Distortion elements for surface homeomorphisms

Militon, Emmanuel

doi:10.2140/gt.2014.18.521

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Distortion elements for surface homeomorphisms

Emmanuel Militon

Geometry & Topology 18 (2014) 521–614

DOI: 10.2140/gt.2014.18.521

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 $\tilde{f}$ 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

lim_{n \to + \infty} \frac{1}{n} d_{n} log (d_{n}) = 0,

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

Keywords

homeomorphism, surface, group, distortion

Mathematical Subject Classification 2010

Primary: 37C85

References

Publication

Received: 2 November 2012

Revised: 27 June 2013

Accepted: 28 July 2013

Published: 29 January 2014

Proposed: Leonid Polterovich

Seconded: Danny Calegari, Jean-Pierre Otal

Authors

Emmanuel Militon
Centre de Mathématiques Laurent Schwartz École Polytechnique 91128 Palaiseau Cedex France
http://www.math.u-psud.fr/~militon

Volume 18, issue 1 (2014)