# 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
##### References:
 [1] V. I. Arnold and B. A. Khesin, "Topological Methods in Hydrodynamics," Applied Mathematical Sciences, 125,, Springer-Verlag, (1998). Google Scholar [2] M. Burger, A. Iozzi and A. Wienhard, Surface group representations with maximal Toledo invariant,, Ann. of Math. (2), 172 (2010), 517. doi: 10.4007/annals.2010.172.517. Google Scholar [3] D. Calegari, "scl," MSJ Memoirs, 20,, Mathematical Society of Japan, (2009). Google Scholar [4] D. Calegari and M. H. Freedman, Distortion in transformation groups,, With an appendix by Yves de Cornulier, 10 (2006), 267. doi: 10.2140/gt.2006.10.267. Google Scholar [5] J. Franks, Rotation vectors and fixed points of area-preserving surface diffeomorphisms,, Trans. Amer. Math. Soc., 348 (1996), 2637. doi: 10.1090/S0002-9947-96-01502-4. Google Scholar [6] J. Franks and M. Handel, Distortion elements in group actions on surfaces,, Duke Math. J., 131 (2006), 441. doi: 10.1215/S0012-7094-06-13132-0. Google Scholar [7] Ś. R. Gal and J. Kędra, A cocycle on the group of symplectic diffeomorphisms,, Advances in Geometry, 11 (2011), 73. doi: 10.1515/ADVGEOM.2010.039. Google Scholar [8] Ś. R. Gal and J. Kędra, A two-cocycle on the group of symplectic diffeomorphisms,, Math. Z., (). Google Scholar [9] J.-M. Gambaudo and É. Ghys, Enlacements asymptotiques,, Topology, 36 (1997), 1355. doi: 10.1016/S0040-9383(97)00001-3. Google Scholar [10] J.-M. Gambaudo and É. Ghys, Commutators and diffeomorphisms of surfaces,, Ergodic Theory Dynam. Systems, 24 (2004), 1591. doi: 10.1017/S0143385703000737. Google Scholar [11] É. Ghys, Groups acting on the circle,, Enseign. Math. (2), 47 (2001), 329. Google Scholar [12] M. Gromov, Volume and bounded cohomology,, Inst. Hautes Études Sci. Publ. Math., 56 (1982), 5. Google Scholar [13] R. S. Ismagilov, M. Losik and P. W. Michor, A 2-cocycle on a symplectomorphism group,, Mosc. Math. J., 6 (2006), 307. Google Scholar [14] A. Lubotzky, S. Mozes and M. S. Raghunathan, The word and Riemannian metrics on lattices of semisimple groups,, Inst. Hautes Études Sci. Publ. Math. No., 91 (2000), 5. Google Scholar [15] N. Monod, "Continuous Bounded Cohomology of Locally Compact Groups," Lecture Notes in Mathematics, 1758,, Springer-Verlag, (2001). Google Scholar [16] L. Polterovich, Growth of maps, distortion in groups and symplectic geometry,, Invent. Math., 150 (2002), 655. doi: 10.1007/s00222-002-0251-x. Google Scholar

show all references

##### References:
 [1] V. I. Arnold and B. A. Khesin, "Topological Methods in Hydrodynamics," Applied Mathematical Sciences, 125,, Springer-Verlag, (1998). Google Scholar [2] M. Burger, A. Iozzi and A. Wienhard, Surface group representations with maximal Toledo invariant,, Ann. of Math. (2), 172 (2010), 517. doi: 10.4007/annals.2010.172.517. Google Scholar [3] D. Calegari, "scl," MSJ Memoirs, 20,, Mathematical Society of Japan, (2009). Google Scholar [4] D. Calegari and M. H. Freedman, Distortion in transformation groups,, With an appendix by Yves de Cornulier, 10 (2006), 267. doi: 10.2140/gt.2006.10.267. Google Scholar [5] J. Franks, Rotation vectors and fixed points of area-preserving surface diffeomorphisms,, Trans. Amer. Math. Soc., 348 (1996), 2637. doi: 10.1090/S0002-9947-96-01502-4. Google Scholar [6] J. Franks and M. Handel, Distortion elements in group actions on surfaces,, Duke Math. J., 131 (2006), 441. doi: 10.1215/S0012-7094-06-13132-0. Google Scholar [7] Ś. R. Gal and J. Kędra, A cocycle on the group of symplectic diffeomorphisms,, Advances in Geometry, 11 (2011), 73. doi: 10.1515/ADVGEOM.2010.039. Google Scholar [8] Ś. R. Gal and J. Kędra, A two-cocycle on the group of symplectic diffeomorphisms,, Math. Z., (). Google Scholar [9] J.-M. Gambaudo and É. Ghys, Enlacements asymptotiques,, Topology, 36 (1997), 1355. doi: 10.1016/S0040-9383(97)00001-3. Google Scholar [10] J.-M. Gambaudo and É. Ghys, Commutators and diffeomorphisms of surfaces,, Ergodic Theory Dynam. Systems, 24 (2004), 1591. doi: 10.1017/S0143385703000737. Google Scholar [11] É. Ghys, Groups acting on the circle,, Enseign. Math. (2), 47 (2001), 329. Google Scholar [12] M. Gromov, Volume and bounded cohomology,, Inst. Hautes Études Sci. Publ. Math., 56 (1982), 5. Google Scholar [13] R. S. Ismagilov, M. Losik and P. W. Michor, A 2-cocycle on a symplectomorphism group,, Mosc. Math. J., 6 (2006), 307. Google Scholar [14] A. Lubotzky, S. Mozes and M. S. Raghunathan, The word and Riemannian metrics on lattices of semisimple groups,, Inst. Hautes Études Sci. Publ. Math. No., 91 (2000), 5. Google Scholar [15] N. Monod, "Continuous Bounded Cohomology of Locally Compact Groups," Lecture Notes in Mathematics, 1758,, Springer-Verlag, (2001). Google Scholar [16] L. Polterovich, Growth of maps, distortion in groups and symplectic geometry,, Invent. Math., 150 (2002), 655. doi: 10.1007/s00222-002-0251-x. Google Scholar
 [1] Richard Sharp. Distortion and entropy for automorphisms of free groups. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 347-363. doi: 10.3934/dcds.2010.26.347 [2] Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517 [3] Paweł G. Walczak. Expansion growth, entropy and invariant measures of distal groups and pseudogroups of homeo- and diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4731-4742. doi: 10.3934/dcds.2013.33.4731 [4] 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 [5] 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 [6] 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 [7] 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 [8] Michel Laurent, Arnaldo Nogueira. Rotation number of contracted rotations. Journal of Modern Dynamics, 2018, 12: 175-191. doi: 10.3934/jmd.2018007 [9] Deissy M. S. Castelblanco. Restrictions on rotation sets for commuting torus homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5257-5266. doi: 10.3934/dcds.2016030 [10] Abdelhamid Adouani, Habib Marzougui. Computation of rotation numbers for a class of PL-circle homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3399-3419. doi: 10.3934/dcds.2012.32.3399 [11] Ludovic Rifford. Ricci curvatures in Carnot groups. Mathematical Control & Related Fields, 2013, 3 (4) : 467-487. doi: 10.3934/mcrf.2013.3.467 [12] 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 [13] Sergei V. Ivanov. On aspherical presentations of groups. Electronic Research Announcements, 1998, 4: 109-114. [14] 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 [15] Robert McOwen, Peter Topalov. Groups of asymptotic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6331-6377. doi: 10.3934/dcds.2016075 [16] Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725 [17] Elon Lindenstrauss. Pointwise theorems for amenable groups. Electronic Research Announcements, 1999, 5: 82-90. [18] Hans Ulrich Besche, Bettina Eick and E. A. O'Brien. The groups of order at most 2000. Electronic Research Announcements, 2001, 7: 1-4. [19] Marc Peigné. On some exotic Schottky groups. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 559-579. doi: 10.3934/dcds.2011.31.559 [20] Uri Bader, Alex Furman. Boundaries, Weyl groups, and Superrigidity. Electronic Research Announcements, 2012, 19: 41-48. doi: 10.3934/era.2012.19.41

2018 Impact Factor: 0.295