Geometry & Topology

Distortion elements for surface homeomorphisms

Emmanuel Militon

Full-text: Open access


Let S be a compact orientable surface and f be an element of the group Homeo0(S) of homeomorphisms of S isotopic to the identity. Denote by f̃ a lift of f to the universal cover S̃ of S. In this article, the following result is proved: If there exists a fundamental domain D of the covering S̃S such that

lim n + 1 n d n log ( d n ) = 0 ,

where dn is the diameter of f̃n(D), then the homeomorphism f is a distortion element of the group Homeo0(S).

Article information

Geom. Topol., Volume 18, Number 1 (2014), 521-614.

Received: 2 November 2012
Revised: 27 June 2013
Accepted: 28 July 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37C85: Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx]

homeomorphism surface group distortion


Militon, Emmanuel. Distortion elements for surface homeomorphisms. Geom. Topol. 18 (2014), no. 1, 521--614. doi:10.2140/gt.2014.18.521.

Export citation


  • R D Anderson, The algebraic simplicity of certain groups of homeomorphisms, Amer. J. Math. 80 (1958) 955–963
  • A Avila, Distortion elements in $\mathrm{Diff}^{\infty}(\mathbb{R} / \mathbb{Z})$
  • F Béguin, S Crovisier, F Le Roux, A Patou, Pseudo-rotations of the closed annulus: Variation on a theorem of J Kwapisz, Nonlinearity 17 (2004) 1427–1453
  • A Bounemoura, Simplicité des groupes de transformations de surfaces, Ensaios Matemáticos 14, Soc. Bras. Mat., Rio de Janeiro (2008)
  • D Calegari, M H Freedman, Distortion in transformation groups, Geom. Topol. 10 (2006) 267–293
  • B Farb, A Lubotzky, Y Minsky, Rank-$1$ phenomena for mapping class groups, Duke Math. J. 106 (2001) 581–597
  • A Fathi, M R Herman, Existence de difféomorphismes minimaux, from: “Dynamical systems, Vol. I: Warsaw”, Astérisque 49, Soc. Math. France, Paris (1977) 37–59
  • G M Fisher, On the group of all homeomorphisms of a manifold, Trans. Amer. Math. Soc. 97 (1960) 193–212
  • J Franks, M Handel, Distortion elements in group actions on surfaces, Duke Math. J. 131 (2006) 441–468
  • É Ghys, Groups acting on the circle, Enseign. Math. 47 (2001) 329–407
  • M-E Hamstrom, Homotopy groups of the space of homeomorphisms on a $2$–manifold, Illinois J. Math. 10 (1966) 563–573
  • P de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press (2000)
  • M R Herman, Construction of some curious diffeomorphisms of the Riemann sphere, J. London Math. Soc. 34 (1986) 375–384
  • T Jäger, The concept of bounded mean motion for toral homeomorphisms, Dyn. Syst. 24 (2009) 277–297
  • A Katok, B Hasselblatt, Introduction to the modern theory of dynamical systems, Encyc. Math. Appl. 54, Cambridge Univ. Press (1995)
  • R C Kirby, Stable homeomorphisms and the annulus conjecture, Ann. of Math. 89 (1969) 575–582
  • A Koropecki, F A Tal, Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion
  • P Le Calvez, J-C Yoccoz, Un théorème d'indice pour les homéomorphismes du plan au voisinage d'un point fixe, Ann. of Math. 146 (1997) 241–293
  • R C Lyndon, P E Schupp, Combinatorial group theory, Ergeb. Math. Grenzgeb. 89, Springer, Berlin (1977)
  • E Militon, Éléments de distorsion de ${\rm Diff}\sp \infty\sb 0(M)$, Bull. Soc. Math. France 141 (2013) 35–46
  • M Misiurewicz, K Ziemian, Rotation sets for maps of tori, J. London Math. Soc. 40 (1989) 490–506
  • C F Novak, Discontinuity-growth of interval-exchange maps, J. Mod. Dyn. 3 (2009) 379–405
  • L Polterovich, Growth of maps, distortion in groups and symplectic geometry, Invent. Math. 150 (2002) 655–686
  • F Quinn, Ends of maps, III: Dimensions $4$ and $5$, J. Differential Geom. 17 (1982) 503–521