Conjugacy growth series for finitary wreath products

TL;DR

This paper studies the conjugacy growth series of wreath products involving finitary symmetric and alternating groups, revealing their asymptotic behavior and a specific ratio limit condition based on their dimensions.

Contribution

It provides a detailed analysis of the conjugacy growth series for all such wreath products and characterizes the limiting behavior between symmetric and alternating cases.

Findings

Asymptotic formulas for conjugacy growth series

Characterization of limit behavior between symmetric and alternating wreath products

Ratio limit condition related to the dimensions of the groups

Abstract

We examine the conjugacy growth series of all wreath products of the finitary permutation groups $\text{Sym}(X)$ and $\text{Alt}(X)$ for an infinite set $X$. We determine their asymptotics, and we characterize the limiting behavior between the $\text{Alt}(X)$ and $\text{Sym}(X)$ wreath products. In particular, their ratios form a limit if and only if the dimension of the symmetric wreath product is twice the dimension of the alternating wreath product.