Bounded conjugacy classes, commutators, and approximate subgroups

TL;DR

This paper extends classical results on finite conjugacy classes to approximate subgroups, showing that bounded conjugacy class sizes imply boundedness of certain commutator subgroups in groups.

Contribution

It introduces bounds on the order of commutator subgroups related to approximate subgroups with bounded conjugacy class sizes, generalizing Neumann's theorem.

Findings

Bounded conjugacy class sizes imply finite bounded commutator subgroups.

Results apply to approximate subgroups, not just finite groups.

Provides explicit bounds depending on approximation parameters.

Abstract

Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing the element $g$. A theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup $G'$ is finite. We establish the following results. Let $K,n$ be positive integers and $G$ a group having a $K$-approximate subgroup $A$. If $|a^G|\leq n$ for each $a\in A$, then the commutator subgroup of $\langle A^G\rangle$ has finite $(K,n)$-bounded order. If $|[g,a]^G|\leq n$ for all $g\in G$ and $a\in A$, then the commutator subgroup of $[G,A]$ has finite $(K,n)$-bounded order.