Automorphisms of profinite and procongruence curve complexes and the Grothendieck–Teichmüller group

This paper is devoted primarily to the identification of the automorphism group for the profinite (and/or procongruence) completion of the curve complex $C(S)$ attached to an orientable hyperbolic surface of finite type $S$. It can be regarded as a sequel to the paper: Algebra i Analiz, 35, no. 3 (2023), 57–137, where the author explored in particular (see Theorem 7.1 there) the rigidity of the completed pants (or maximal multicurve) complex $C_P(S)$. Roughly speaking $\mathrm{Out}(\hat{C}_P(S))=\mathrm{Out}(C_P(S))= \mathbb{Z}/2$, where the outer automorphism group $\mathrm{Out}$ refers to the quotient of the automorphism group by the conjugacy action of the completed (respectively, discrete) Teichmüller (or mapping class) group $\Gamma(S)$. Here by contrast, it will emerge that $\mathrm{Out}(\hat{C}(S))=\widehat{GT}$ (say, if $S$ is a punctured sphere with $n>4$ punctures), the profinite version of the Grothendieck–Teichmüller group. Recall also that in Galois terms the arithmetic Galois group $G_\mathbb{Q}=\mathrm{Gal}({\bar{\mathbb{Q}}}/\mathbb{Q})$ is contained in $\widehat{GT}$ whereas $\mathbb{Z}/2=\mathrm{Gal}(\mathbb{C}/\mathbb{R})$. In passing, the geometric or topological emergence and meaning of the Grothendieck–Teichmüller group itself will be sdisplayed, emphasis on its natural relationship with the deformation theory, possibly also with the string topology.