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 $g<sup>n=id$. We first prove that a finitely generated periodic group acting on the 2-sphere $\SS<sup>2$ by $C<sup>1$-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.