Geometric models and asymptotic dimension for infinite-type surface mapping class groups
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918120697888768 |
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| author | Kopreski, Michael C. Shaji, George |
| author_facet | Kopreski, Michael C. Shaji, George |
| contents | Let $S$ be an infinite-type surface and let $G \leq \operatorname{Map}(S)$ be a locally bounded Polish subgroup. We construct a metric graph $M$ of simple arcs and curves on $S$ preserved by the action of $G$ and for which the vertex orbit map $G \to V(M)$ is a coarse equivalence; if $G$ is boundedly generated, then $M$ is a Cayley--Abels--Rosendal graph for $G$ and the orbit map is a quasi-isometry. In particular, if $S$ contains a non-displaceable subsurface and $G \geq \operatorname{PMap}_c(S)$ is boundedly generated or $G \in \{\overline{\operatorname{PMap}_c(S)}, \operatorname{PMap}(S), \operatorname{Map}(S) \}$ and is locally bounded, then $\operatorname{asdim} M = \operatorname{asdim} G = \infty$. This result completes the classification of the asymptotic dimension of stable boundedly generated infinite-type surface mapping class groups begun by Grant--Rafi--Verberne. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06679 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric models and asymptotic dimension for infinite-type surface mapping class groups Kopreski, Michael C. Shaji, George Geometric Topology General Topology Group Theory Let $S$ be an infinite-type surface and let $G \leq \operatorname{Map}(S)$ be a locally bounded Polish subgroup. We construct a metric graph $M$ of simple arcs and curves on $S$ preserved by the action of $G$ and for which the vertex orbit map $G \to V(M)$ is a coarse equivalence; if $G$ is boundedly generated, then $M$ is a Cayley--Abels--Rosendal graph for $G$ and the orbit map is a quasi-isometry. In particular, if $S$ contains a non-displaceable subsurface and $G \geq \operatorname{PMap}_c(S)$ is boundedly generated or $G \in \{\overline{\operatorname{PMap}_c(S)}, \operatorname{PMap}(S), \operatorname{Map}(S) \}$ and is locally bounded, then $\operatorname{asdim} M = \operatorname{asdim} G = \infty$. This result completes the classification of the asymptotic dimension of stable boundedly generated infinite-type surface mapping class groups begun by Grant--Rafi--Verberne. |
| title | Geometric models and asymptotic dimension for infinite-type surface mapping class groups |
| topic | Geometric Topology General Topology Group Theory |
| url | https://arxiv.org/abs/2508.06679 |