TITLE: How bad is the modular space?
QUESTION [6 upvotes]: I'm wondering if there is some results about the quotient space $\mathbb{H}^{3}/PSL(2,\mathcal{O}_{K})$?
Do we know something about its homology or homotopy groups ?
$\mathbb{H}^{3}$ is the hyperbolic space of dimension 3 and $\mathcal{O}_{K}$ the ring of integers associated to a the number field $K$.

REPLY [5 votes]: I assume you mean for $K$ to be an imaginary quadratic field (see Misha's comment). There have been computations of the homology for a substantial array of small values of the discriminant of $K$, by Alexander Rahm (see here), in fact there are programs to compute cell decompositions of the quotient orbifolds (see loc.  cit. and also Aurel Pages's work here). 
There are also more theoretical works about the asymptotics of the homology when the discriminant goes to infinity, see here, or for local systems of rank going to infinity and finite-index subgroups.