Automorphisms of procongruence mapping class groups

Marco Boggi

Published 2020-11-30Version 1

For $S=S_{g,n}$ a closed orientable differentiable surface of genus $g$ from which $n$ points have been removed, with $\chi(S)=2-2g-n<0$, let $\Gamma(S)$ be the mapping class group of $S$ and $\hat{\Gamma}(S)$ and $\check{\Gamma}(S)$, respectively, its profinite and its congruence completion. The latter is the image of the natural representation $\hat{\Gamma}(S)\to\operatorname{Out}(\hat{\pi}_1(S))$, where $\hat{\pi}_1(S)$ is the profinite completion of the fundamental group of the surface $S$. Let $\operatorname{Out}^{\mathbb{I}_0}(\check{\Gamma}(S))$ be the group of outer automorphisms of $\check{\Gamma}(S)$ which preserve the conjugacy class of a procyclic subgroup generated by a nonseparating Dehn twist and put $d(S)=3g-3+n$. The main result of the paper is that, for $d(S)>1$, there is a natural isomorphism: \[\operatorname{Out}^{\mathbb{I}_0}(\check{\Gamma}(S))\cong\widehat{\operatorname{GT}},\] where $\widehat{\operatorname{GT}}$ is the profinite Grothendieck-Teichm\"uller group. We will actually prove a slightly stronger result which implies that the automorphism group of the procongruence Grothendieck-Teichm\"uller tower is also isomorphic to $\widehat{\operatorname{GT}}$.