Hyperbolic Coxeter groups and minimal growth rates in dimensions four and five

TL;DR

This paper proves that specific cocompact Coxeter groups in four and five dimensions have the smallest growth rates among all Coxeter groups acting cocompactly on hyperbolic space, based on combinatorial and classification methods.

Contribution

It identifies and proves minimal growth rates for particular Coxeter groups in dimensions four and five, extending known results to higher dimensions.

Findings

G_4 and G_5 have minimal growth rates among cocompact Coxeter groups in their dimensions.

The proof uses combinatorial properties and partial classifications of hyperbolic Coxeter polyhedra.

Growth rate monotonicity properties are key to establishing minimality.

Abstract

For small $n$, the known compact hyperbolic $n$-orbifolds of minimal volume are intimately related to Coxeter groups of smallest rank. For $n=2$ and $3$, these Coxeter groups are given by the triangle group $[7,3]$ and the tetrahedral group $[3,5,3]$, and they are also distinguished by the fact that they have minimal growth rate among all cocompact hyperbolic Coxeter groups in $\hbox{Isom}\mathbb H^n$, respectively. In this work, we consider the cocompact Coxeter simplex group $G_4$ with Coxeter symbol $[5,3,3,3]$ in $\hbox{Isom}\mathbb H^4$ and the cocompact Coxeter prism group $G_5$ based on $[5,3,3,3,3]$ in $\hbox{Isom}\mathbb H^5$. Both groups are arithmetic and related to the fundamental group of the minimal volume arithmetic compact hyperbolic $n$-orbifold for $n=4$ and $5$, respectively. Here, we prove that the group $G_n$ is distinguished by having smallest growth rate among all Coxeter groups acting cocompactly on $\mathbb H^n$ for $n=4$ and $5$, respectively. The proof is based on combinatorial properties of compact hyperbolic Coxeter polyhedra, some partial classification results and certain monotonicity properties of growth rates of the associated Coxeter groups.