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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 $ 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.

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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

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.

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