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 $H^n$ for every $n\le 19$ (resp. $n\le 6$). When $n=7$ or $8$, they may be taken to be nonarithmetic. Furthermore, for $2\le n\le 19$, with the possible exceptions $n=16$ and $17$, the number of essentially distinct Coxeter groups in $H^n$ with noncompact fundamental domain of volume$\le V$ grows at least exponentially with respect to $V$. The same result holds for cocompact groups for $n\le 6$. The technique is a doubling trick and variations on it; getting the most out of the method requires some work with the Leech lattice.