A rigidity theorem for the mapping class group action on the space of unmeasured foliations on a surface

Author:
Athanase Papadopoulos

Journal:
Proc. Amer. Math. Soc. **136** (2008), 4453-4460

MSC (2000):
Primary 57M60; Secondary 57M50, 20F65, 57R30

Published electronically:
June 17, 2008

MathSciNet review:
2431062

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Abstract: Let 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 be the space of equivalence classes of measured foliations of compact support on and let be the quotient space of 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 of acts by homeomorphisms on . We show that the restriction of the action of the whole homeomorphism group of on some dense subset of coincides with the action of on that subset. More precisely, let be the natural image in of the set of homotopy classes of not necessarily connected essential disjoint and pairwise non-homotopic simple closed curves on . The set is dense in , it is invariant by the action of on , and the restriction of the action of on is faithful. We prove that the restriction of the action on of the group coincides with the action of on that subspace.

**1.***Travaux de Thurston sur les surfaces*, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR**568308****2.**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****3.**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****4.**E. Klarreich,*The boundary at infinity of the curve complex and the relative Teichmüller space*, preprint.**5.**Mustafa Korkmaz,*Automorphisms of complexes of curves on punctured spheres and on punctured tori*, Topology Appl.**95**(1999), no. 2, 85–111. MR**1696431**, 10.1016/S0166-8641(97)00278-2**6.**Feng Luo,*Automorphisms of the complex of curves*, Topology**39**(2000), no. 2, 283–298. MR**1722024**, 10.1016/S0040-9383(99)00008-7**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**, 10.1007/s002220050343**8.**Athanase Papadopoulos,*Foliations of surfaces and semi-Markovian subsets of subshifts of finite type*, Topology Appl.**66**(1995), no. 2, 171–183. MR**1358819**, 10.1016/0166-8641(95)00020-H**9.**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**, 10.1090/S0273-0979-1988-15685-6

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Additional Information

**Athanase Papadopoulos**

Affiliation:
Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg cedex, France

Email:
papadopoulos@math.u-strasbg.fr

DOI:
http://dx.doi.org/10.1090/S0002-9939-08-09433-1

Keywords:
Measured foliation,
unmeasured foliation space,
curve complex,
mapping class group.

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

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.