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 $K$.
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$\begingroup$ 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. $\endgroup$– MishaCommented Nov 12, 2015 at 23:44
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$\begingroup$ @Misha Do you mean that $PSL(2,\mathcal{O}_{K})$ is dense in $PSL(2,\mathbf{C})$ if $[K:\mathbf{Q}]>2$ ? $\endgroup$– sphereCommented Nov 13, 2015 at 0:05
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$\begingroup$ @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 $\mathrm{M}_2(\mathcal{O}_K)$. $\endgroup$– user1073Commented Nov 13, 2015 at 17:15
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2 Answers
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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.
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Well, in many cases they are quite beautiful, and quite well-understood. See the wikipedia article https://en.wikipedia.org/wiki/Bianchi_group and references therein.
answered Nov 12, 2015 at 22:37
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2$\begingroup$ 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... $\endgroup$– sphereCommented Nov 12, 2015 at 22:46