Burnside Problem for Groups of Homeomorphisms of Compact Surfaces

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 ${\mathbb S}^2$ by $C^1$-diffeomorphisms with a finite orbit, is finite and $C^1$-conjugate to a subgroup of $\mathrm {O}(3,{\mathbb R})$. This result is obtained by proving the more general statement: a finitely generated periodic group acting on any compact manifold by $C^1$-diffeomorphisms with a finite orbit, is finite. We use it for proving that a countable 2-group of spherical diffeomorphisms with bounded orders is finite. This gives a negative partial answer to a question posed by D. Fisher. 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 is finite.