[Allcock(2006)](http://arxiv.org/abs/0903.0138) proved that 



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> there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space $H^n$ for every $n\le 19$ (resp. $n\le 6$).
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His main technique of construction is a "doubling trick". If a wall of the Coxeter polyhedron meets all the neighbor walls at even submultiples of $\pi$, reflection in this wall creates a larger polyhedron. Such walls are called doubling walls. If there are disjoint doubling walls, reflections in them generate infinitely many hyperbolic Coxeter polyhedra.


I noticed that the doubling trick constructs hyperbolic Coxeter subgroups **of finite index**. In terms of Coxeter complex (with polyhedral cells), it correspond to a subcomplex with the same vertices. I then wonder, what if we quotient the set of finite-covolume hyperbolic Coxeter groups by commensurable classes? Therefore the questions:



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> **Up to commensurability**, are there infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on lower dimensional hyperbolic spaces?
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It forces another proof of Allcock's theorem without using the doubling trick. For Coxeter groups whose fundamental domain is a hyperbolic simplex, [Johnson et al.(2002)](http://www.sciencedirect.com/science/article/pii/S0024379501004773) found all the commensurable classes (in the wide sense).



Are there infinitely many commensurable classes of finite-covolume hyperbolic Coxeter groups?
There is a finiteness result of commensurability classes in certain cases. First, note that [Vinberg showed](http://www.ams.org/mathscinet-getitem?mr=774946) there are no cocompact hyperbolic reflection groups in dimension $\geq 30$. This was extended by Prokhorov to dimensions $\geq 996$ for co-finite volume reflection groups. 


As Neil notes, there are infinitely many commensurability classes in dimensions 2 and 3. 


However, considering cofinite arithmetic reflection groups, there are finitely many commensurability classes in any dimension. This was proved in dimension $\geq 10$ by [Nikulin](http://www.ams.org/mathscinet-getitem?mr=607579), then in dimension 2 by [Long-MacLachlan-Reid](http://www.ams.org/mathscinet-getitem?mr=2251482), and 3 by [Agol](http://www.ams.org/mathscinet-getitem?mr=2275630), then independently in all dimensions by Nikulin and [Agol-Belolipetsky-Storm-Whyte](http://www.ams.org/mathscinet-getitem?mr=2442945). 


So one should consider in general non-arithmetic groups to get infinitely many commensurability classes. Of the reflection groups known (in the literature), I believe there are only finitely many commensurability classes in dimensions $\geq 4$. One could try to produce infinite families by "interbreeding", using the method of Gromov & Piatetskii-Shapiro, by gluing together isometric faces of incommensurable reflection groups. However, of the known examples, there would need to be coincidences between faces of reflection groups which are incommensurable, but meet adjacent faces at even submultiples of $\pi$, and as far as I know, these don't exist. 


