Normal closure of finite subgroups of $\mathrm{Aut}(F_n)$ and $\mathrm{Out}(F_n)$
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911308578816000 |
|---|---|
| author | Shen, Jiayi |
| author_facet | Shen, Jiayi |
| contents | For $n\geq 3$, let $G$ be a nontrivial finite subgroup of $\mathrm{Aut}(F_n)$ with $|G|$ not a power of $2$. We prove that the normal closure $N(G)$ is $\mathrm{SAut}(F_n)$ if $G\subset\mathrm{SAut}(F_n)$ and $N(G)$ is $\mathrm{Aut}(F_n)$ otherwise. When $|G|$ is a power of $2$, we have a partial theorem. Similarly, let $G'$ be a nontrivial finite subgroup of $\mathrm{Out}(F_n)$ with $|G'|$ not a power of $2$. Then the normal closure $N(G')$ is $\mathrm{SOut}(F_n)$ if $G'\subset\mathrm{SOut}(F_n)$ and $N(G')$ is $\mathrm{Out}(F_n)$ otherwise. When $|G'|$ is a power of $2$, we have a partial theorem as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07759 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normal closure of finite subgroups of $\mathrm{Aut}(F_n)$ and $\mathrm{Out}(F_n)$ Shen, Jiayi Group Theory Geometric Topology For $n\geq 3$, let $G$ be a nontrivial finite subgroup of $\mathrm{Aut}(F_n)$ with $|G|$ not a power of $2$. We prove that the normal closure $N(G)$ is $\mathrm{SAut}(F_n)$ if $G\subset\mathrm{SAut}(F_n)$ and $N(G)$ is $\mathrm{Aut}(F_n)$ otherwise. When $|G|$ is a power of $2$, we have a partial theorem. Similarly, let $G'$ be a nontrivial finite subgroup of $\mathrm{Out}(F_n)$ with $|G'|$ not a power of $2$. Then the normal closure $N(G')$ is $\mathrm{SOut}(F_n)$ if $G'\subset\mathrm{SOut}(F_n)$ and $N(G')$ is $\mathrm{Out}(F_n)$ otherwise. When $|G'|$ is a power of $2$, we have a partial theorem as well. |
| title | Normal closure of finite subgroups of $\mathrm{Aut}(F_n)$ and $\mathrm{Out}(F_n)$ |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2512.07759 |