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Simplicial embeddings between pants graphs. (English) Zbl 1194.57020
Let \(\Sigma\) be a compact orientable surface such that every connected component has negative Euler characteristic. When the genus of \(\Sigma\) is \(g\) and the number of boundary components of \(\Sigma\) is \(b\), we define the complexity of \(\Sigma\) by \(\kappa(\Sigma) = 3g-3+b\). A curve on \(\Sigma\) is a homotopy class of nontrivial and nonperipheral simple closed curves on \(\Sigma\). A multicurve \(Q\) on \(\Sigma\) is a collection of pairwise distinct and pairwise disjoint curves on \(\Sigma\). A pants decomposition \(Q\) of \(\Sigma\) is a multicurve \(Q\) such that \(|Q| = \kappa(\Sigma)\), i.e. each connected component of \(\Sigma - Q\) is a \(3\)-holed sphere. The pants graph \({\mathcal P}(\Sigma)\) of \(\Sigma\) is the simplicial graph whose vertex set is the set of all pants decompositions of \(\Sigma\) and where two vertices are connected by an edge if the corresponding pants decompositions are related by an elementary move.
The pants graph was introduced by A. E. Hatcher and W. Thurston [Topology 19, 221–237 (1980; Zbl 0447.57005)], who proved it is connected. D. Margalit [Duke Math. J. 121, No. 3, 457–479 (2004; Zbl 1055.57024)] showed that if \(\Sigma\) is a compact connected oriented surface with \(\kappa(\Sigma) > 0\), then the natural homomorphism from the mapping class group of \(\Sigma\) to the automorphism group of \({\mathcal P}(\Sigma)\) is injective, moreover, if \(\kappa(\Sigma)>3\), then this homomorphism is an isomorphism.
In the paper under review, as an extension of the above result by Margalit to injective simplicial maps between pants graphs, it is shown: let \(\Sigma_1\) and \(\Sigma_2\) be compact orientable surfaces of negative Euler characteristic such that each connected component of \(\Sigma_1\) has complexity at least \(2\), and let \(\phi : {\mathcal P}(\Sigma_1) \to {\mathcal P}(\Sigma_2)\) be an injective simplicial map, then there exists a \(\pi_1\)-injective embedding \(h : \Sigma_1 \to \Sigma_2\) and a multicurve \(Q\) on \(\Sigma_2\) such that \(\phi(v) = h(v) \cup Q\) for all vertices \(v\) of \({\mathcal P}(\Sigma_1)\). This result is proved by investigation on a stratified structure \(\{ {\mathcal P}_Q ; Q \text{ is a multicurve on } \Sigma \}\) of \({\mathcal P}(\Sigma)\), where \({\mathcal P}_Q\) is a subgraph of \({\mathcal P}(\Sigma)\) spanned by vertices of \({\mathcal P}(\Sigma)\) that contains \(Q\), especially on Farey graphs (\({\mathcal P}_Q\) for the multicurve \(Q\) of cardinality \(\kappa(\Sigma)-1\)) in \({\mathcal P}(\Sigma)\).

57M50 General geometric structures on low-dimensional manifolds
05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
Full Text: DOI arXiv
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