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

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 MF be the space of equivalence classes of measured foliations of compact support on S and let UMF be the quotient space of 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 Γ of S acts as by homeomorphisms of UMF . We show that the restriction of the action of the whole homeomorphism group of UMF on some dense subset of UMF coincides with the action of Γ on that subset. More precisely, let D be the natural image in UMF of the set of homotopy classes of not necessarily connected essential disjoint and pairwise nonhomotopic simple closed curves on S. The set D is dense in UMF, it is invariant by the action of Γ on UMF and the restriction of the action of Γ on D is faithful. We prove that the restriction of the action on D of the group Homeo(UMF) coincides with the action of Γ(S) on that subspace. AMS Mathematics Subject Classification: 57M50 ; 57M60 ; 20F65 ; 20F38.