Galois actions on surfaces and a higher genus Grothendieck-Teichmüller group

Abstract

We construct an operadic model for the higher-genus Teichmüller tower. More precisely, we define a modular operad $\mathbf{S}$ in groupoids built from mapping class groups, with compositions and contractions encoding gluing operations on surfaces. We prove a presentation theorem for maps out of $\mathbf{S}$, showing that they are determined by a small number of genus-zero and genus-one generators and relations. Using this presentation and the work of Nakamura--Schneps, we construct a faithful action of the Nakamura--Schneps subgroup $\widehatΓ\subseteq\widehat{\mathsf{GT}}$ on the profinite completion $\widehat{\mathbf{S}}$, and hence an action of $\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$. The genus-zero truncation of $\mathbf{S}$ recovers the cyclic operad of parenthesized ribbon braids, and its group of object-fixing profinite automorphisms recovers $\widehat{\mathsf{GT}}$. Finally, the profinite completion of the classifying spaces of $\mathbf{S}$ assemble into a modular $\infty$-operad in profinite spaces whose values identify with the étale homotopy types of moduli stacks of curves with marked tangent vectors, and the $\widehatΓ$-action extends to this homotopy-coherent Teichmüller tower.