The Burnside problem for Diffω(S2)

Abstract

A group $G$ is periodic of bounded exponent if there exists $k\in \mathbb{N}$ such that every element of $G$ has order at most $k$. We show that every finitely generated periodic group of bounded exponent $G<{Diff}_{\omega }\left({\mathbb{S}}^2\right)$ is finite, where ${Diff}_{\omega }\left({\mathbb{S}}^2\right)$ denotes the group of diffeomorphisms of ${\mathbb{S}}^2$ that preserve an area form $\omega$.