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 construct 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 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).

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).