# Are there infinitely many commensurable classes of finite-covolume hyperbolic Coxeter groups?

Allcock(2006) proved that

there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space $H^n$ for every $n\le 19$ (resp. $n\le 6$).

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:

Up to commensurability, are there infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on lower dimensional hyperbolic spaces?

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) found all the commensurable classes (in the wide sense).

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There is a finiteness result of commensurability classes in certain cases. First, note that Vinberg showed 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, then in dimension 2 by Long-MacLachlan-Reid, and 3 by Agol, then independently in all dimensions by Nikulin and Agol-Belolipetsky-Storm-Whyte.

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.

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Thank you for the very interesting references. It seems that one can not expect a complete answer in the near future, so I would like to accept this one. – Hao Chen Apr 24 '14 at 8:27
FYI, Tumarkin pointed me to the 1985 paper of Vinberg "Hyperbolic reflection groups". In Part 4 of Sec. 5 (Ch. II), he described a construction due to Makarov, which seems to be a sequence of incommensurable Coxeter polytopes in dimension 4 and 5. – Hao Chen Jul 20 '14 at 17:14
@HaoCHEN: I see, yes, this is similar to the technique of Gromov-Piatetskii-Shapiro I was alluding to. I wasn't aware of these examples (even though I downloaded a copy of Vinberg's paper :). – Ian Agol Jul 22 '14 at 21:35

Just a partial answer, for dimension 2 and 3, the answer is yes, there are infinitely many cocompact hyperbolic Coxeter groups. Furthermore, in these dimensions the commensurability classes of the groups can be distinguished by their respective invariant trace fields.

For $n=3$, it follows from $\S 4.7.3$ of Maclachlan and Reid's "The Arithmetic of Hyperbolic 3-manifolds." In this section, the authors explicitly construct infinitely many distinct hyperbolic Coxeter groups generated by reflections in the faces of a compact prism. Then they have as an exercise (4.7.4 on page 151) that the invariant trace fields of their examples only depend on a free parameter. In particular, the invariant trace fields can be arbitrarily high degree extensions of $\mathbb{Q}$.

For $n=2$, one could use hyperbolic triangle groups. To set notation, the groups can be distinguished by their orientation preserving index 2 subgroups of the form $\langle x,y | x^m=y^n=(xy)^p=1 \rangle$. For a fixed $(m,n,p)$, the invariant trace field is $\mathbb{Q}(\cos \frac{2\pi}{m},cos \frac{2\pi}{n}, \cos \frac{2\pi}{p},\cos \frac{\pi}{m} \cos \frac{\pi}{n} cos \frac{\pi}{p})$. Again, if the triple $(m,n,p)$ is varied, this field can be made an arbitrarily high degree extension of $\mathbb{Q}$.

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