### 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∈ 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.

We investigate the distortion in Homeo(X,a). Let g∈ 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.

Original language | English |
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Pages (from-to) | 609-622 |

Number of pages | 14 |

Journal | Journal of Modern Dynamics |

Volume | 5 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jul 2011 |

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## Cite this

Gal, S., & Kedra, J. (2011). On distortion in groups of homeomorphisms.

*Journal of Modern Dynamics*,*5*(3), 609-622. https://doi.org/10.3934/jmd.2011.5.609