On distortion in groups of homeomorphisms

Światosław R. Gal; Jarek Kędra

doi:10.3934/jmd.2011.5.609

Article Contents

2011, Volume 5, Issue 3: 609-622. Doi: 10.3934/jmd.2011.5.609

This issue Previous Article Bernoulli equilibrium states for surface diffeomorphisms Next Article

On distortion in groups of homeomorphisms

Światosław R. Gal^1, and
Jarek Kędra^2,

1.
Instytut Matematyczny Uniwersytetu Wrocławskiego, pl. Grunwaldzki 2/4, 50-384 Wrocław
2.
University of Aberdeen, Institute of Mathematics, Fraser Noble Building, Aberdeen AB24 3UE

Received: April 30, 2011

Revised: August 31, 2011

Published: October 2011

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Abstract

Let $X$ be a path-connected topological space admitting a universal cover. Let Homeo$(X, a)$ denote the group of homeomorphisms of $X$ preserving a degree one cohomology class $ a$.
We investigate the distortion in Homeo$(X, a)$. Let $g\in$ Homeo$(X, a)$. We define a Nielsen-type equivalence relation on the space of $g$-invariant Borel probability measures on $X$ and prove that if a homeomorphism $g$ admits two nonequivalent invariant measures then it is undistorted. We also define a local rotation number of a homeomorphism generalizing the notion of the rotation of a homeomorphism of the circle. Then we prove that a homeomorphism is undistorted if its rotation number is nonconstant.

Keywords:

Distortion in groups; rotation number; groups of homeomorphisms; invariant measures.

Mathematics Subject Classification: 54h20; 20f, 37b, 57s.

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