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W2002636844 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." @default.
W2002636844 created "2016-06-24" @default.
W2002636844 creator A5077143673 @default.
W2002636844 date "2008-06-17" @default.
W2002636844 modified "2023-10-18" @default.
W2002636844 title "A rigidity theorem for the mapping class group action on the space of unmeasured foliations on a surface" @default.
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W2002636844 doi "https://doi.org/10.1090/s0002-9939-08-09433-1" @default.
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