Area preserving group actions on surfaces

Abstract

Suppose $\mathcal{G}$ is an almost simple group containing a subgroup isomorphic to the three-dimensional integer Heisenberg group. For example any finite index subgroup of $SL\left(3,\mathbb{Z}\right)$ is such a group. The main result of this paper is that every action of $\mathcal{G}$ on a closed oriented surface by area preserving diffeomorphisms factors through a finite group.