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, New York, 1998.

[2]

M. Burger, A. Iozzi and A. Wienhard, Surface group representations with maximal Toledo invariant, Ann. of Math. (2), 172 (2010), 517-566. doi: 10.4007/annals.2010.172.517.

[3]

D. Calegari, "scl," MSJ Memoirs, 20, Mathematical Society of Japan, Tokyo, 2009.

[4]

D. Calegari and M. H. Freedman, Distortion in transformation groups, With an appendix by Yves de Cornulier, Geom. Topol., 10 (2006), 267-293. doi: 10.2140/gt.2006.10.267.

[5]

J. Franks, Rotation vectors and fixed points of area-preserving surface diffeomorphisms, Trans. Amer. Math. Soc., 348 (1996), 2637-2662. doi: 10.1090/S0002-9947-96-01502-4.

[6]

J. Franks and M. Handel, Distortion elements in group actions on surfaces, Duke Math. J., 131 (2006), 441-468. doi: 10.1215/S0012-7094-06-13132-0.

[7]

Ś. R. Gal and J. Kędra, A cocycle on the group of symplectic diffeomorphisms, Advances in Geometry, 11 (2011), 73-88. doi: 10.1515/ADVGEOM.2010.039.

[8]

Ś. R. Gal and J. Kędra, A two-cocycle on the group of symplectic diffeomorphisms,, Math. Z., (). 

[9]

J.-M. Gambaudo and É. Ghys, Enlacements asymptotiques, Topology, 36 (1997), 1355-1379. doi: 10.1016/S0040-9383(97)00001-3.

[10]

J.-M. Gambaudo and É. Ghys, Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Systems, 24 (2004), 1591-1617. doi: 10.1017/S0143385703000737.

[11]

É. Ghys, Groups acting on the circle, Enseign. Math. (2), 47 (2001), 329-407.

[12]

M. Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math., 56 (1982), 5-99.

[13]

R. S. Ismagilov, M. Losik and P. W. Michor, A 2-cocycle on a symplectomorphism group, Mosc. Math. J., 6 (2006), 307-315, 407.

[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), 5-53.

[15]

N. Monod, "Continuous Bounded Cohomology of Locally Compact Groups," Lecture Notes in Mathematics, 1758, Springer-Verlag, Berlin, 2001.

[16]

L. Polterovich, Growth of maps, distortion in groups and symplectic geometry, Invent. Math., 150 (2002), 655-686. doi: 10.1007/s00222-002-0251-x.

show all references

References:
[1]

V. I. Arnold and B. A. Khesin, "Topological Methods in Hydrodynamics," Applied Mathematical Sciences, 125, Springer-Verlag, New York, 1998.

[2]

M. Burger, A. Iozzi and A. Wienhard, Surface group representations with maximal Toledo invariant, Ann. of Math. (2), 172 (2010), 517-566. doi: 10.4007/annals.2010.172.517.

[3]

D. Calegari, "scl," MSJ Memoirs, 20, Mathematical Society of Japan, Tokyo, 2009.

[4]

D. Calegari and M. H. Freedman, Distortion in transformation groups, With an appendix by Yves de Cornulier, Geom. Topol., 10 (2006), 267-293. doi: 10.2140/gt.2006.10.267.

[5]

J. Franks, Rotation vectors and fixed points of area-preserving surface diffeomorphisms, Trans. Amer. Math. Soc., 348 (1996), 2637-2662. doi: 10.1090/S0002-9947-96-01502-4.

[6]

J. Franks and M. Handel, Distortion elements in group actions on surfaces, Duke Math. J., 131 (2006), 441-468. doi: 10.1215/S0012-7094-06-13132-0.

[7]

Ś. R. Gal and J. Kędra, A cocycle on the group of symplectic diffeomorphisms, Advances in Geometry, 11 (2011), 73-88. doi: 10.1515/ADVGEOM.2010.039.

[8]

Ś. R. Gal and J. Kędra, A two-cocycle on the group of symplectic diffeomorphisms,, Math. Z., (). 

[9]

J.-M. Gambaudo and É. Ghys, Enlacements asymptotiques, Topology, 36 (1997), 1355-1379. doi: 10.1016/S0040-9383(97)00001-3.

[10]

J.-M. Gambaudo and É. Ghys, Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Systems, 24 (2004), 1591-1617. doi: 10.1017/S0143385703000737.

[11]

É. Ghys, Groups acting on the circle, Enseign. Math. (2), 47 (2001), 329-407.

[12]

M. Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math., 56 (1982), 5-99.

[13]

R. S. Ismagilov, M. Losik and P. W. Michor, A 2-cocycle on a symplectomorphism group, Mosc. Math. J., 6 (2006), 307-315, 407.

[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), 5-53.

[15]

N. Monod, "Continuous Bounded Cohomology of Locally Compact Groups," Lecture Notes in Mathematics, 1758, Springer-Verlag, Berlin, 2001.

[16]

L. Polterovich, Growth of maps, distortion in groups and symplectic geometry, Invent. Math., 150 (2002), 655-686. doi: 10.1007/s00222-002-0251-x.

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