Distortion elements for surface homeomorphisms

Abstract

Let S be a compact orientable surface and f be an element of the group Homeo_{0}(S) of homeomorphisms of S isotopic to the identity. Denote by F a lift of f to the universal cover of S. In this article, the following result is proved: if there exists a fundamental domain D of the universal cover of S such that the sequence (d_{n}log(d_{n})/n) converges to 0 where d_{n} is the diameter of F^{n}(D), then the homeomorphism f is a distortion element of the group Homeo_{0}(S).