[Paper Review] Actions of the Absolute Galois Group

Norbert A’Campo, Lizhen Ji+1 more|arXiv (Cornell University)|Mar 10, 2016

Algebraic Geometry and Number Theory64 references3 citations

TL;DR

This paper surveys Grothendieck's program to understand the absolute Galois group of the rationals via its actions on geometric and topological objects, particularly the Teichmüller tower, dessins d'enfants, and the Grothendieck-Teichmüller group. It presents how nonlinear actions of Galois groups on profinite fundamental groups of moduli spaces of curves provide a pathway to understanding arithmetic via geometry, with the central conjecture being that the Galois action on the profinite Teichmüller tower is isomorphic to the Grothendieck-Teichmüller group.

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.

Motivation & Objective

• To survey Grothendieck's vision of understanding the absolute Galois group $\Gamma_{\mathbb{Q}}$ through its nonlinear actions on geometric objects like moduli spaces and fundamental groups.
• To explain how dessins d'enfants and the Teichmüller tower serve as combinatorial and geometric models for understanding Galois actions.
• To examine the conjecture that the homomorphism from $\Gamma_{\mathbb{Q}}$ to the profinite Grothendieck-Teichmüller group is an isomorphism.
• To clarify the role of the cartographic group and its relation to Teichmüller theory and surface topology.
• To assess the current status of key conjectures and the challenges in proving them, especially in light of Deligne’s skepticism and recent progress.

Proposed method

• Analyzes the action of $\Gamma_{\mathbb{Q}} = \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the étale fundamental groups of moduli spaces $\mathcal{M}_{g,n}$ and their stable compactifications.
• Explores the theory of dessins d'enfants as combinatorial realizations of Belyi maps, linking algebraic curves defined over $\overline{\mathbb{Q}}$ to embedded graphs on Riemann surfaces.
• Introduces the Teichmüller tower as a system of profinite fundamental groups of $\mathcal{M}_{0,n}$ with natural morphisms, forming a projective system under degenerations.
• Applies the reconstruction principle to show that surfaces are determined by stabilizers of group actions on bi-arcs, linking topology to group-theoretic data.
• Uses the cartographic group as a combinatorial model of Teichmüller space, where subgroups of finite index correspond to compact surfaces.
• Examines the Grothendieck-Teichmüller group as a profinite group acting on the Teichmüller tower, with the conjecture that $\Gamma_{\mathbb{Q}}$ maps isomorphically onto it.

Experimental results

Research questions

• RQ1 Can the absolute Galois group $\Gamma_{\mathbb{Q}}$ be fully understood through its action on the étale fundamental groups of moduli spaces of curves?
• RQ2 Is the homomorphism from $\Gamma_{\mathbb{Q}}$ to the profinite Grothendieck-Teichmüller group an isomorphism, as conjectured by Grothendieck?
• RQ3 How do dessins d'enfants encode arithmetic information about algebraic curves defined over $\overline{\mathbb{Q}}$?
• RQ4 What is the precise relationship between the cartographic group and Teichmüller theory, and can it realize Nielsen’s realization problem combinatorially?
• RQ5 Why has the proof of key results in Grothendieck-Teichmüller theory remained elusive, and what progress has been made despite the complexity of the language?

Key findings

• The absolute Galois group $\Gamma_{\mathbb{Q}}$ acts faithfully on the profinite fundamental group of the moduli space $\mathcal{M}_{0,n}$ via Belyi's theorem, linking arithmetic to geometry.
• Dessins d'enfants provide a combinatorial model for algebraic curves defined over $\overline{\mathbb{Q}}$, with Galois group acting by permuting the dessins.
• The Teichmüller tower, a projective system of profinite fundamental groups of $\mathcal{M}_{g,n}$, is conjectured to be a universal object encoding the action of $\Gamma_{\mathbb{Q}}$.
• The cartographic group acts transitively on bi-arcs of a surface, and the stabilizers of this action classify surfaces, providing a combinatorial realization of Teichmüller space.
• The profinite Grothendieck-Teichmüller group is conjectured to be isomorphic to $\Gamma_{\mathbb{Q}}$, though this remains unproven and is considered one of the central open problems.
• Despite Deligne’s skepticism about the utility of the language of dessins, the subject has matured with recent surveys and books, indicating growing consensus and progress.