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

Abstract

Let $S$ be a 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 as by homeomorphisms of $\mathcal{UMF}$. 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 nonhomotopic 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 $\Gamma^*(S)$ on that subspace.