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

TL;DR

This paper proves a rigidity theorem showing that the homeomorphism group of the unmeasured foliation space acts essentially the same as the extended mapping class group on a dense subset, revealing deep symmetry properties.

Contribution

It establishes that the full homeomorphism group of the unmeasured foliation space coincides with the extended mapping class group on a dense subset, demonstrating a rigidity phenomenon.

Findings

The action of the extended mapping class group on a dense subset is faithful.

The homeomorphism group of the unmeasured foliation space coincides with the extended mapping class group on that subset.

The set of simple closed curves maps densely into the unmeasured foliation space.

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.