Hyperbolic Coxeter groups of minimal growth rates in higher dimensions

TL;DR

This paper proves that certain Coxeter groups associated with hyperbolic n-orbifolds have the smallest growth rates among non-cocompact Coxeter groups in hyperbolic space, extending previous results to higher dimensions.

Contribution

It establishes minimal growth rates for non-cocompact Coxeter groups in higher-dimensional hyperbolic spaces, generalizing earlier two- and three-dimensional results.

Findings

Coxeter groups $ Gamma_n$ have minimal growth rates among non-cocompact Coxeter groups in $ ext{Isom}\mathbb{H}^n$

Extension of Floyd's and Kellerhals' results to higher dimensions

Generalization of methods for cocompact cases to non-cocompact groups

Abstract

The cusped hyperbolic n-orbifolds of minimal volume are well known for $n \leq 9$. 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 $\hbox{Isom}\mathbb H^n$. 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.