# How bad is the modular space?

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 $K$.

• When the subgroup is dense in SL(2,C), the quotient space has trivial topology. When the subgroup is dense in SL(2, R), the quotient is the product of a continuum with trivial topology and the real line. In both cases,the quotient will be contractible, if this is what you wanted to know. Nov 12, 2015 at 23:44
• @Misha Do you mean that $PSL(2,\mathcal{O}_{K})$ is dense in $PSL(2,\mathbf{C})$ if $[K:\mathbf{Q}]>2$ ? Nov 13, 2015 at 0:05
• @sphere - There is a discussion about when $\mathrm{SL}_2(\mathcal{O}_K)$ is discrete or dense in $\mathrm{SL}_2(\textbf{C})$ (or in $\mathrm{SL}_2(\textbf{R})$) in Chapter 8.1 of Maclachlan and Reid's book "The arithmetic of hyperbolic 3-manifolds". In particular see Theorem 8.1.2. In your case you will be taking your quaternion algebra to be $\mathrm{M}_2(K)$ and your order to be $\mathrm{M}_2(\mathcal{O}_K)$.
– user1073
Nov 13, 2015 at 17:15

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).
• In the case of quadratic exertions $K=\mathbf{Q}(\sqrt{-d})$, $PSL(2,\mathcal{O}_{K})$ is a discrete subgroup... for higher field extension $PSL(2,\mathcal{O}_{K})$ is not discrete, no ? it seems to be more complicated... Nov 12, 2015 at 22:46