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 Hn 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.