Automorphisms of profinite mapping class groups

TL;DR

This paper determines automorphism groups of profinite mapping class groups of surfaces, revealing a deep connection with the profinite Grothendieck-Teichmüller group and establishing isomorphisms under certain conditions.

Contribution

It provides the first explicit description of automorphism groups of procongruence completions of mapping class groups, linking them to the profinite Grothendieck-Teichmüller group.

Findings

Automorphism groups are explicitly determined under a natural rigidity condition.

A natural isomorphism relates automorphisms to symmetric groups and the profinite Grothendieck-Teichmüller group.

A faithful representation of the Grothendieck-Teichmüller group into automorphisms is established.

Abstract

For $S=S_{g,n}$ a closed orientable differentiable surface of genus $g$ from which $n$ points have been removed, such that $\chi(S)=2-2g-n<0$, let $\mathrm{P}\Gamma(S)$ be the pure mapping class group of $S$ and $\mathrm{P}\widehat\Gamma(S)$ and $\mathrm{P}\check\Gamma(S)$ be, respectively, its profinite and its congruence completions, the latter being identified with the image of the natural representation $\mathrm{P}\widehat\Gamma(S)\to\operatorname{Out}({\widehat\pi}_1(S))$ (where ${\widehat\pi}_1(S)$ 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-Teichm\"uller group embeds into the outer automorphism group of the profinite completion. Let $\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\widehat\Gamma(S))$ and $\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\check\Gamma(S))$ 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 $\chi(S)<g-2$ and $(g,n)\neq (1,2)$, there is a natural isomorphism: \[\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\check\Gamma(S))\cong\Sigma_n\times\widehat{\operatorname{GT}},\] where $\Sigma_{n}$ is the symmetric group on $n$ letters and $\widehat{\operatorname{GT}}$ denotes the profinite Grothendieck-Teichm\"uller group. We also prove that, for $\chi(S)<g-2$, there is a natural faithful representation $\widehat{\operatorname{GT}}\hookrightarrow\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\widehat\Gamma(S))$.