Automorphisms of profinite mapping class groups
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910157027409920 |
|---|---|
| author | Boggi, Marco |
| author_facet | Boggi, Marco |
| contents | For $S=S_{g,n}$ a closed orientable differentiable surface of genus $g$ from which $n$ points have been removed, such that $χ(S)=2-2g-n<0$, let $\mathrm{P}Γ(S)$ be the pure mapping class group of $S$ and $\mathrm{P}\widehatΓ(S)$ and $\mathrm{P}\checkΓ(S)$ be, respectively, its profinite and its congruence completions, the latter being identified with the image of the natural representation $\mathrm{P}\widehatΓ(S)\to\operatorname{Out}({\widehatπ}_1(S))$ (where ${\widehatπ}_1(S)$ is the profinite completion of the fundamental group of $S$). We determine the automorphism groups of procongruence completions under a natural rigidity condition, and show that the profinite Grothendieck-Teichmüller group embeds into the outer automorphism group of the profinite completion.
Let $\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\widehatΓ(S))$ and $\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\checkΓ(S))$ be the groups of outer automorphisms which preserve the conjugacy class of a procyclic subgroup generated by a nonseparating Dehn twist (a condition trivially satisfied for $g=0$). Our main result gives that, for $χ(S)<g-2$ and $(g,n)\neq (1,2)$, there is a natural isomorphism: \[\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\checkΓ(S))\congΣ_n\times\widehat{\operatorname{GT}},\] where $Σ_{n}$ is the symmetric group on $n$ letters and $\widehat{\operatorname{GT}}$ denotes the profinite Grothendieck-Teichmüller group. We also prove that, for $χ(S)<g-2$, there is a natural faithful representation $\widehat{\operatorname{GT}}\hookrightarrow\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\widehatΓ(S))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_15075 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Automorphisms of profinite mapping class groups Boggi, Marco Geometric Topology Algebraic Geometry Number Theory 20F34 For $S=S_{g,n}$ a closed orientable differentiable surface of genus $g$ from which $n$ points have been removed, such that $χ(S)=2-2g-n<0$, let $\mathrm{P}Γ(S)$ be the pure mapping class group of $S$ and $\mathrm{P}\widehatΓ(S)$ and $\mathrm{P}\checkΓ(S)$ be, respectively, its profinite and its congruence completions, the latter being identified with the image of the natural representation $\mathrm{P}\widehatΓ(S)\to\operatorname{Out}({\widehatπ}_1(S))$ (where ${\widehatπ}_1(S)$ is the profinite completion of the fundamental group of $S$). We determine the automorphism groups of procongruence completions under a natural rigidity condition, and show that the profinite Grothendieck-Teichmüller group embeds into the outer automorphism group of the profinite completion. Let $\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\widehatΓ(S))$ and $\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\checkΓ(S))$ be the groups of outer automorphisms which preserve the conjugacy class of a procyclic subgroup generated by a nonseparating Dehn twist (a condition trivially satisfied for $g=0$). Our main result gives that, for $χ(S)<g-2$ and $(g,n)\neq (1,2)$, there is a natural isomorphism: \[\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\checkΓ(S))\congΣ_n\times\widehat{\operatorname{GT}},\] where $Σ_{n}$ is the symmetric group on $n$ letters and $\widehat{\operatorname{GT}}$ denotes the profinite Grothendieck-Teichmüller group. We also prove that, for $χ(S)<g-2$, there is a natural faithful representation $\widehat{\operatorname{GT}}\hookrightarrow\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\widehatΓ(S))$. |
| title | Automorphisms of profinite mapping class groups |
| topic | Geometric Topology Algebraic Geometry Number Theory 20F34 |
| url | https://arxiv.org/abs/2011.15075 |