Conjugacy Growth and Conjugacy Width of Certain Branch Groups

TL;DR

This paper investigates the conjugacy growth in certain branch groups, establishing lower bounds and introducing the concept of bounded conjugacy width, which relates to other algebraic properties and palindromic width.

Contribution

It provides new bounds for conjugacy growth and introduces the property of bounded conjugacy width in branch groups, connecting these to algebraic and combinatorial properties.

Findings

Lower bounds for conjugacy growth in branch groups including the Grigorchuk group.

Introduction of bounded conjugacy width property for certain groups.

Analysis of the relationship between bounded conjugacy width and palindromic width.

Abstract

The conjugacy growth function counts the number of distinct conjugacy classes in a ball of radius $n$. We give a lower bound for the conjugacy growth of certain branch groups, among them the Grigorchuk group. This bound is a function of intermediate growth. We further proof that certain branch groups have the property that every element can be expressed as a product of uniformly boundedly many conjugates of the generators. We call this property bounded conjugacy width. We also show how bounded conjugacy width relates to other algebraic properties of groups and apply these results to study the palindromic width of some branch groups.