曲面上不可测叶理空间上映射类群作用的刚性定理
A rigidity theorem for the mapping class group action on the space of unmeasured foliations on a surface
摘要：
设$S$是有限型曲面，它不是至多有四个穿孔的球面，至多有两个穿孔的环面，也不是亏格为二的闭曲面。设$\mathcal{MF}$是$S$上紧支集的测量叶理的等价类空间，$\mathcal{UMF}$是$\mathcal{MF}$的商空间，它是通过识别两个等价类而得到的，只要它们可以由拓扑等价的叶理表示，也就是说，忽略横测度. $S$的扩展映射类群$Γ^*$通过$\mathcal{UMF}$的同胚充当.证明了$\mathcal{UMF}$的全同胚群在$\mathcal{UMF}$的某个稠密子集上的作用的限制与$Γ^*$在该子集上的作用一致.更准确地说，令$\mathcal{D}$是$S$上不一定连通的本质不相交且成对非同伦的简单闭曲线的同伦类集合在$\mathcal{UMF}$中的自然像。集合$\mathcal{D}$在$\mathcal{UMF}$中是稠密的，它不受$Γ^*$作用于$\mathcal{UMF}$的影响，并且$Γ^*$作用于$\mathcal{D}$的限制是忠实的.证明了群$\mathrm{Homeo}(\mathcal{UMF})$对$\mathcal{D}$作用的限制与群$Γ^*(S)$对该子空间的作用一致.
英文摘要：
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 $Γ^*$ 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 $Γ^*$ 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 $Γ^*$ on $\mathcal{UMF}$ and the restriction of the action of $Γ^*$ 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 $Γ^*(S)$ on that subspace.
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