Infinitely many hyperbolic Coxeter groups through dimension 19

TL;DR

This paper demonstrates the existence of infinitely many hyperbolic Coxeter groups in various dimensions, with growth rates of their counts, using advanced geometric and lattice techniques.

Contribution

It establishes the infinite existence of hyperbolic Coxeter groups in dimensions up to 19, including nonarithmetic cases, and analyzes their growth in number with respect to volume.

Findings

Infinite Coxeter groups exist in dimensions up to 19.

Number of such groups grows exponentially with volume in most dimensions.

Construction uses doubling trick and Leech lattice techniques.

Abstract

We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space H^n for every n < 20 (resp. n < 7). When n=7 or 8, they may be taken to be nonarithmetic. Furthermore, for 1 < n < 20, with the possible exceptions n=16 and 17, the number of essentially distinct Coxeter groups in H^n with noncompact fundamental domain of volume less than or equal to V grows at least exponentially with respect to V. The same result holds for cocompact groups for n < 7. The technique is a doubling trick and variations on it; getting the most out of the method requires some work with the Leech lattice.