The growth of abelian sections

TL;DR

This paper investigates the growth of abelian sections in groups, establishing polynomial bounds for groups without abelian composition factors and deriving implications for their representation growth.

Contribution

It introduces a new function to measure abelian sections and provides sharp bounds for groups lacking abelian composition factors, linking to their representation growth.

Findings

ab_n(G) is polynomially bounded for certain groups

sharp upper bounds for representation growth are established

connects abelian section growth to representation theory

Abstract

Given an abstract group $G$, we study the function $ab_n(G) := \sup_{|G:H| \leq n} |H/[H,H]|$. If $G$ has no abelian composition factors, then $ab_n(G)$ is bounded by a polynomial: as a consequence, we find a sharp upper bound for the representation growth of these groups.