Growth rates of Coxeter groups and Perron numbers

TL;DR

This paper introduces a broad class of Coxeter groups called ∞-spanned, demonstrating that their growth rates are Perron numbers and that geodesic growth exceeds word growth, supporting existing conjectures.

Contribution

The paper defines the ∞-spanned class of Coxeter groups and proves their growth rates are Perron numbers, also showing geodesic growth strictly dominates word growth within this class.

Findings

Growth rates are Perron numbers for ∞-spanned Coxeter groups.

Geodesic growth rate strictly exceeds word growth rate.

Supports conjecture relating hyperbolic Coxeter groups and Perron numbers.

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.