Topological normal generation of big mapping class groups
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914302069309440 |
|---|---|
| author | Baik, Juhun |
| author_facet | Baik, Juhun |
| contents | A topological group $G$ is topologically normally generated if there exists $g \in G$ such that the normal closure of $g$ is dense in $G$. Let $S$ be a tame, infinite type surface whose mapping class group $\mathrm{Map}(S)$ is generated by a coarsely bounded set (CB generated). We prove that if the end space of $S$ is countable, then $\mathrm{Map}(S)$ is topologically normally generated if and only if $S$ is uniquely self-similar. Moreover, when the end space of $S$ is uncountable, we provide a sufficient condition under which $\mathrm{Map}(S)$ is topologically normally generated. As a consequence, we construct uncountably many examples of surfaces that are not telescoping yet have topologically normally generated mapping class groups. Additionally, we establish the semidirect product structure of $\mathrm{FMap}(S)$, the subgroup of $\mathrm{Map}(S)$ that pointwisely fixes all maximal ends that each is isolated in the set of maximal ends of $S$. This result leads to a proof that the minimum number of topological normal generators of $\mathrm{Map}(S)$ is bounded both above and below by constants that depend only on the topology of $S$. Furthermore, we demonstrate that the upper bound grows quadratically with respect to this constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12700 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topological normal generation of big mapping class groups Baik, Juhun Group Theory Geometric Topology (Primary) 57K20, 20F65, 54H11, (Secondary) 20E45 A topological group $G$ is topologically normally generated if there exists $g \in G$ such that the normal closure of $g$ is dense in $G$. Let $S$ be a tame, infinite type surface whose mapping class group $\mathrm{Map}(S)$ is generated by a coarsely bounded set (CB generated). We prove that if the end space of $S$ is countable, then $\mathrm{Map}(S)$ is topologically normally generated if and only if $S$ is uniquely self-similar. Moreover, when the end space of $S$ is uncountable, we provide a sufficient condition under which $\mathrm{Map}(S)$ is topologically normally generated. As a consequence, we construct uncountably many examples of surfaces that are not telescoping yet have topologically normally generated mapping class groups. Additionally, we establish the semidirect product structure of $\mathrm{FMap}(S)$, the subgroup of $\mathrm{Map}(S)$ that pointwisely fixes all maximal ends that each is isolated in the set of maximal ends of $S$. This result leads to a proof that the minimum number of topological normal generators of $\mathrm{Map}(S)$ is bounded both above and below by constants that depend only on the topology of $S$. Furthermore, we demonstrate that the upper bound grows quadratically with respect to this constant. |
| title | Topological normal generation of big mapping class groups |
| topic | Group Theory Geometric Topology (Primary) 57K20, 20F65, 54H11, (Secondary) 20E45 |
| url | https://arxiv.org/abs/2409.12700 |