TITLE: Cohomology of certain arithmetic groups
QUESTION [7 upvotes]: This is a question on literature about cohomology of arithmetic groups.
Let $M$ denote a quaternion algebra over $\mathbb Q$ and assume it is non-split over $\mathbb R$. Fix a maximal order $\Lambda$ in $M$ and for any ring $R$ let
$$
M(R)=\Lambda\otimes_{\mathbb Z}R,\qquad G(R)=M(R)^\times/R^\times.
$$
Let $p$ be a prime number at which $M$ splits and set 
$$
\Gamma_p=G({\mathbb Z}[1/p])
$$
 I need to know how the first Betti-number 
$$
\dim H^1(\Gamma_p,{\mathbb C}),
$$
depends on $p$. I would be grateful if you could point me towards some literature on this.

REPLY [6 votes]: We can write $G(\mathbb Q_p)/ G(\mathbb Z_p) =GL_2(\mathbb Q_p)/GL_2(\mathbb Z_p)$ as the set of vertices of a tree (the Bruhat-Tits tree) of degree $p+1$. 
The group $\Gamma_p$ acts on this tree with finite stabilizers. Hence the cohomology of the quotient $\Gamma_p \backslash G(\mathbb Q_p)/ G(\mathbb Z_p) $ is equal to the group cohomology of $\Gamma_p$.
That's some graph with $n$ vertices and at most $n(p+1)/2$ edges, where $n$ is the cardinality of $\Gamma_p \backslash G(\mathbb Q_p)/ G(\mathbb Z_p) $, and thus Euler characteristic at least $-n (p-1)/2$ (and at most $1$).
$n$ is bounded independently of $p$ because any class in $G(\mathbb Q_p)/G(\mathbb Z_p)$ defines a locally free left $M(\mathbb Z)$-submodule inside $M(\mathbb Z[1/p])$, with two modules $\Gamma_p$-conjugate if and only if they are isomorphic, and because there are finitely many isomorphism classes of locally free left $M(\mathbb Z)$-modules.