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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2307.15161 |
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| _version_ | 1866910258871402496 |
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| author | Hernández, Jesús Hernández |
| author_facet | Hernández, Jesús Hernández |
| contents | This work is the extension of the results by the author in [7] and [6] for low-genus surfaces. Let $S$ be an orientable, connected surface of finite topological type, with genus $g \leq 2$, empty boundary, and complexity at least $2$; as a complement of the results of [6], we prove that any graph endomorphism of the curve graph of $S$ is actually an automorphism. Also, as a complement of the results in [6] we prove that under mild conditions on the complexity of the underlying surfaces any graph morphism between curve graphs is induced by a homeomorphism of the surfaces.
To prove these results, we construct a finite subgraph whose union of iterated rigid expansions is the curve graph $\mathcal{C}(S)$. The sets constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid sets in [2]. Similarly to [7], a consequence of our proof is that Aramayona and Leininger's rigid set also exhausts the curve graph via rigid expansions, and the combinatorial rigidity results follow as an immediate consequence, based on the results in [6]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_15161 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Graph morphisms and exhaustion of curve graphs of low-genus surfaces Hernández, Jesús Hernández Geometric Topology Group Theory 57K20 This work is the extension of the results by the author in [7] and [6] for low-genus surfaces. Let $S$ be an orientable, connected surface of finite topological type, with genus $g \leq 2$, empty boundary, and complexity at least $2$; as a complement of the results of [6], we prove that any graph endomorphism of the curve graph of $S$ is actually an automorphism. Also, as a complement of the results in [6] we prove that under mild conditions on the complexity of the underlying surfaces any graph morphism between curve graphs is induced by a homeomorphism of the surfaces. To prove these results, we construct a finite subgraph whose union of iterated rigid expansions is the curve graph $\mathcal{C}(S)$. The sets constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid sets in [2]. Similarly to [7], a consequence of our proof is that Aramayona and Leininger's rigid set also exhausts the curve graph via rigid expansions, and the combinatorial rigidity results follow as an immediate consequence, based on the results in [6]. |
| title | Graph morphisms and exhaustion of curve graphs of low-genus surfaces |
| topic | Geometric Topology Group Theory 57K20 |
| url | https://arxiv.org/abs/2307.15161 |