Hyperbolic Coxeter groups of minimal growth rates in higher dimensions

Abstract

The cusped hyperbolic n-orbifolds of minimal volume are well known for $n\le 9$. Their fundamental groups are related to the Coxeter n-simplex groups ${\Gamma }_n$ listed in Table 1. In this work, we prove that ${\Gamma }_n$ has minimal growth rate among all non-cocompact Coxeter groups of finite covolume in $Isom{\mathbb{H}}^n$. In this way, we extend previous results of Floyd for $n=2$ and of Kellerhals for $n=3$ respectively. Our proof is a generalisation of the methods developed in [2] for the cocompact case.

13 pages