Is every finitely generated Kleinian group commensurable to a Coxeter group?

Question

Or, to a finitely generated reflection group? Here, I do not insist that the Coxeter group is represented as a hyperbolic reflection group.

If not, what is an counterexample? And what is a characterisation of Kleinian groups that are commensurable to a Coxeter group?

Answer 1

Here is a partial answer.

1. Suppose that $\Gamma< PSL(2,C)$ is a lattice which is commensurable to a reflection group. Then (by looking at the $\Gamma$-stabilizers of the mirrors of reflections), $\Gamma$ contains Fuchsian subgroups $\Lambda$. In this context, Fuchsian means that $\Lambda$ preserves a hyperplane $H<H^3$ and acts as a lattice on $H$. However, there are examples of arithmetic lattices without Fuchsian subgroups, see the book by Maclachlan and Reid "The Arithmetic of Hyperbolic 3-Manifolds", discussion on page 174, end of section 5.3.1.

2. A generic Fuchsian group uniformizing a genus 3 hyperbolic surface will be maximal discrete Fuchsian groups, hence , not commensurable to a group with torsion, hence, not commensurable to a Coxeter group.

3. Having a criterion seems very hard/hopeless (to me): Each time you have a non-tautological criterion for something complicated, you probably proved a deep theorem.