Groups of small period growth

TL;DR

This paper constructs finitely generated groups with very slow growth in the maximum order of elements relative to their word length, addressing a question about group lawlessness growth and linking to the restricted Burnside problem.

Contribution

It introduces new examples of groups with small period growth, providing insights into the lawlessness growth and its relation to the restricted Burnside problem.

Findings

Existence of finitely generated groups with slow period growth

Answers to Bradford's question on lawlessness growth

Connections to the restricted Burnside problem

Abstract

We construct finitely generated groups of small period growth, i.e. groups where the maximum order of an element of word length $n$ grows very slowly in $n$. This answers a question of Bradford related to the lawlessness growth of groups and is connected to an approximative version of the restricted Burnside problem.