# Rational growth and degree of commutativity of graph products

## TL;DR

This paper investigates the growth properties of certain infinite groups with rational growth and calculates the degree of commutativity, confirming a conjecture for specific classes of groups with exponential growth.

## Contribution

It establishes bounds on sphere sizes in Cayley graphs of groups with rational growth and verifies the conjecture that groups of exponential growth have zero degree of commutativity in specific cases.

## Key findings

- Sphere sizes grow roughly as n^α λ^n with bounds
- Degree of commutativity is zero for certain graph product groups
- Verifies conjecture for right-angled Artin groups with specific properties

## 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".