GrothendieckTeichmüller

Grothendieck–Teichmüller theory is a program in number theory and algebraic geometry that aims to understand the absolute Galois group of the rational numbers through the geometry of moduli spaces of curves and the anabelian geometry of their étale fundamental groups. Originating from Alexander Grothendieck’s Esquisse d’un Programme, the approach envisions a universal symmetry group governing these geometric objects and encoding arithmetic information in their topological and algebraic structure.

Central objects are the Grothendieck–Teichmüller group GT̂ and its variants. GT̂ is a profinite group defined by Drinfeld in terms of automorphisms of the profinite free group on two generators subject to relations known as pentagon and hexagon equations. These relations reflect coherence conditions for braiding and associativity in braided monoidal categories and mirror the combinatorics of moduli spaces of punctured spheres. A closely related group, GT, is the discrete or profinite version used in various formalisms; GT1 denotes the subgroup fixing a distinguished central element.

An important feature is the natural homomorphism from the absolute Galois group Gal(Q̄/Q) into GT̂, obtained by considering how Galois automorphisms act on the étale fundamental groups of the projective line minus three points and on profinite braid groups. This map provides a bridge between arithmetic and geometry, suggesting that GT̂ captures universal symmetry governing these actions.

Grothendieck–Teichmüller theory connects to Drinfeld associators, mapping class groups, moduli spaces M_{0,n}, and anabelian geometry. It yields a framework to study representations of fundamental groups and braid groups, and to formulate conjectures about the structure of Gal(Q̄/Q).

While influential, the program remains largely conjectural. Progress includes concrete descriptions of the defining relations and partial results on the image of Gal within GT̂; many questions about the full structure and consequences of GT̂ remain active areas of research.