A note on the Burnside problem for homeomorphism groups of manifolds

TL;DR

This paper investigates the Burnside problem for homeomorphism groups of manifolds, establishing conditions under which finitely generated periodic subgroups are finite, and extends previous results to non-orientable surfaces and the circle.

Contribution

It proves the torsion-free property of the identity component for certain surfaces and extends finiteness results of periodic subgroups to non-orientable surfaces and the circle.

Findings

Identity component of homeomorphism group is torsion-free except for specific surfaces.

Finitely generated periodic subgroups are finite for most surfaces.

Every finitely generated periodic subgroup of circle homeomorphisms is finite and cyclic.

Abstract

This note studies the Burnside problem for homeomorphism groups of compact connected manifolds. For surfaces, we prove that the identity component of the homeomorphism group is torsion-free precisely when the surface is not the sphere, torus, projective plane, or Klein bottle. An extension argument based on the Tits alternative for mapping class groups then implies that every finitely generated periodic subgroup of the full homeomorphism group is finite for all surfaces outside this exceptional list, recovering and extending a theorem of Guelman and Liousse to non-orientable surfaces. For the circle, we prove that every finitely generated periodic subgroup of its homeomorphism group is finite and cyclic. We close with remarks on manifolds with boundary and open questions on the Burnside problem for hyperbolic three-manifolds and doubled handlebodies.