Journal of Modern Dynamics (JMD)

On distortion in groups of homeomorphisms

Pages: 609 - 622, Issue 3, July 2011      doi:10.3934/jmd.2011.5.609

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Światosław R. Gal - Instytut Matematyczny Uniwersytetu Wrocławskiego, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland (email)
Jarek Kędra - University of Aberdeen, Institute of Mathematics, Fraser Noble Building, Aberdeen AB24 3UE, Scotland (email)

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

Received: May 2011;      Revised: September 2011;      Available Online: November 2011.