Burnside problem for groups of homeomorphisms of compact surfaces

TL;DR

This paper proves that finitely generated periodic groups of homeomorphisms on certain compact surfaces are finite, extending classical results and providing new classifications for groups acting on surfaces like the sphere and other orientable surfaces.

Contribution

It establishes finiteness of finitely generated periodic groups acting on various compact surfaces, including the sphere and other orientable surfaces, with new results for groups on the sphere.

Findings

Finitely generated periodic groups on the 2-sphere are finite and conjugate to subgroups of O(3)

Groups with bounded orders acting on the sphere are finite

Groups on other orientable surfaces are finite

Abstract

A group $\Gamma$ is said to be periodic if for any $g$ in $\Gamma$ there is a positive integer $n$ with $g^n=id$. We first prove that a finitely generated periodic group acting on the 2-sphere $\SS^2$ by $C^1$-diffeomorphisms with a finite orbit, is finite and conjugate to a subgroup of $\mathrm{O}(3,\R)$ and we use it for proving that a finitely generated periodic group of spherical diffeomorphisms with even bounded orders is finite. Finally, we show that a finitely generated periodic group of homeomorphisms of any orientable compact surface other than the 2-sphere or the 2-torus (which is the purpose of a previous paper of the authors) is finite.