On the growth rate of ideal Coxeter groups in hyperbolic 3-space

TL;DR

This paper investigates the growth rates of ideal Coxeter groups in hyperbolic 3-space, revealing their algebraic properties, boundedness, and interval localization based on the number of generators.

Contribution

It establishes that all such growth rates are Perron numbers, unbounded above, and confined within specific intervals depending on the number of generators.

Findings

Growth rates are real algebraic integers greater than 1.

The set of growth rates is unbounded above.

Growth rates with n generators lie in [n-3, n-1].

Abstract

We study the set G of growth rates of of ideal Coxeter groups in hyperbolic 3-space which consists of real algebraic integers greater than 1. We show that (1) G is unbounded above while it has the minimum, (2) any element of G is a Perron number, and (3) growth rates of of ideal Coxeter groups with $n$ generators are located in the closed interval $[n-3, n-1]$.