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

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.

Geodesic Growth of some 3-dimensional RACGs

“…Concrete formulas for the standard growth series of Coxeter groups, proved without the use of automata theory, can be found in [10,11]. Recently, it was shown that the growth rate of the geodesic and the standard growth function are either one or a Perron number [8],…”