Roots of hyperelliptic involutions and braid groups modulo their center inside mapping class groups
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915194797555712 |
|---|---|
| author | Lamy, Ryan |
| author_facet | Lamy, Ryan |
| contents | Let $n,k\in\mathbb{N}$ and let $S$ be the closed surface of genus $nk$. A copy of the braid group on $2k+2$ strands modulo its center is found inside $\mathrm{Mod}(S)$, provided $n\geq 3$. In particular, for $k=1$ the class of the half-twist braid inside $B_4/Z(B_4)$ is identified with a hyperelliptic involution inside $\mathrm{Mod}(S)$. As a consequence, we can show that each hyperelliptic involution inside $\mathrm{Mod}(S)$ has infinitely many square roots, and discuss their conjugacy classes. Furthermore, a copy of $\mathrm{Mod}(S_1)\cong\mathrm{SL}_2(\mathbb{Z})$ is found inside $\mathrm{Mod}(S_2)$. This subgroup contains the unique hyperelliptic involution on $S_2$. As a result, we can show that the latter admits infinitely many square and cubic roots, and discuss their conjugacy classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09552 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Roots of hyperelliptic involutions and braid groups modulo their center inside mapping class groups Lamy, Ryan Geometric Topology 57K20 (Primary), 20F36 (Secondary) Let $n,k\in\mathbb{N}$ and let $S$ be the closed surface of genus $nk$. A copy of the braid group on $2k+2$ strands modulo its center is found inside $\mathrm{Mod}(S)$, provided $n\geq 3$. In particular, for $k=1$ the class of the half-twist braid inside $B_4/Z(B_4)$ is identified with a hyperelliptic involution inside $\mathrm{Mod}(S)$. As a consequence, we can show that each hyperelliptic involution inside $\mathrm{Mod}(S)$ has infinitely many square roots, and discuss their conjugacy classes. Furthermore, a copy of $\mathrm{Mod}(S_1)\cong\mathrm{SL}_2(\mathbb{Z})$ is found inside $\mathrm{Mod}(S_2)$. This subgroup contains the unique hyperelliptic involution on $S_2$. As a result, we can show that the latter admits infinitely many square and cubic roots, and discuss their conjugacy classes. |
| title | Roots of hyperelliptic involutions and braid groups modulo their center inside mapping class groups |
| topic | Geometric Topology 57K20 (Primary), 20F36 (Secondary) |
| url | https://arxiv.org/abs/2503.09552 |