Automorphisms of profinite mapping class groups

Abstract

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 $\widehat{\operatorname{GT}}\hookrightarrow\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\widehatΓ(S))$.