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

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.