Burnside problem for measure preserving groups of toral homeomorphisms and for 2-groups of toral homeomorphisms

TL;DR

This paper proves that finitely generated periodic groups of measure-preserving toral homeomorphisms are finite, and explores their structure, showing abelianity and free action under certain conditions, with a focus on 2-groups.

Contribution

It establishes finiteness of finitely generated periodic measure-preserving groups of toral homeomorphisms and characterizes their structure, especially for groups isotopic to the identity and 2-groups.

Findings

Finitely generated periodic measure-preserving groups on the 2-torus are finite.

Groups isotopic to the identity are abelian and act freely.

Every finitely generated 2-group of toral homeomorphisms is finite.

Abstract

A group $G$ is said to be periodic if for any $g\in G$ there exists a positive integer $n$ with $g^n=id$. We prove that a finitely generated periodic group of homeomorphisms on the 2-torus that preserves a measure $\mu$ is finite. Moreover if the group consists in homeomorphisms isotopic to the identity, then it is abelian and acts freely on $\mathbb{T}^2$. In the Appendix, we show that every finitely generated 2-group of toral homeomorphisms is finite.