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.



Cohomology of certain arithmetic groups
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.


