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

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 $[n-3, n-1]$.