Rational growth and degree of commutativity of graph products

Abstract

Let G be an infinite group and let X be a finite generating set for G such that the growth series of G with respect to X is a rational function; in this case G is said to have rational growth with respect to X. In this paper a result on sizes of spheres (or balls) in the Cayley graph Γ(G,X) is obtained: namely, the size of the sphere of radius n is bounded above and below by positive constant multiples of n^αλⁿ for some integer α≥0 and some λ≥1. As an application of this result, a calculation of degree of commutativity (d. c.) is provided: for a finite group F, its d. c. is defined as the probability that two randomly chosen elements in F commute, and Antolín, Martino and Ventura have recently generalised this concept to all finitely generated groups. It has been conjectured that the d. c. of a group G of exponential growth is zero. This paper verifies the conjecture (for certain generating sets) when G is a right-angled Artin group or, more generally, a graph product of groups of rational growth in which centralisers of non-trivial elements are “uniformly small”.

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