Growth Rates of Coxeter Groups and Perron Numbers

Abstract

Abstract We define a large class of abstract Coxeter groups that we call $\infty $–spanned, and for which the word growth rate and the geodesic growth rate appear to be Perron numbers. This class contains a fair amount of Coxeter groups acting on hyperbolic spaces, thus corroborating a conjecture by Kellerhals and Perren. We also show that for this class the geodesic growth rate strictly dominates the word growth rate.