Spherical and geodesic growth rates of right-angled Coxeter and Artin groups are Perron numbers

TL;DR

This paper proves that the growth rates of infinite right-angled Coxeter and Artin groups are either Perron numbers or one, and calculates the average number of geodesics for elements of given length.

Contribution

It establishes that the spherical and geodesic growth rates are Perron numbers or one, and provides a computation of average geodesics in these groups.

Findings

Growth rates are Perron numbers or equal to 1.

Average number of geodesics for elements of given length is computed.

Provides new insights into the algebraic and geometric properties of these groups.

Abstract

We prove that for any infinite right-angled Coxeter or Artin group, its spherical and geodesic growth rates (with respect to the standard generating set) either take values in the set of Perron numbers, or equal $1$. Also, we compute the average number of geodesics representing an element of given word length in such groups.