Actions of the Absolute Galois Group

TL;DR

This paper reviews Grothendieck's ideas on the actions of the absolute Galois group of Q, exploring its connections to surface topology, Teichmüller theory, and related geometric structures, highlighting conjectures and various research directions.

Contribution

It synthesizes Grothendieck's conjectures and ideas on Galois actions on geometric objects, connecting algebraic and topological perspectives in a comprehensive overview.

Findings

Discussion of Grothendieck's conjectures on Galois actions

Overview of dessins d'enfant and Teichmüller tower

Summary of nonlinear Galois actions in homotopy theory

Abstract

We review some ideas of Grothendieck and others on actions of the absolute Galois group {\Gamma} Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichm{\"u}ller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation came in part from his desire to understand the absolute Galois group. But he was also interested in Thurston's work on surfaces, and he expressed this in his Esquisse d'un programme, his R{\'e}coltes et semailles and on other occasions. He introduced the notions of dessin d'enfant, Teichm{\"u}ller tower, and other related objects, he considered the actions of {\Gamma} Q on them or on their etale fundamental groups, and he made conjectures on some natural homomorphisms between the absolute Galois group and the automor-phism groups (or outer automorphism groups) of these objects. We mention several ramifications of these ideas, due to various authors. We also report on the works of Sullivan and others on nonlinear actions of {\Gamma} Q , in particular in homotopy theory. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichm{\"u}ller theory. This volume is dedicated to the memory of Alexander Grothendieck.