#### Volume 18, issue 1 (2014)

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Distortion elements for surface homeomorphisms

### Emmanuel Militon

Geometry & Topology 18 (2014) 521–614
##### 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 $\stackrel{̃}{f}$ a lift of $f$ to the universal cover $\stackrel{̃}{S}$ of $S$. In this article, the following result is proved: If there exists a fundamental domain $D$ of the covering $\stackrel{̃}{S}\to S$ such that

$\underset{n\to +\infty }{lim}\frac{1}{n}\phantom{\rule{0.3em}{0ex}}{d}_{n}log\left({d}_{n}\right)=0,$

where ${d}_{n}$ is the diameter of ${\stackrel{̃}{f}}^{n}\left(D\right)$, then the homeomorphism $f$ is a distortion element of the group ${Homeo}_{0}\left(S\right)$.

##### Keywords
homeomorphism, surface, group, distortion
Primary: 37C85