[Paper Review] An elementary approach to dessins d'enfants and the Grothendieck-Teichm\\"uller group

Pierre Guillot|arXiv (Cornell University)|Sep 8, 2013

History and Theory of Mathematics6 references17 citations

TL;DR

This paper presents an elementary, self-contained treatment of dessins d'enfants, establishing an equivalence of categories between embedded graphs, permutation sets, field extensions, and algebraic curves. It proves the faithfulness of the Galois action on dessins and constructs an explicit injective homomorphism from the absolute Galois group of ℚ into the Grothendieck-Teichmüller group 𝔾𝕋₀, using explicit approximations via finite groups.

ABSTRACT

We give an account of the theory of dessins d'enfants which is both elementary and self-contained. We describe the equivalence of many categories (graphs embedded nicely on surfaces, finite sets with certain permutations, certain field extensions, and some classes of algebraic curves), some of which are naturally endowed with an action of the absolute Galois group of the rational field. We prove that the action is faithful. Eventually we prove that this absolute Galois group embeds into the Grothendieck-Teichm\\"uller group $GT_0$ introduced by Drinfel'd. There are explicit approximations of $GT_0$ by finite groups, and we hope to encourage computations in this area. Our treatment includes a result which has not appeared in the literature yet: the Galois action on the subset of regular dessins - that is, those exhibiting maximal symmetry -- is also faithful.

Motivation & Objective

• To provide a self-contained, elementary account of dessins d'enfants without relying on advanced tools like Weil's rigidity criterion.
• To establish the equivalence of multiple mathematical categories: embedded graphs, permutation sets, field extensions, and algebraic curves.
• To prove that the action of the absolute Galois group Gal(ℚ̄/ℚ) on dessins is faithful, including on the subset of regular dessins.
• To construct an explicit injective homomorphism from Gal(ℚ̄/ℚ) into the Grothendieck-Teichmüller group 𝔾𝕋₀ using finite approximations.
• To encourage computational exploration by showing that 𝔾𝕋₀ can be approximated by finite groups derived from regular dessins.

Proposed method

• Uses elementary methods from 19th-century topology and group theory, including Riemann surface theory and fundamental group properties.
• Establishes categorical equivalences via combinatorial data: graphs on surfaces correspond to pairs of permutations generating a transitive subgroup.
• Defines the Galois action on dessins via field extensions and lifts it to automorphisms of the profinite completion of the free group on two generators, F̂₂.
• Constructs the group 𝔾𝕋₀ as the group of pairs (k, f) ∈ ℤ̂× × [F̂₂, F̂₂] such that the associated self-homomorphism β is an automorphism and commutes with δ and θ in Out(F̂₂).
• Proves that the map Γ: Gal(ℚ̄/ℚ) → Out(F̂₂) lifts to an injective homomorphism into Aut(F̂₂), leading to an embedding into 𝔾𝕋₀.
• Uses the fact that regular dessins correspond to finite groups generated by two elements of order less than n, enabling finite computational approximations of 𝔾𝕋₀.

Experimental results

Research questions

• RQ1 Can the theory of dessins d'enfants be developed using only elementary methods, avoiding advanced tools like Weil's rigidity criterion?
• RQ2 Is the action of the absolute Galois group Gal(ℚ̄/ℚ) on the set of dessins d'enfants faithful, including on the subset of regular dessins?
• RQ3 Can an explicit injective homomorphism from Gal(ℚ̄/ℚ) into the Grothendieck-Teichmüller group 𝔾𝕋₀ be constructed using combinatorial and profinite group methods?
• RQ4 How can the Grothendieck-Teichmüller group 𝔾𝕋₀ be approximated by finite groups, and what does this imply for computational exploration of Gal(ℚ̄/ℚ)?
• RQ5 What is the precise relationship between the group 𝔾𝕋₀ and the profinite completion of the free group on two generators, and how does it relate to Galois actions?

Key findings

• The action of Gal(ℚ̄/ℚ) on the set of dessins d'enfants is faithful, meaning the Galois group acts nontrivially on the combinatorial data of embedded graphs.
• The action of Gal(ℚ̄/ℚ) on the subset of regular dessins—those with maximal symmetry—is also faithful, a result not previously published in the literature.
• An explicit injective homomorphism is constructed from Gal(ℚ̄/ℚ) into the Grothendieck-Teichmüller group 𝔾𝕋₀ via pairs (k, f) in ℤ̂× × [F̂₂, F̂₂] defining automorphisms of the profree group F̂₂.
• The group 𝔾𝕋₀ is isomorphic to the standard Grothendieck-Teichmüller group 𝔾𝕋, and the embedding into 𝔾𝕋₀ is compatible with the cyclotomic character via the projection to ℤ̂×.
• Finite approximations of 𝔾𝕋₀ are obtained by restricting to finite groups generated by two elements of order less than n, which correspond exactly to regular dessins.
• The construction provides a concrete, computable framework for studying Gal(ℚ̄/ℚ) through finite groups, enabling potential algorithmic exploration of the Galois group.