Infinitely many hyperbolic Coxeter groups through dimension 19

Abstract

We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space Hn for every n ≤ 19 (resp. n ≤ 6). When n = 7 or 8, they may be taken to be nonarithmetic. Furthermore, for 2 ≤ n ≤ 19, with the possible exceptions n = 16 and 17, the number of essentially distinct Coxeter groups in H n with noncompact fundamental domain of volume ≤ V grows at least exponentially with respect to V. The same result holds for cocompact groups for n ≤ 6. The technique is a doubling trick and variations on it; getting the most out of the method requires some work with the Leech lattice.