# American Institute of Mathematical Sciences

July  2011, 5(3): 609-622. doi: 10.3934/jmd.2011.5.609

## On distortion in groups of homeomorphisms

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

Received  May 2011 Revised  September 2011 Published  November 2011

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.
Citation: Światosław R. Gal, Jarek Kędra. On distortion in groups of homeomorphisms. Journal of Modern Dynamics, 2011, 5 (3) : 609-622. doi: 10.3934/jmd.2011.5.609
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##### References:
 [1] Or Landesberg. Horospherically invariant measures and finitely generated Kleinian groups. Journal of Modern Dynamics, 2021, 17: 337-352. doi: 10.3934/jmd.2021012 [2] Richard Sharp. Distortion and entropy for automorphisms of free groups. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 347-363. doi: 10.3934/dcds.2010.26.347 [3] Javier Pérez Álvarez. Invariant structures on Lie groups. Journal of Geometric Mechanics, 2020, 12 (2) : 141-148. doi: 10.3934/jgm.2020007 [4] Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517 [5] Paweł G. Walczak. Expansion growth, entropy and invariant measures of distal groups and pseudogroups of homeo- and diffeomorphisms. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4731-4742. doi: 10.3934/dcds.2013.33.4731 [6] Kanat Abdukhalikov. On codes over rings invariant under affine groups. Advances in Mathematics of Communications, 2013, 7 (3) : 253-265. doi: 10.3934/amc.2013.7.253 [7] Firas Hindeleh, Gerard Thompson. Killing's equations for invariant metrics on Lie groups. Journal of Geometric Mechanics, 2011, 3 (3) : 323-335. doi: 10.3934/jgm.2011.3.323 [8] Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141 [9] Fawwaz Batayneh, Cecilia González-Tokman. On the number of invariant measures for random expanding maps in higher dimensions. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5887-5914. doi: 10.3934/dcds.2021100 [10] Dennis I. Barrett, Rory Biggs, Claudiu C. Remsing, Olga Rossi. Invariant nonholonomic Riemannian structures on three-dimensional Lie groups. Journal of Geometric Mechanics, 2016, 8 (2) : 139-167. doi: 10.3934/jgm.2016001 [11] Michel Laurent, Arnaldo Nogueira. Rotation number of contracted rotations. Journal of Modern Dynamics, 2018, 12: 175-191. doi: 10.3934/jmd.2018007 [12] Deissy M. S. Castelblanco. Restrictions on rotation sets for commuting torus homeomorphisms. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5257-5266. doi: 10.3934/dcds.2016030 [13] Abdelhamid Adouani, Habib Marzougui. Computation of rotation numbers for a class of PL-circle homeomorphisms. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3399-3419. doi: 10.3934/dcds.2012.32.3399 [14] Ludovic Rifford. Ricci curvatures in Carnot groups. Mathematical Control & Related Fields, 2013, 3 (4) : 467-487. doi: 10.3934/mcrf.2013.3.467 [15] Eduard Duryev, Charles Fougeron, Selim Ghazouani. Dilation surfaces and their Veech groups. Journal of Modern Dynamics, 2019, 14: 121-151. doi: 10.3934/jmd.2019005 [16] Sergei V. Ivanov. On aspherical presentations of groups. Electronic Research Announcements, 1998, 4: 109-114. [17] Benjamin Weiss. Entropy and actions of sofic groups. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3375-3383. doi: 10.3934/dcdsb.2015.20.3375 [18] Neal Koblitz, Alfred Menezes. Another look at generic groups. Advances in Mathematics of Communications, 2007, 1 (1) : 13-28. doi: 10.3934/amc.2007.1.13 [19] Robert McOwen, Peter Topalov. Groups of asymptotic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6331-6377. doi: 10.3934/dcds.2016075 [20] Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725

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