mathbf{q=4:}$ Since $\nu$ and $\sigma$ are both isomorphisms, their
composition gives an isomorphism of Bredon chain complexes
$C_{\ast}^{\mathcal{FIN}}(X;\mathcal{R}_{\mathbb{R}})\cong
C_{\ast}^{\mathcal{FIN}}(X;\mathcal{R}_{\mathbb{H}})$. The result now follows
as in the $q=0$ case.

$\mathbf{q=5:}$ Since $\sigma$ is an isomorphism, the cokernel of the map

$\sigma_{\ast}:C^{\mathcal{FIN}}_{\ast}(X;\mathcal{R}_{\mathbb{C}})\rightarrow
C^{\mathcal{FIN}}_{\ast}(X;\mathcal{R}_{\mathbb{H}})$

vanishes. The result follows.

$\mathbf{q=6:}$ Since $\nu$ is an isomorphism, the cokernel of the map

$\nu_{\ast}:C^{\mathcal{FIN}}_{\ast}(X;\mathcal{R}_{\mathbb{R}})\rightarrow
C^{\mathcal{FIN}}_{\ast}(X;\mathcal{R}_{\mathbb{C}})$

vanishes. The result follows.

$\mathbf{q=7:}$ Immediate since $KO^{\Gamma}_{7}(-)=0$. ∎

######  Theorem 3.2 (Theorem 1.3).

Let $\Gamma=\mathrm{SL}_{3}(\mathbb{Z})$, then for $n=0,\dots,7$ we have

$KO^{\Gamma}_{n}(\underline{\emph{E}}\Gamma)=\mathbb{Z}^{8},\quad\mathbb{Z}_{2}^{8},\quad\mathbb{Z}_{2}^{8},\quad
0,\quad\mathbb{Z}^{8},\quad 0,\quad 0,\quad 0$

and the remaining groups are given by $8$-fold Bott-periodicity.

###### Proof.

Let $\Gamma=\mathrm{SL}_{3}(\mathbb{Z})$, $\mathcal{F}=\mathcal{FIN}$ and
$X=\underline{E}\mathrm{SL}_{3}(\mathbb{Z})$. We can now complete the
calculation for the equivariant $KO$-homology groups. First, we recap the
calculation of the Bredon chain complex with complex representation ring
coefficients due to Sánchez-García. We have a chain complex

${0}$${\mathbb{Z}}$${\mathbb{Z}^{11}}$${\mathbb{Z}^{28}}$${\mathbb{Z}^{26}}$${0}$$\scriptstyle{\partial_{3}}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\partial_{1}}$

where

$\partial_{3}\sim\begin{bmatrix}1&\mathbf{0}_{1\times
10}\end{bmatrix},\quad\partial_{2}\sim\begin{bmatrix}I_{10}&\mathbf{0}_{10\times
18}\\\ \mathbf{0}_{10\times 1}&\mathbf{0}_{1\times
18}\end{bmatrix},\quad\text{and}\quad\partial_{1}\sim\begin{bmatrix}I_{18}&\mathbf{0}_{18\times
8}\\\ \mathbf{0}_{10\times 18}&\mathbf{0}_{10\times 8}\end{bmatrix}.$

Therefore, the homology groups of the chain complex are isomorphic to
$\mathbb{Z}^{8}$ in dimension $0$ and to $0$ in every other dimension.

Now, the cell stabiliser subgroups of $\mathrm{SL}_{3}(\mathbb{Z})$ acting on
$X$ are isomorphic to $\\{1\\}$, $\mathbb{Z}_{2}$, $\mathbb{Z}_{2}^{2}$,
$D_{3}$, $D_{4}$, ${\rm Sym}(4)$ and $D_{6}$. Each of these satisfies the
conditions of the proposition above. This is easily checked by computing the
Schur indicators of each of the irreducible characters of each group. Since
the Schur indicator equals $1$ in every case we conclude the three character
tables for each group are equal (see for instance [19, Exercise 3.38]).
Applying this to the previous calculation we obtain a single non-trivial
column when $p=0$ in the Atiyah-Hirzebruch spectral sequence and so it
collapses trivially. ∎

######  Corollary 3.3 (Corollary 1.4).

Let $\Gamma=\mathrm{GL}_{3}(\mathbb{Z})$, then for $n=0,\dots,7$ we have

$KO^{\Gamma}_{n}(\underline{\emph{E}}\Gamma)=\mathbb{Z}^{16},\quad\mathbb{Z}_{2}^{16},\quad\mathbb{Z}_{2}^{16},\quad
0,\quad\mathbb{Z}^{16},\quad 0,\quad 0,\quad 0$

and the remaining groups are given by $8$-fold Bott-periodicity.

###### Proof.

First, note that the direct product of $\mathbb{Z}_{2}$ with any of the cell
stabiliser subgroups of $\mathrm{SL}_{3}(\mathbb{Z})$ still satisfies the
conditions of the Proposition 3.1. Now, we may compute the $E^{2}$-page of the
associated Atiyah Hirzebruch spectral sequence by applying the Künneth formula
[45, Theorem 3.6] to the calculation of each row of the $E^{2}$-page for
$\underline{E}\mathrm{SL}_{3}(\mathbb{Z})$. Since the spectral sequence is
concentrated in a single column we have isomorphisms

$KO^{\mathrm{GL}_{3}(\mathbb{Z})}_{n}(\underline{E}\mathrm{GL}_{3}(\mathbb{Z}))\cong
KO^{\mathrm{SL}_{3}(\mathbb{Z})}_{n}(\underline{E}\mathrm{SL}_{3}(\mathbb{Z}))\otimes
KO^{\mathbb{Z}_{2}}_{n}(\ast),$

from which the result is immediate. ∎

##  4\. Equivariant $K$-homology of Fuchsian groups

In this section we compute the equivariant $K$-homology of every finitely
generated Fuchsian group, that is, a finitely generated discrete subgroup of
$\mathrm{PSL}_{2}(\mathbb{R})$. The reason for this apparent detour is that we
will later split the groups $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ as
amalgamated free products of certain Fuchsian subgroups. Thus, we can use a
Mayer-Vietoris type argument to compute their $K$-homology.

Note that Theorem 4.1(a) was computed in [31] along with a more general result
for cocompact NEC groups. Moreover, their integral cohomology was determined
by the author in [23]. An introduction to Fuchsian groups is provided by [27].

The computation is made easier by the fact that every finitely generated
Fuchsian group is described by piece of combinatorial data called a signature
[27, Chapter 4.3]. Indeed, a Fuchsian group of signature
$[g,s;m_{1},\dots,m_{r}]$ has presentation

$\left\langle a_{1},\dots,a_{2g},c_{1},\dots,c_{s},d_{1}.\dots,d_{r}\ |\
\prod_{i=1}^{g}[a_{i},a_{g+i}]\prod_{j=1}^{r}d_{j}\prod_{k=1}^{s}c_{k}=d_{1}^{m_{1}}=\dots=d_{r}^{m_{r}}=1\right\rangle$

and acts on the hyperbolic plane $\mathbb{R}\textbf{H}^{2}$ with a $4g+2s+2r$
sided fundamental polygon. The tessellation of the polygon under the group
action has $1+s+r$ orbits of vertices, $s$ of which are on the boundary
$\partial\mathbb{R}\textbf{H}^{2}$, $2g+s+r$ orbits of edges and $1$ orbit of
faces. All edge and face stabilisers are trivial. All vertex stabilisers are
trivial except for $r$ orbits of vertices, each of which is stabilised by some
$\mathbb{Z}_{m_{j}}$. Note that if $s=0$ we say $\Gamma$ is _cocompact_.

The signature also describes a quotient $2$-orbifold which is homeomorphic to
a genus $g$ surface with $s$ points removed. The orbifold data is then given
by the $r$ marked points, each corresponding to one of the $m_{j}$, or
equivalently a maximal conjugacy of finite subgroups.

If $r=0$, we do not write any $m_{j}$ in the signature. In which case $\Gamma$
has signature $[g,s;]$, is torsionfree, and isomorphic to either the
fundamental group of a genus $g$ surface, or a free group of rank $2g+s-1$.

######  Theorem 4.1.

Let $\Gamma$ be a Fuchsian group of signature $[g,s;m_{1},\dots,m_{r}]$.

  1. (a)

If $s=0$ then,

$K^{\Gamma}_{n}(\underline{\emph{E}}\Gamma)=K_{n}(C_{r}^{\ast}(\Gamma))=\begin{cases}\mathbb{Z}^{2-r+\sum_{j=1}^{r}m_{j}}&n\text{
even,}\\\ \mathbb{Z}^{2g}&n\text{ odd.}\end{cases}$

  2. (b)

If $s>0$ then,

$K^{\Gamma}_{n}(\underline{\emph{E}}\Gamma)=K_{n}(C_{r}^{\ast}(\Gamma))=\begin{cases}\mathbb{Z}^{1-r+\sum_{j=1}^{r}m_{j}}&n\text{
even,}\\\ \mathbb{Z}^{2g+s-1}&n\text{ odd.}\end{cases}$

###### Proof of (a).

Let $\Gamma$ be a Fuchsian group of signature $[g,s;m_{1},\dots,m_{r}]$ with
$s=0$ and $\mathcal{F}=\mathcal{FIN}$. Since $\Gamma$ satisfies the Baum-
Connes conjecture [21] it is enough to compute the equivariant $K$-homology.
The hyperbolic plane with the induced cell structure of the $\Gamma$ action is
a model for $\underline{E}\Gamma$ (see for instance [32]). Recall that the
cell structure has $r+1$ orbits of vertices, $2g+r$ orbits of edges and
exactly $1$ orbit of $2$-cells. One vertex $v_{0}$ is stabilised by the
trivial group and for $j=1,\dots,r$ the vertex $v_{j}$ is stabilised by
$\mathbb{Z}_{m_{j}}$. Thus, we have a Bredon chain complex

${0}$${\mathbb{Z}\oplus\left(\bigoplus_{j=1}^{r}\mathcal{R}_{\mathbb{C}}(\mathbb{Z}_{m_{j}})\right)}$${\mathbb{Z}^{2g+r}}$${\mathbb{Z}}$${0,}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{2}}$

and substituting in
$\mathcal{R}_{\mathbb{C}}(\mathbb{Z}_{m_{j}})=\mathbb{Z}^{m_{j}}$ we obtain

${0}$${\mathbb{Z}\oplus\left(\bigoplus_{j=1}^{r}\mathbb{Z}^{m_{j}}\right)}$${\mathbb{Z}^{2g+r}}$${\mathbb{Z}}$${0.}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{2}}$

We fix the following basis for each chain group: In degree $0$ we have
generators $x_{j,l}$, for $j=1,\dots,r$ and $l=1,\dots,m_{j}$, and the
generator $z$. In degree $1$ we have $a_{1},\dots,a_{2g}$ and
$y_{1},\dots,y_{r}$, and in degree $2$, the generator $w$. An easy calculation
yields that $\partial_{2}(w)=0$, $\partial_{1}(a_{i})=0$, and
$\partial_{1}(y_{j})=\sum_{l=1}^{m_{j}}x_{j,l}-z$. Thus,

$H_{n}^{\mathcal{F}}(\underline{E}\Gamma;\mathcal{R}_{\mathbb{C}})=\begin{cases}\mathbb{Z}^{1+\sum_{j=1}^{r}(m_{j}-1)}&\text{if
}n=0;\\\ \mathbb{Z}^{2g}&\text{if }n=1;\\\ \mathbb{Z}&\text{if }n=2;\\\
0&\text{otherwise}.\end{cases}$

The result now follows from the collapsed Atiyah-Hirzebruch spectral sequence
given in [34, Theorem 5.27] and we obtain
$K^{\Gamma}_{0}(\underline{E}\Gamma)=H^{\mathcal{F}}_{0}(\underline{E}\Gamma;\mathcal{R}_{\mathbb{C}})\oplus
H^{\mathcal{F}}_{2}(\underline{E}\Gamma;\mathcal{R}_{\mathbb{C}})$ and
$K^{\Gamma}_{1}(\underline{E}\Gamma)=H^{\mathcal{F}}_{1}(\underline{E}\Gamma;\mathcal{R}_{\mathbb{C}})$.
∎

$\mathbb{Z}_{m_{1}}$$\mathbb{Z}_{m_{r}}$$2g+s-1$ Figure 1. A graph of groups
for a non-cocompact Fuchsian group.

###### Proof of (b).

Let $\Gamma$ be a Fuchsian group of signature $[g,s;m_{1},\dots,m_{r}]$ with
$s>0$ and let $\mathcal{F}=\mathcal{FIN}$. In this case we can rearrange the
presentation of $\Gamma$ such that we have a splitting of $\Gamma$ as an
amalgamated free product
$\Gamma\cong\mathbb{Z}^{s-1}\ast\mathbb{Z}_{m_{1}}\ast\dots\ast\mathbb{Z}_{m_{r}}$.
Now, $\Gamma$ splits as a finite graph of finite groups (Figure 1) and it is
easy to see that the Bass-Serre tree of $\Gamma$ is a model for
$\underline{E}\Gamma$.

We will first compute the Bredon homology
$H_{\ast}^{\mathcal{F}}(\underline{E}\Gamma;\mathcal{R}_{\mathbb{C}})$ with
coefficients in the representation ring, then apply the equivariant Atiyah-
Hirzebruch spectral sequence. We have a Bredon chain complex

${0}$${\mathbb{Z}\oplus\left(\bigoplus_{j=1}^{r}\mathcal{R}_{\mathbb{C}}(\mathbb{Z}_{m_{j}})\right)}$${\mathbb{Z}^{2g+s-1}}$${0,}$$\scriptstyle{\partial}$

substituting in
$\mathcal{R}_{\mathbb{C}}(\mathbb{Z}_{m_{j}})=\mathbb{Z}^{m_{j}}$ we obtain

${0}$${\mathbb{Z}\oplus\left(\bigoplus_{j=1}^{r}\mathbb{Z}^{m_{j}}\right)}$${\mathbb{Z}^{2g+s-1}}$${0.}$$\scriptstyle{\partial}$

Let the first non-zero term have generating set $\langle x_{j,l},z\ |\
j=1,\dots,r,\ l=1,\dots,m_{j}\rangle$ and the second term $\langle
a_{1},\dots,a_{2g},c_{1},\dots,c_{s-1},d_{1},\dots,d_{r}\rangle$. It is easy
to see the differential $\partial$ is given by
$\partial(a_{i})=\partial(b_{i})=\partial(c_{k})=0$ and
$\partial(d_{j})=\sum_{l=1}^{m_{j}}x_{j,l}-z$. It follows that
$H^{\mathcal{F}}_{0}(\underline{E}\Gamma;\mathcal{R}_{\mathbb{C}})=\mathbb{Z}^{1+\sum_{j=1}^{r}(m_{j}-1)}$,
$H^{\mathcal{F}}_{1}(\underline{E}\Gamma;\mathcal{R}_{\mathbb{C}})=\mathbb{Z}^{2g+s-1}$
and $0$ otherwise. The result now follows from the collapsed Atiyah-Hirzebruch
spectral sequence [34, Theorem 5.27]. In particular, we have
$K^{\Gamma}_{n}(\underline{E}\Gamma)=H^{\mathcal{F}}_{n}(\underline{E}\Gamma;\mathcal{R}_{\mathbb{C}})$
for $n=0,1$. ∎

##  5\. Computations for $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ and
$\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$

###  5.1. Preliminaries

In an abuse of notation, throughout this section we will denote the image
$\\{\pm A\\}$ of a matrix $A\in\mathrm{SL}_{2}(\mathbb{R})$ in
$\mathrm{PSL}_{2}(\mathbb{R})$ by the matrix $A$. Recall that for $p$ a prime
we have
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])=\mathrm{PSL}_{2}(\mathbb{Z})\ast_{\Gamma_{0}(p)}\mathrm{PSL}_{2}(\mathbb{Z})$,
where $\Gamma_{0}(p)$ is the level $p$ Hecke principle congruence subgroup
(see for instance Serre’s book “Trees” [47]). The amalgamation is specified by
two embeddings of the congruence subgroup $\Gamma_{0}(p)$ into
$\mathrm{PSL}_{2}(\mathbb{Z})$. The first is given by

$\Gamma_{0}(p):=\left\\{\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\in\mathrm{PSL}_{2}(\mathbb{Z})\colon c\equiv
0\pmod{p}\right\\}$

and the second via

$\begin{bmatrix}a&b\\\ c&d\end{bmatrix}\mapsto\begin{bmatrix}a&pb\\\
p^{-1}c&d\end{bmatrix}.$

In light of this we will collect some facts about each of the groups in the
amalgamation. We begin by recording (Table 3) the Fuchsian signatures and the
associated Bredon homology for each of the groups $\Gamma_{0}(p)$ and
$\mathrm{PSL}_{2}(\mathbb{Z})$. Note that when $p\equiv 11\pmod{12}$ the group
$\Gamma_{0}(p)$ is free.

######  Lemma 5.1.

The signatures and Bredon homology groups of $\Gamma_{0}(p)$ and
$\mathrm{PSL}_{2}(\mathbb{Z})$ are given in Table 3.

$p$ | Signature of $\Gamma_{0}(p)$ | $H^{\mathcal{F}}_{0}(\underline{E}\Gamma_{0}(p);R_{\mathbb{C}})$ |  $H^{\mathcal{F}}_{1}(\underline{E}\Gamma_{0}(p);R_{\mathbb{C}})$
---|---|---|---
$2$ | $[0,2;2]$ | $\mathbb{Z}^{2}$ | $\mathbb{Z}$
$3$ | $[0,2;3]$ | $\mathbb{Z}^{3}$ | $\mathbb{Z}$
$p\equiv 1\pmod{12}$ | $[0,\frac{1}{6}(p-7)+1;2,2,3,3]$ | $\mathbb{Z}^{7}$ | $\mathbb{Z}^{\frac{1}{6}(p-7)}$
$p\equiv 5\pmod{12}$ | $[0,\frac{1}{6}(p+1)+1;2,2]$ | $\mathbb{Z}^{3}$ | $\mathbb{Z}^{\frac{1}{6}(p+1)}$
$p\equiv 7\pmod{12}$ | $[0,\frac{1}{6}(p-1)+1;3,3]$ | $\mathbb{Z}^{5}$ | $\mathbb{Z}^{\frac{1}{6}(p-1)}$
$p\equiv 11\pmod{12}$ | $[0,\frac{1}{6}(p+7)+1;]$ | $\mathbb{Z}$ | $\mathbb{Z}^{\frac{1}{6}(p+7)}$
$\mathrm{PSL}_{2}(\mathbb{Z})$ | $[0,1;2,3]$ | $\mathbb{Z}^{4}$ | $0$
Table 3. Fuchsian signatures and Bredon homology groups of ${\rm
P}\Gamma_{0}(p)$.

###### Proof.

Let $[0,s;m_{1},\dots,m_{r}]$ be the signature of $\Gamma_{0}(p)$. We will
first compute the ordinary cohomology groups of $\Gamma_{0}(p)$, then using
these we will deduce the signatures, finally the Bredon homology may then be
read off of Theorem 4.1(b). Our computation of the cohomology will be near
identical to the computation in [1, Section 2]. The key difference is that the
modules in [1] are for the lifts of $\Gamma_{0}(p)$ in
$\mathrm{SL}_{2}(\mathbb{R})$ whereas we are always working with the
projectivised groups (see the discussion after [1, Proposition 2.2]). Note
that the fact the signature of $\mathrm{PSL}_{2}(\mathbb{Z})$ is $[0,1;2,3]$
is well known.

Let $G=\mathrm{PSL}_{2}(p)$ and let $Q$ be the subgroup of equivalence classes
of matrices with lower left hand entry equal to zero. Clearly,
$Q\cong\mathbb{Z}_{p}\rtimes\mathbb{Z}_{\frac{1}{2}(p-1)}$ (unless $p=2,3$
where $Q\cong\mathbb{Z}_{p}$). Each $\Gamma_{0}(p)$ fits into a short exact
sequence with normal subgroup a congruence subgroup $\Gamma(p)$ isomorphic to
a free group and quotient $Q$.

Now, recall [47, Example 4.2(c)] that $\mathrm{PSL}_{2}(\mathbb{Z})$ acts on a
tree $\mathcal{T}$ with fundamental domain an edge. Moreover, $G$ acts on
$\mathcal{T}/\Gamma(p)$. The stabiliser subgroups for both actions are
$\mathbb{Z}_{2}$ and $\mathbb{Z}_{3}$ for the vertices and trivial for the
edges. It follows $\Gamma_{0}(p)$ acts freely on $EQ\times\mathcal{T}$ and so
$EQ\times_{Q}\mathcal{T}/\Gamma_{0}(p)$ is a model for $B\Gamma_{0}(p)$.

Let $C^{\ast}$ denote the cellular cochains on the $B$-CW complex
$\mathcal{T}/\Gamma(p)$, then by [1, Section 2] we have the following
isomorphisms of $B$-modules

$C^{0}:=\mathbb{Z}[G/\mathbb{Z}_{2}]\ |_{B}\oplus\mathbb{Z}[G/\mathbb{Z}_{3}]\
|_{B}\quad\text{and}\quad C^{1}:=\mathbb{Z}[G]\ |_{B}.$

As in [1, Section 2] we have a long exact sequence

$\cdots\rightarrow H^{n}(\Gamma_{0}(p);\mathbb{Z})\rightarrow
H^{n}(B;C^{0})\rightarrow H^{n}(B;C^{1})\rightarrow
H^{n+1}(\Gamma_{0}(p);\mathbb{Z})\rightarrow\cdots$

Since $C^{0}$ is a permutation module, $H^{1}(B;C^{0})=0$. Calculating ranks
yields that $H^{1}(\Gamma_{0}(p);\mathbb{Z})=\mathbb{Z}^{N(p)}$ where

$N(p)=1,\quad
1,\quad\frac{1}{6}(p-7),\quad\frac{1}{6}(p+1),\quad\frac{1}{6}(p-1),\quad\frac{1}{6}(p+7),$

ordered as in Table 3. In [23, Theorem 1.4(b)] it is shown that a Fuchsian
group $\Gamma$ of signature $[0,s;m_{1},\dots,m_{r}]$ has
$H^{1}(\Gamma;\mathbb{Z})\cong\mathbb{Z}^{s-1}$. In particular, we deduce the
signature of $\Gamma_{0}(p)$ must have the form
$[0,N(p)+1;m_{1},\dots,m_{r}]$.

Now, $C^{1}$ is a free $B$-module and so we have an isomorphism
$H^{2n}(\Gamma_{0}(p);\mathbb{Z})\cong H^{2}(B;C^{0})$ for all $n\geq 1$. As
in [1, Proposition 2.3] we obtain that

$H^{2n}(\Gamma_{0}(p);\mathbb{Z})\cong\mathbb{Z}_{2},\quad\mathbb{Z}_{3},\quad\mathbb{Z}_{6}^{2},\quad\mathbb{Z}_{2}^{2},\quad\mathbb{Z}_{3}^{2},\quad
0,$

ordered as in Table 3. The result now follows from the following three facts.
Firstly, for a Fuchsian group $\Gamma$ of signature $[0,s;m_{1},\dots,m_{r}]$,
each $m_{j}$ corresponds to a conjugacy class of maximal finite cyclic
subgroups $\mathbb{Z}_{m_{j}}$. Secondly, by the proof of [23, Theorem
1.4(b)], each maximal conjugacy class of finite cyclic subgroups
$\mathbb{Z}_{m_{j}}$ contributes a
$\widetilde{H}^{\ast}(\mathbb{Z}_{m_{j}};\mathbb{Z})$ summand to
$\widetilde{H}^{\ast}(\Gamma;\mathbb{Z})$. Thirdly,
$\mathrm{PSL}_{2}(\mathbb{Z})$ and hence $\Gamma_{0}(p)$ has no elements of
order $6$. ∎

######  Remark 5.2.

Let $\widetilde{\Gamma_{0}}(p)$ denote the lift of $\Gamma_{0}(p)$ in
$\mathrm{SL}_{2}(\mathbb{R})$. An alternative computation of
$H^{\ast}(\Gamma_{0}(p);\mathbb{Z})$ can be achieved by back solving the
Lyndon-Hochschild-Serre spectral sequence (see for instance [7, Chapter
VII.6]) for the group extension
$\mathbb{Z}_{2}\rightarrowtail\widetilde{\Gamma_{0}}(p)\twoheadrightarrow\Gamma_{0}(p)$
which takes the form

$E_{2}^{\ast,\ast}=H^{\ast}(\Gamma_{0}(p);H^{\ast}(\mathbb{Z}_{2};\mathbb{Z}))\Rightarrow
H^{\ast}(\widetilde{\Gamma_{0}}(p);\mathbb{Z})$

using the cohomology calculations for $\widetilde{\Gamma_{0}}(p)$ in [1].

We shall also record the conjugacy classes of finite order elements of
$\Gamma_{0}(p)$ and $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$. Note that the
only conjugacy classes of finite subgroups of $\mathrm{PSL}_{2}(\mathbb{Z})$
are one class of groups isomorphic to $\mathbb{Z}_{2}$ and one to
$\mathbb{Z}_{3}$ since
$\mathrm{PSL}_{2}(\mathbb{Z})\cong\mathbb{Z}_{2}\ast\mathbb{Z}_{3}$. The
conjugacy classes of finite subgroups of $\Gamma_{0}(p)$ can be read off of
the signature, there is exactly one of order $m_{j}$ for each $j=1,\dots,r$.

######  Lemma 5.3.

The number of conjugacy classes of finite order elements in
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ are those given in Table 4.

| $p=2$ | $p=3$ | $p\equiv 1\pmod{12}$ | $p\equiv 5\pmod{12}$ | $p\equiv 7\pmod{12}$ | $p\equiv 11\pmod{12}$
---|---|---|---|---|---|---
Identity | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
Order $2$ | 1 | 2 | 1 | 1 | 2 | 2
Order $3$ | 4 | 2 | 2 | 4 | 2 | 4
Total | $6$ | 5 | 4 | 6 | 5 | 7
Table 4. Number of conjugacy classes of finite order elements of
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$.

###### Proof.

The result follows from the following observation: If there is a conjugacy of
elements of order $2$ (resp. $3$) in $\Gamma_{0}(p)$, then each of class of
elements of order $2$ (resp. $3$) in $\mathrm{PSL}_{2}(\mathbb{Z})$ fuses in
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$. To see this, consider an element
in the first copy of $\mathrm{PSL}_{2}(\mathbb{Z})$, conjugate it to an
element in $\Gamma_{0}(p)$, and then conjugate it to an element in the other
copy of $\mathrm{PSL}_{2}(\mathbb{Z})$. ∎

######  Lemma 5.4.

Both $\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$ and
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ satisfy the Baum-Connes
Conjecture.

###### Proof.

Since
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])=\mathrm{PSL}_{2}(\mathbb{Z})\ast_{\Gamma_{0}(p)}\mathrm{PSL}_{2}(\mathbb{Z})$,
the Bass-Serre tree of the amalgamation is a locally-finite $1$-dimensional
contractible $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$-CW complex. Moreover,
each of the stabilisers $\Gamma_{c}$ have ${\rm
cd}_{\mathbb{Q}}(\Gamma_{c})=1$, being a graph of finite groups. Now, we apply
[34, Corollary 5.14] to see that the stabilisers satisfy Baum-Connes and [34,
Theorem 5.13] to see that $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ does.
The proof is identical for $\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$. ∎

###  5.2. Computations

There is a long exact Mayer-Vietoris sequence for computing the Bredon
homology of an amalgamated free product.

######  Theorem 5.5.

_[34, Corollary 3.32]_ Let $\Gamma=H\ast_{L}K$ and let $M$ be a Bredon module.
There is a long exact Mayer-Vietoris sequence:

${\cdots}$${H_{n}^{\mathcal{FIN}}(L;M)}$${H_{n}^{\mathcal{FIN}}(H;M)\oplus
H_{n}^{\mathcal{FIN}}(K;M)}$${\cdots}$${H_{n-1}^{\mathcal{FIN}}(L;M)}$${H_{n}^{\mathcal{FIN}}(\underline{\emph{E}}\Gamma;M)}$

We are now ready to compute the $K$-theory of
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$.

######  Theorem 5.6 (Theorem 1.5).

Let $p$ be a prime and $\Gamma=\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$,
then $K_{n}^{\Gamma}(\underline{{\emph{E}}}\Gamma)$ is a free abelian group
with rank as given in Table 1. Moreover, since the Baum-Connes Conjecture
holds for $\Gamma$ we have
$K^{\Gamma}_{\ast}(\underline{{\emph{E}}}\Gamma)\cong
K^{\textrm{top}}_{\ast}(C^{\ast}_{r}(\Gamma))$.

###### Proof.

There are $6$ cases to consider, the two cases when $p=2,3$ and the four cases
given by $p\equiv 1,5,7,11\pmod{12}$. Let
$\Gamma=\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ and
$\mathcal{F}=\mathcal{FIN}$. In each case we have the following long exact
Mayer-Vietoris sequence

${0}$${H_{2}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})}$${H_{1}^{\mathcal{F}}(\Gamma_{0}(p);\mathcal{R}_{\mathbb{C}})}$${\left(H_{1}^{\mathcal{F}}(\mathrm{PSL}_{2}(\mathbb{Z});\mathcal{R}_{\mathbb{C}})\right)^{2}}$${\left(H_{0}^{\mathcal{F}}(\mathrm{PSL}_{2}(\mathbb{Z});\mathcal{R}_{\mathbb{C}})\right)^{2}}$${H_{0}^{\mathcal{F}}(\Gamma_{0}(p);\mathcal{R}_{\mathbb{C}})}$${H_{1}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})}$${H_{0}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})}$${0.}$

We have computed the Bredon homology groups of $\mathrm{PSL}_{2}(\mathbb{Z})$
and $\Gamma_{0}(p)$ in Table 3. Thus, we can separate the above sequence into
two sequences. Indeed,
$H_{1}^{\mathcal{F}}(\mathrm{PSL}_{2}(\mathbb{Z});\mathcal{R}_{\mathbb{C}}))=0$,
so it follows that $H_{2}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})\cong
H_{1}^{\mathcal{F}}(\Gamma_{0}(p);\mathcal{R}_{\mathbb{C}})$. The other
sequence is then given by the remaining terms.

We will treat the case $p=2$, the other cases proceed identically. We have
$H_{2}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})=\mathbb{Z}$ and an exact
sequence

${0}$${H_{1}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})}$${\mathbb{Z}^{2}}$${\mathbb{Z}^{8}}$${H_{0}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})}$${0.}$

We now compute the colimit
$H_{0}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})=\textrm{colim}_{\Gamma/H\in\mathbf{Or}_{\mathcal{F}}(\Gamma)}\mathcal{R}_{\mathbb{C}}(H)$.
Since we have a complete description of the conjugacy classes of finite
subgroups of $\Gamma$ and the only inclusions are given by
$\\{1\\}\hookrightarrow\mathbb{Z}_{2}$ and
$\\{1\\}\hookrightarrow\mathbb{Z}_{3}$, it follows that
$H_{0}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})=\mathbb{Z}^{6}$.
Moreover, for the sequence to be exact, it follows the map
$\mathbb{Z}^{2}\rightarrow\mathbb{Z}^{8}$ must be an isomorphism onto the
kernel of the first map. In particular
$H_{1}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})=0$.

The result now follows from the collapsed Atiyah-Hirzebruch spectral sequence
given in [34, Theorem 5.27] and we obtain
$K^{\Gamma}_{0}(\underline{E}\Gamma)=H_{0}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})\oplus
H_{2}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})$ and
$K^{\Gamma}_{1}(\underline{E}\Gamma)=H_{1}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})$.
We record the Bredon homology groups for the remaining cases in Table 5, the
reader can easily verify these. Note that they are always torsion-free and so
are completely determined by their $\mathbb{Z}$-rank. ∎

| $p=2$ | $p=3$ | $p\equiv 1\pmod{12}$ | $p\equiv 5\pmod{12}$ | $p\equiv 7\pmod{12}$ | $p\equiv 11\pmod{12}$
---|---|---|---|---|---|---
$n=0$ | $6$ | $5$ | $4$ | $6$ | $5$ | $7$
$n=1$ | $0$ | $0$ | $3$ | $1$ | $2$ | $0$
$n=2$ | $1$ | $1$ | $\frac{1}{6}(p-7)$ | $\frac{1}{6}(p+1)$ | $\frac{1}{6}(p-1)$ |  $\frac{1}{6}(p+7)$
Table 5. $\mathbb{Z}$-rank of the Bredon homology of
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$.

The computation for $\mathrm{SL}_{2}(\mathbb{Z}[1/p])$ is almost entirely
analogous. We highlight the differences below.

######  Theorem 5.7 (Theorem 1.6).

Let $p$ be a prime and $\Gamma=\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$. Then
$K_{n}^{\Gamma}(\underline{\emph{E}}\Gamma)$ is additively isomorphic to the
direct sum of two copies of the corresponding equivariant $K$-homology group
of $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$.

###### Proof (Sketch)..

Let $\mathcal{F}=\mathcal{FIN}$. First, we must compute the Bredon homology of
the lifts of $\mathrm{PSL}_{2}(\mathbb{Z})$ and $\Gamma_{0}(p)$ to
$\mathrm{SL}_{2}(\mathbb{R})$. For this we use the graph of groups in Figure 1
and note that now every edge group is the same central copy of
$\mathbb{Z}_{2}$ and the vertex groups change as follows: The vertices with
trivial vertex group now have vertex group the same central copy of
$\mathbb{Z}_{2}$. The vertex groups isomorphic to $\mathbb{Z}_{2}$ are now
$\mathbb{Z}_{4}$ and the vertex groups isomorphic to $\mathbb{Z}_{3}$ are now
$\mathbb{Z}_{6}$ (each extended by the central $\mathbb{Z}_{2}$). Computing
the Bredon homology we find that in each case it is isomorphic to the direct
sum of two copies of the corresponding Bredon homology group in the projective
case.

Now, we apply the long exact Mayer-Vietoris sequence to the amalgamated free
product decomposition of $\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$. Since
$H^{\mathcal{F}}_{1}(\mathrm{SL}_{2}(\mathbb{Z});\mathcal{R}_{\mathbb{C}})=0$,
like in the projective case, the sequence splits into two exact sequences. The
computation of
$H^{\mathcal{F}}_{2}(\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])))$ is immediate
as before and is additively isomorphic to the direct sum of two copies of
$H_{2}^{\mathcal{F}}(\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}]);\mathcal{R}_{\mathbb{C}})$.

To compute the zeroth and first homology groups we will again use the colimit
isomorphism
$H_{0}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})=\textrm{colim}_{\Gamma/H\in\mathbf{Or}_{\mathcal{F}}(\Gamma)}\mathcal{R}_{\mathbb{C}}(H)$.
To use this we obtain a count of the total number of conjugacy classes of
finite order elements in $\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$. To do
this use a near identical argument to Lemma 5.3 that takes into account the
central $\mathbb{Z}_{2}$ subgroup. It follows that the number of conjugacy
classes of elements of finite order in
$\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$ is equal to twice the number for
the corresponding projective group. It follows that the colimit computation
for
$H_{0}^{\mathcal{F}}(\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}]);\mathcal{R}_{\mathbb{C}})$
is additively isomorphic to the direct sum of two copies of
$H_{0}^{\mathcal{F}}(\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}]);\mathcal{R}_{\mathbb{C}})$,
where $\Gamma=\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$.

From here one computes
$H_{1}^{\mathcal{F}}(\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}]);\mathcal{R}_{\mathbb{C}})$
in an identical manner to the projective case. The resulting groups are
isomorphic to the direct sum of two copies of the corresponding Bredon
homology groups in the projective case. The result now follows from the
collapsed Atiyah-Hirzebruch spectral sequence given in [34, Theorem 5.27] and
we obtain
$K^{\Gamma}_{0}(\underline{E}\Gamma)=H_{0}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})\oplus
H_{2}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})$ and
$K^{\Gamma}_{1}(\underline{E}\Gamma)=H_{1}^{\mathcal{F}}(\Gamma;\mathcal{R}_{\mathbb{C}})$.
∎

##  6\. The Unstable Gromov-Lawson-Rosenberg Conjecture

Given a smooth closed $n$-manifold $M$ a classical question is to ask whether
$M$ admits a Riemannian metric of positive scalar curvature. In a vast
generalisation of the Atiyah-Singer index theorem, Rosenberg [40] exhibits a
class in $KO_{n}^{\rm{top}}(C^{\ast}_{r}(\pi_{1}(M)))$ which is an obstruction
to $M$ admitting a metric of positive scalar curvature.

More precisely, let $M$ be a closed spin $n$-manifold and $f\colon
M\rightarrow B\Gamma$ be a continuous map for some discrete group $\Gamma$.
Let $\alpha\colon\Omega^{\textrm{Spin}}_{n}(B\Gamma)\rightarrow
KO_{n}^{\rm{top}}(C_{r}^{\ast}(\Gamma))$ be the index of the Dirac operator.
If $M$ admits a metric of positive scalar curvature, then $\alpha[M,f]=0\in
KO_{n}^{\rm{top}}(C_{r}^{\ast}(\Gamma))$

######  Conjecture 6.1 (The Unstable GLR Conjecture).

Let $M$ be a closed spin $n$-manifold and $\Gamma=\pi_{1}(M)$. If $f\colon
M\rightarrow B\Gamma$ is a continuous map which induces the identity on the
fundamental groups, then $M$ admits a metric positive scalar curvature if and
only if $\alpha[M,f]=0\in KO_{n}^{\rm{top}}(C_{r}^{\ast}(\Gamma))$.

The conjecture has been verified in the case of some finite groups [50, 41,
28, 39], when the group has periodic cohomology, torsion-free groups for which
the dimension of $B\Gamma$ is less than $9$ [24], and cocompact Fuchsian
groups [15]. However, there are counterexamples: The first is due to Schick
[48] who disproves the conjecture for the direct product
$\mathbb{Z}^{4}\times\mathbb{Z}_{n}$ when $n$ is odd; while other instances
have been constructed in [24]. For more information on the Unstable GLR
Conjecture the reader should consult [24] and the references therein.

###  6.1. Proof of Theorem 1.7

We will now prove the conjecture for a large class of groups. Our proof is
structurally similar to the proof by Davis-Pearson [15] so we will summarise
their method and highlight any differences.

Let $ko$ be the connective cover of $KO$ with covering map $p$ and let $D$ be
the $ko$-orientation of spin bordism. The map $\alpha$ (from above) is
obtained by the following composition

${\Omega_{n}^{\textrm{Spin}}(B\Gamma)}$${ko_{n}(B\Gamma)}$${KO_{n}(B\Gamma)}$${KO_{n}^{\rm{top}}(C_{r}^{\ast}(\Gamma))}$$\scriptstyle{D}$$\scriptstyle{p}$$\scriptstyle{\mu_{\mathbb{R}}}$

We note that $ko_{n}(\ast)=0$ for $n<0$ and that $p$ is an isomorphism for
$n\geq 0$ on the one point space.

Recall from the introduction that a group $\Gamma$ satisfies:

  1. (M)

If every finite subgroup is contained in a unique maximal finite subgroup.

  2. (NM)

If $M$ is a maximal finite subgroup of $\Gamma$, then the normaliser
$N_{\Gamma}(M)$ of $M$ is equal to $M$.

  3. (BC)

If $\Gamma$ satisfies the Baum-Connes conjecture.

  4. (PFS)

If all maximal finite subgroups of $\Gamma$ are odd order and have periodic
cohomology.

######  Proposition 6.2.

Let $\Gamma$ be a group satisfying (BC), (M), and (NM). Let $\Lambda$ be a set
of conjugacy classes of maximal finite subgroups of $\Gamma$. There is a
commutative diagram with exact rows

${\widetilde{KO}_{n+1}(\underline{\emph{B}}\Gamma)}$${\underset{(H)\in\Lambda}{\bigoplus}\widetilde{KO}_{n}(BH)}$${\widetilde{KO}_{n}(B\Gamma)}$${\widetilde{KO}_{n}(\underline{\emph{B}}\Gamma)}$${\widetilde{KO}_{n+1}(\underline{\emph{B}}\Gamma)}$${\underset{(H)\in\Lambda}{\bigoplus}\widetilde{KO}_{n}^{\rm{top}}(C_{r}^{\ast}(H))}$${\widetilde{KO}_{n}^{\rm{top}}(C_{r}^{\ast}(\Gamma))}$${\widetilde{KO}_{n}(\underline{\emph{B}}\Gamma).}$$\scriptstyle{id}$$\scriptstyle{\mu_{\mathbb{R}}}$$\scriptstyle{\mu_{\mathbb{R}}}$$\scriptstyle{id}$

###### Proof.

First, since $\Gamma$ satisfies (BC), (M) and (NM) by either [13, Corollary
3.13] or the proof of [13, Theorem 4.1] for any constant functor
$\mathbf{E}_{c}\colon\mathbf{Or}_{\mathcal{F}}(\Gamma)\rightarrow\mathbf{Spectra}$
by $\Gamma/H\mapsto\mathbf{E}$ there are long exact sequences

${\cdots}$${\underset{(H)\in\Lambda}{\bigoplus}H_{n}(BH;\mathbf{E})}$${\left(\underset{(H)\in\Lambda}{\bigoplus}\pi_{n}(\mathbf{E})\right)\oplus
H_{n}(B\Gamma;\mathbf{E})}$${H_{n}(\underline{B}\Gamma;\mathbf{E})}$${\cdots}$

and

${\cdots}$${\underset{(H)\in\Lambda}{\bigoplus}\widetilde{H}_{n}(BH;\mathbf{E})}$${\widetilde{H}_{n}(B\Gamma;\mathbf{E})}$${\widetilde{H}_{n}(\underline{B}\Gamma;\mathbf{E})}$${\cdots}$

The result then follows a diagram chase exactly as in [15, Proposition 4],
taking $\mathbf{E}=\mathbf{KO}$ and the isomorphism

$\pi_{n}\left(\underset{\mathbf{Or}_{\mathcal{F}}(\Gamma)}{\operatorname{hocolim}}(\mathbf{E}_{c})\right)\cong
H_{n}(\underline{B}\Gamma;\mathbf{E}).$

∎

######  Theorem 6.3 (Theorem 1.7).

Let $\Gamma$ be a group satisfying (BC), (M), (NM) and (PFS). If
$\underline{\emph{B}}\Gamma$ is finite and has dimension at most $9$, then the
Unstable Gromov-Lawson-Rosenberg Conjecture holds for $\Gamma$.

###### Proof.

Let $\mathbf{ko}$ be the spectrum of the connective cover of $\mathbf{KO}$.
Via the cover we obtain a natural transformation
$p:\mathbf{ko}_{c}\rightarrow\mathbf{KO}_{c}$ of constant
$\mathbf{Or}_{\mathcal{F}}(\Gamma)$-$\mathbf{Spectra}$. From the previous
proposition we obtain a commutative diagram

($\dagger$)
${\widetilde{ko}_{n+1}(\underline{B}\Gamma)}$${\underset{(H)\in\Lambda}{\bigoplus}\widetilde{ko}_{n}(BH)}$${\widetilde{ko}_{n}(B\Gamma)}$${\widetilde{ko}_{n}(\underline{B}\Gamma)}$${\widetilde{KO}_{n+1}(\underline{B}\Gamma)}$${\underset{(H)\in\Lambda}{\bigoplus}\widetilde{KO}_{n}^{\rm{top}}(C_{r}^{\ast}(H))}$${\widetilde{KO}_{n}^{\rm{top}}(C_{r}^{\ast}(\Gamma))}$${\widetilde{KO}_{n}(\underline{B}\Gamma).}$$\scriptstyle{p}$$\scriptstyle{\mu_{\mathbb{R}}\circ
p}$$\scriptstyle{\mu_{\mathbb{R}}\circ p}$$\scriptstyle{p}$

By Joachim-Schick [24, Lemma 2.6] $p$ is an isomorphism for $n\geq 6$ and an
injection for $n=5$. Now, suppose that $n\geq 5$ so we are in the setting of
the GLR conjecture. Consider an element

$\beta\in K:=\mathrm{Ker}\left(\mu_{\mathbb{R}}\circ
p:ko_{n}(B\Gamma)\rightarrow KO_{n}(C_{r}^{\ast}(\Gamma;\mathbb{R})\right)$

and note that $K\cong\mathrm{Ker}(\mu_{\mathbb{R}}\circ
p:\widetilde{ko}_{n}(B\Gamma)\rightarrow\widetilde{KO}_{n}^{\rm{top}}(C_{r}^{\ast}(\Gamma))$.
Combining the diagram ($\dagger$ ‣ 6.1) with the isomorphism
$p\colon\widetilde{ko}_{n}(\underline{B}\Gamma)\rightarrow\widetilde{KO}_{n}^{\rm{top}}(\underline{B}\Gamma)$
for $n\geq 6$ (injection for $n=5$), we can deduce that there exists

$\gamma\in\mathrm{Ker}\left(\underset{(H)\in\Lambda}{\bigoplus}ko_{n}(BH)\rightarrow\underset{(H)\in\Lambda}{\bigoplus}KO_{n}(BH)\right)$

which maps to $\beta$.

For a group $L$ let $ko^{+}_{n}(BL)$ be the subgroup of $ko_{n}(BL)$ given by
$D[M,f]$ where $M$ is a positively curved spin manifold and $f$ is a
continuous map. In [6] the authors prove for any finite group of odd order
with periodic cohomology $H$, that
$ko^{+}_{n}(BH)=\mathrm{Ker}(\mu_{\mathbb{R}}\circ p:ko_{n}(BH)\rightarrow
KO_{n}^{\rm{top}}(C_{r}^{\ast}(H))$. Thus, we have $\gamma\in ko^{+}_{n}(BH)$
and $\beta\in ko^{+}_{n}(B\Gamma)$. Now, in [51] it is proven that if
$D[M,f]\in ko^{+}_{n}(BG)$, then $M$ admits a metric of positive scalar
curvature. In particular, we are done. ∎

######  Remark 6.4.

It is unclear whether the assumption that the finite subgroups having odd
order can be dropped. Indeed, it was pointed out to the author by J.F. Davis
that the statement of [6, Corollary 2.2] contains a misprint. One should
instead (in the notation of [6]) define $\mathcal{Y}_{n}(B\pi)$ to be the
kernel of $A\circ p$ restricted to the subgroup $D(\Omega^{n}(B\pi))\subseteq
ko_{n}(B\pi)$. After making this correction, the statements and proofs in the
paper are correct. This error caused a mistake in the main theorem of [15]
which is only correct if one restricts to Fuchsian groups whose torsion only
has odd order.

######  Corollary 6.5.

Let $M$ be a connected closed spin $n$-manifold and let $\Gamma=\pi_{1}(M)$ be
a group satisfying (M), (NM) and (PFS). Suppose the assembly map
$\mu_{\mathbb{R}}$ is injective and $\underline{\emph{B}}\Gamma$ is finite of
dimension $N$. If $n\geq\max\\{5,N-4\\}$, then $M$ admits a metric with
positive scalar curvature if and only if $\alpha[M,f]=0$.

###  6.2. Some examples

In this section we will detail some applications of Theorem 1.7 to various
families of groups. These results are new whenever the groups involved are
infinite and have torsion.

####  6.2.1. Graphs of groups

In [42, Theorem 3.1] it is shown that the fundamental groups of graphs of
groups with vertex groups satisfying (M) and (NM) and with torsion-free edge
groups, satisfy (M) and (NM). It follows that we have the following
combination theorem:

######  Corollary 6.6.

The $\Gamma$ be a finitely presented fundamental group of a graph of groups
such that the vertex groups satisfy (BC), (M) and (NM) and the edge groups are
torsion-free and satisfy (BC). If the vertex groups satisfy (PFS) and
$\underline{\emph{B}}\Gamma$ has dimension at most $9$, then $\Gamma$
satisfies the Unstable GLR Conjecture.

####  6.2.2. $3$-manifold groups

In [42, Section 3.3] it is shown that $3$-manifold groups satisfy (M) and (NM)
and there is a well known classification of finite subgroups of orientable
connected $3$-manifold groups. These are exactly the groups which act freely
on the $3$-sphere and so have periodic cohomology by [7, Chapter VI.9]. In
[33] it is shown $3$-manifold groups satisfy (BC). Applying Theorem 1.7 we
obtain the following result (which is new whenever $\Gamma$ is infinite and
contains torsion):

######  Corollary 6.7.

Let $M$ be a closed orientable connected $3$-manifold with fundamental group
$\Gamma$. If $\Gamma$ has no elements of order $2$ then $\Gamma$ satisfies the
Unstable GLR Conjecture.

####  6.2.3. One-relator groups

In [13, Page 32] it is shown that one-relator groups satisfy (BC), (M) and
(NM) and admit a two dimensional model for $\underline{E}\Gamma$. Applying
Theorem 1.7 we obtain the following result (which to the authors knowledge is
new whenever $\Gamma$ is infinite and contains torsion):

######  Corollary 6.8.

Let $\Gamma=\langle X\ |\ w\rangle$ be a finitely generated one-relator group
and suppose $w$ has odd order when interpreted in $\Gamma$, then $\Gamma$
satisfies the Unstable GLR Conjecture.

####  6.2.4. Hilbert modular groups

Let $k$ be a totally real number field of degree $n$ and $\mathcal{O}_{k}$ be
its ring of integers. The Hilbert modular group of $k$ is defined to be
$\mathrm{PSL}_{2}(\mathcal{O}_{k})$ and is a lattice in
$\mathrm{PSL}_{2}(\mathbb{R})^{n}$. Note that if $k=\mathbb{Q}$ then we
recover the classical modular group $\mathrm{PSL}_{2}(\mathbb{Z})$. Properties
(BC), (M) and (NM) are given in [11, Lemma 4.3]. Applying Theorem 1.7 we
obtain the following new result:

######  Corollary 6.9.

Let $k$ be a totally real number field with degree less than or equal to $4$.
Let $\Gamma\leq\mathrm{PSL}_{2}(\mathcal{O}_{k})$ be finitely presented. If
all finite subgroups of $\Gamma$ are cyclic of odd order then $\Gamma$
satisfies the Unstable GLR Conjecture.

###### Proof.

This follows from the fact every finite subgroup of
$\mathrm{PSL}_{2}(\mathbb{R})^{n}$ is a product of finite cyclic groups. ∎

####  6.2.5. Subgroups of $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ for
$p\equiv 11\pmod{12}$

In this section we will prove the result that many subgroups of
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ for $p\equiv 11\pmod{12}$ satisfy
the conditions of Theorem 1.7. The result is new whenever the subgroup is
infinite, has torsion, and is not isomorphic to a Fuchsian group. We will also
compute the $KO$-theory of
$C_{r}^{\ast}(\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}]);\mathbb{R})$.

######  Corollary 6.10.

Let $p\equiv 11\pmod{12}$ and let
$\Gamma<\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ be finitely presented. If
$\Gamma$ has no elements of order $2$, then $\Gamma$ satisfies the Unstable
GLR Conjecture.

There are many finite index subgroups of
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ for $p\equiv 11\pmod{12}$
satisfying the hypothesis of the corollary. Indeed, by the amalgamated free
product decomposition, $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ is
generated by the two subgroups isomorphic to
$\mathrm{PSL}_{2}(\mathbb{Z})\cong\mathbb{Z}_{2}\ast\mathbb{Z}_{3}$. Thus,
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ is a four generated group $\langle
a,b,c,d\rangle$ where $a$ and $c$ have order $2$, and $b$ and $d$ have order
$3$. The kernel of the homomorphism
$\phi:\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])\rightarrow\mathbb{Z}_{2}$ by
$a,c\mapsto 1$ and $b,d\mapsto 0$ is a finite index subgroup of
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ with no $2$-torsion.

###### Proof.

The proof follows from applying Theorem 1.7 to the observation that every
finite subgroup of $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ is cyclic and
hence has periodic cohomology and the following lemma. Note that $\Gamma$ is
necessarily a proper subgroup since
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ always contains elements of order
$2$. ∎

######  Lemma 6.11.

Let $p$ be a prime, then $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ satisfies
(M). Moreover, if $p\equiv 11\pmod{12}$ then
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$ satisfies (NM).

###### Proof.

Since each non-trivial finite subgroup of $\Gamma$ is of order $2$ or $3$ it
is obvious that $\Gamma$ satisfies (M). Now, assume $p\equiv 11\pmod{12}$ and
note that $\mathrm{PSL}_{2}(\mathbb{Z})\cong\mathbb{Z}_{2}\ast\mathbb{Z}_{3}$
satisfies (NM). Recall the amalgamated free product decomposition,
$\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])=\mathrm{PSL}_{2}(\mathbb{Z})\ast_{\Gamma_{0}(p)}\mathrm{PSL}_{2}(\mathbb{Z})$.
The amalgamated subgroup $\Gamma_{0}(p)$ is torsion-free so we may apply [42,
Theorem 3.1]. ∎

An alternative direct proof of the calculations of the $K$-groups of
$C_{r}^{\ast}(\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}]))$ when $p\equiv
11\pmod{12}$ is as follows. Note that this bypasses the computation of the
Bredon homology but does not give us a way to compute either invariant for
$\mathrm{SL}_{2}(\mathbb{Z}[\frac{1}{p}])$.

######  Lemma 6.12.

Let $\Gamma=\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$, then
$\underline{\emph{B}}\Gamma\simeq\bigvee_{b_{1}(\Gamma_{0}(p))}S^{2}$

###### Proof.

We have

$\displaystyle\underline{B}\Gamma$
$\displaystyle\simeq\underset{\mathbf{Top}}{\operatorname{hocolim}}\left((\underline{E}\mathrm{PSL}_{2}(\mathbb{Z})\times_{\mathrm{PSL}_{2}(\mathbb{Z})}\Gamma)\leftarrow(\underline{E}\Gamma_{0}(p)\times_{\Gamma_{0}(p)}\Gamma)\rightarrow(\underline{E}\mathrm{PSL}_{2}(\mathbb{Z})\times_{\mathrm{PSL}_{2}(\mathbb{Z})}\Gamma)\right)/\Gamma,$
$\displaystyle\simeq\underset{\mathbf{Top}}{\operatorname{hocolim}}\left((\underline{E}\mathrm{PSL}_{2}(\mathbb{Z})\times_{\mathrm{PSL}_{2}(\mathbb{Z})}\Gamma)/\Gamma\leftarrow(\underline{E}\Gamma_{0}(p)\times_{\Gamma_{0}(p)}\Gamma)/\Gamma\rightarrow(\underline{E}\mathrm{PSL}_{2}(\mathbb{Z})\times_{\mathrm{PSL}_{2}(\mathbb{Z})}\Gamma)/\Gamma\right).$
Since $\underline{E}\mathrm{PSL}_{2}(\mathbb{Z})/\mathrm{PSL}_{2}(\mathbb{Z})$
is an interval and $\underline{E}\Gamma_{0}(p)/\Gamma_{0}(p)$ is a finite
graph, we have $\displaystyle\underline{B}\Gamma$
$\displaystyle\simeq\underset{\mathbf{Top}}{\operatorname{hocolim}}\left(I\leftarrow\bigvee_{b_{1}(\Gamma_{0}(p))}S^{1}\rightarrow
I\right),$

but $I$ is contractible, so up to homotopy this becomes a suspension of a
wedge of circles. In particular,
$\underline{B}\Gamma\simeq\bigvee_{b_{1}(\Gamma_{0}(p))}S^{2}$. ∎

######  Theorem 6.13.

Let $p\equiv 11\pmod{12}$ and let
$\Gamma=\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$, then

$K_{0}^{\rm{top}}(C_{r}^{\ast}(\Gamma))=\mathbb{Z}^{7+\frac{1}{6}(p+7)}\quad\text{and}\quad
K_{1}^{\rm{top}}(C_{r}^{\ast}(\Gamma))=0.$

###### Proof.

Let $\Lambda$ be a set of representatives of finite subgroups of $\Gamma$. By
[13, Theorem 4.1(a)] we have a short exact sequence

$0\rightarrow\underset{(H)\in\Lambda}{\bigoplus}\widetilde{K}_{n}^{\rm{top}}(C_{r}^{\ast}(H))\rightarrow
K_{n}^{\rm{top}}(C_{r}^{\ast}(\Gamma))\rightarrow
K_{n}(\underline{B}\Gamma)\rightarrow 0.$

The only nontrivial part now is computing $K_{n}(\underline{B}\Gamma)$, but we
have already shown that $\underline{B}\Gamma$ is homotopy equivalent to a
wedge of $\bigvee_{b_{1}(\Gamma_{0}(p))}S^{2}$, i.e. a wedge of $2$-spheres.
Thus, we can simply apply the homological Atiyah-Hirzebruch spectral sequence
(which collapses trivially) to obtain that
$K_{0}(\underline{B}\Gamma)=\mathbb{Z}^{\frac{1}{6}(p+7)+1}$ and
$K_{1}(\underline{B}\Gamma)=0$. ∎

A near identical argument can be used to compute the $KO$-groups of
$C_{r}^{\ast}(\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}]))$ when $p\equiv
11\pmod{12}$.

######  Theorem 6.14.

Let $p\equiv 11\pmod{12}$ be a prime and
$\Gamma=\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$. Except for an extension
problem in dimensions congruent to $1,3$ and $4$ modulo $8$, we have for
$n=0,\dots,7$ that

$KO_{n}^{\rm{top}}(C_{r}^{\ast})=\mathbb{Z}^{5},\quad\mathbb{Z}_{2}^{3},\quad\mathbb{Z}^{2+\frac{1}{6}(p+7)}\oplus\mathbb{Z}_{2}^{3},\quad\mathbb{Z}_{2}^{\frac{1}{6}(p+7)},\quad\mathbb{Z}^{5}\oplus\mathbb{Z}_{2}^{\frac{1}{6}(p+7)},\quad
0,\quad\mathbb{Z}^{2+\frac{1}{6}(p+7)},\quad 0$

and the remaining groups are given by $8$-fold Bott-periodicity.

###### Proof.

Let $\Lambda$ be a set of representatives of finite subgroups of $\Gamma$. As
before, by [13, Theorem 4.1(a)] we have a short exact sequence

$0\rightarrow\underset{(H)\in\Lambda}{\bigoplus}\widetilde{KO}_{n}^{\rm{top}}(C_{r}^{\ast}(H))\rightarrow
KO_{n}^{\rm{top}}(C_{r}^{\ast}(\Gamma))\rightarrow
KO_{n}(\underline{B}\Gamma)\rightarrow 0.$

Now, $KO_{n}(\underline{B}\Gamma)\cong KO_{n}(\ast)\oplus
KO_{n-2}(\ast)^{\frac{1}{6}(p+7)}$ and the groups
$KO_{n}^{\mathbb{Z}_{m}}(\ast)\cong
KO^{\rm{top}}_{n}(C_{r}^{\ast}(\mathbb{Z}_{m}))$ are given in [17, Section
2.1]. ∎

Our methods leave open the following.

######  Question 6.15.

Let $p$ be a prime, then does $\mathrm{PSL}_{2}(\mathbb{Z}[\frac{1}{p}])$
satisfy the Unstable GLR Conjecture? What about a lattice in
$\mathrm{PSL}_{2}(\mathbb{R})\times\mathrm{PSL}_{2}(\mathbb{Q}_{p})$?

## References

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