The sphere complex of a locally finite graph
| Autor: | Udall, Brian |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | For a locally finite graph $\Gamma$, we consider its mapping class group $\text{Map}(\Gamma)$ as defined by Algom-Kfir-Bestvina. For these groups, we prove a generalization of the results of Laudenbach and Brendle-Broaddus-Putman, producing a $3$-manifold $M_{\Gamma}$ whose mapping class group surjects onto $\text{Map}(\Gamma)$ with kernel a compact abelian group of sphere twists so that the corresponding short exact sequence splits. Along the way we obtain an induced faithful action of $\text{Map}(\Gamma)$ on the sphere complex $\mathcal{S}(M_{\Gamma})$ of $M_{\Gamma}$, which is the simplicial complex whose simplices are isotopy classes of finite collections of spheres in $M_{\Gamma}$ which are pairwise disjoint. When $\Gamma$ has finite rank, we further show that the action of $\text{Map}(\Gamma)$ on a certain natural subcomplex has elements with positive translation length, and also consider a candidate for an Outer space of such a graph. As another application, we prove that for many $\Gamma$, $\text{Map}(\Gamma)$ is quasi-isometric to a particular subgraph of $\mathcal{S}(M_{\Gamma})$, following Schaffer-Cohen. We also deduce analogs of the results of Domat-Hoganson-Kwak. Comment: 57 pages, 8 figures. Fixed the hypotheses of the first theorem, excluding finitely many sporadic cases from one part of the result. Expanded the proofs of all the main results for clarity. Comments are welcome! |
| Databáze: | arXiv |
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