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), 44534460
MSC (2000):
Primary 57M60; Secondary 57M50, 20F65, 57R30
Published electronically:
June 17, 2008
MathSciNet review:
2431062
Fulltext PDF Free Access
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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 nonhomotopic 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.
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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.ustrasbg.fr
DOI:
http://dx.doi.org/10.1090/S0002993908094331
PII:
S 00029939(08)094331
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.
