A rigidity theorem for the mapping class group action on the space of unmeasured foliations on a surface
HTML articles powered by AMS MathViewer
- by Athanase Papadopoulos PDF
- Proc. Amer. Math. Soc. 136 (2008), 4453-4460 Request permission
Abstract:
Let $S$ be a connected oriented surface of finite type which is not a sphere with at most four punctures, a torus with at most two punctures, or a closed surface of genus two. Let $\mathcal {MF}$ be the space of equivalence classes of measured foliations of compact support on $S$ and let $\mathcal {UMF}$ be the quotient space of $\mathcal {MF}$ obtained by identifying two equivalence classes whenever they can be represented by topologically equivalent foliations, that is, forgetting the transverse measure. The extended mapping class group $\Gamma ^*$ of $S$ acts by homeomorphisms on $\mathcal {UMF}$. We show that the restriction of the action of the whole homeomorphism group of $\mathcal {UMF}$ on some dense subset of $\mathcal {UMF}$ coincides with the action of $\Gamma ^*$ on that subset. More precisely, let $\mathcal {D}$ be the natural image in $\mathcal {UMF}$ of the set of homotopy classes of not necessarily connected essential disjoint and pairwise non-homotopic simple closed curves on $S$. The set $\mathcal {D}$ is dense in $\mathcal {UMF}$, it is invariant by the action of $\Gamma ^*$ on $\mathcal {UMF}$, and the restriction of the action of $\Gamma ^*$ on $\mathcal {D}$ is faithful. We prove that the restriction of the action on $\mathcal {D}$ of the group $\mathrm {Homeo}(\mathcal {UMF})$ coincides with the action of $\Gamma ^*$ on that subspace.References
- Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR 568308
- W. J. Harvey, Geometric structure of surface mapping class groups, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 255–269. MR 564431
- Nikolai V. Ivanov, Automorphisms of complexes of curves and of Teichmüller spaces, Progress in knot theory and related topics, Travaux en Cours, vol. 56, Hermann, Paris, 1997, pp. 113–120. MR 1603146
- E. Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space, preprint.
- Mustafa Korkmaz, Automorphisms of complexes of curves on punctured spheres and on punctured tori, Topology Appl. 95 (1999), no. 2, 85–111. MR 1696431, DOI 10.1016/S0166-8641(97)00278-2
- Feng Luo, Automorphisms of the complex of curves, Topology 39 (2000), no. 2, 283–298. MR 1722024, DOI 10.1016/S0040-9383(99)00008-7
- Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103–149. MR 1714338, DOI 10.1007/s002220050343
- Athanase Papadopoulos, Foliations of surfaces and semi-Markovian subsets of subshifts of finite type, Topology Appl. 66 (1995), no. 2, 171–183. MR 1358819, DOI 10.1016/0166-8641(95)00020-H
- William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6
Additional Information
- Athanase Papadopoulos
- Affiliation: Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg cedex, France
- MR Author ID: 135835
- Email: papadopoulos@math.u-strasbg.fr
- Received by editor(s): June 11, 2007
- Received by editor(s) in revised form: October 27, 2007
- Published electronically: June 17, 2008
- Communicated by: Alexander N. Dranishnikov
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 4453-4460
- MSC (2000): Primary 57M60; Secondary 57M50, 20F65, 57R30
- DOI: https://doi.org/10.1090/S0002-9939-08-09433-1
- MathSciNet review: 2431062