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 $\Gamma(G,X)$ is obtained: namely, the size of the sphere of radius $n$ is bounded above and below by positive constant multiples of $n^\alpha \lambda^n$ for some integer $\alpha \geq 0$ and some $\lambda \geq 1$. 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 Antol\'in, 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".