Distortion elements for surface homeomorphisms

Abstract

Let $S$ be a compact orientable surface and $f$ be an element of the group ${Homeo}_0\left(S\right)$ of homeomorphisms of $S$ isotopic to the identity. Denote by $\tilde{f}$ a lift of $f$ to the universal cover $\tilde{S}$ of $S$. In this article, the following result is proved: If there exists a fundamental domain $D$ of the covering $\tilde{S}\to S$ such that

$$\underset{n\to +\infty }{lim}\frac{1}{n}{d}_{n}log\left(d_n\right)=0,$$

where $d_n$ is the diameter of ${\tilde{f}}^n\left(D\right)$, then the homeomorphism $f$ is a distortion element of the group ${Homeo}_0\left(S\right)$.