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

Abstract

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