Conjugacy growth of commutators

TL;DR

This paper determines the asymptotic count of conjugacy classes of commutators in free groups and free products, extending previous work by leveraging classifications of commutators and asymptotic counting techniques.

Contribution

It provides the first asymptotic formulas for conjugacy classes of commutators in free groups and free products, using Wicks' classification and building on Rivin and Sharp's counting methods.

Findings

Asymptotic formulas for conjugacy classes of commutators in free groups

Asymptotic formulas for conjugacy classes of commutators in free products of finite groups

Extension of previous counting results to broader algebraic structures

Abstract

For the free group $F_r$ on $r>1$ generators (respectively, the free product $G_1 * G_2$ of two nontrivial finite groups $G_1$ and $G_2$), we obtain the asymptotic for the number of conjugacy classes of commutators in $F_r$ (respectively, $G_1 * G_2$) with a given word length in a fixed set of free generators (respectively, the set of generators given by the nontrivial elements of $G_1$ and $G_2$). Our result is proven by using the classification of commutators in free groups and in free products by Wicks, and builds on the works of Rivin and Sharp, who asymptotically counted the conjugacy classes of commutator-subgroup elements in $F_r$ with a given word length.