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Finite rigid sets and the separating curve complex. (English) Zbl 1493.57006

Let \(\Sigma\) be a closed oriented surface of genus \(g \geq 3\) and let \(\mathcal{C}^{s}(\Sigma)\) denote the complex of separating curves on the surface \(\Sigma\). A simplicial map \(\phi: Z \to \mathcal{C}^{s}(\Sigma)\) is called incidence-preserving if the geometric intersection number \(i(\alpha,\beta) \neq 0\) implies that \(i(\phi(\alpha),\phi(\beta)) \neq 0\) for vertices \(\alpha\), \(\beta\) of a subcomplex \(Z\) of \(\mathcal{C}^{s}(\Sigma)\). In this paper, the authors prove that there is a finite subcomplex \(Y^{s}\) of \(\mathcal{C}^{s}(\Sigma)\) such that an incidence-preserving map \(\phi: Y^{s} \to \mathcal{C}^{s}(\Sigma)\) is induced by an element \(h\) of the extended mapping class group \(Mod^{\pm}(\Sigma)\) of the surface \(\Sigma\). Moreover, \(h\) is unique. Here, the extended mapping class group \(Mod^{\pm}(\Sigma)\) is the group of isotopy classes of all self-diffeomorphisms of \(\Sigma\).
The proof of the above result contains three steps. The first step concerns the rigidity of a subcomplex \(Y\) of the curve complex \(\mathcal{C}(\Sigma)\). In other words, the authors prove that a locally injective map \(Y \to \mathcal{C}(\Sigma)\) is induced by an extended mapping class, where \(Y=C \cup S \cup B \cup U \cup V\) such that \(C,\, S,\, B,\, U,\) and \(V\) are certain collections of curves on the surface \(\Sigma\). Later, to show that an incidence-preserving map \(\phi: Y^{s} \to \mathcal{C}^{s}(\Sigma)\) extends to a locally injective map \(\tilde{\phi}: Y \to \mathcal{C}(\Sigma)\), the authors use sharing pairs. Namely, they show that sharing pairs are preserved under incidence-preserving maps in the second step. Finally, extending from \(Y^{s}\) to \(Y\) is the last step.

MSC:

57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
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References:

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