Hidden symmetries of the Grothendieck--Teichm\"uller group

TL;DR

This paper explores hidden dihedral symmetries within the Grothendieck--Teichmüller group using real algebraic geometry and web theory, linking these symmetries to fundamental groupoids and Galois groups.

Contribution

It introduces a new perspective on the Grothendieck--Teichmüller group by identifying dihedral symmetries through geometric methods, advancing understanding of its structure.

Findings

Demonstrates preservation of dihedral symmetry relations

Connects symmetries to fundamental groupoids of configuration spaces

Suggests implications for the absolute Galois group

Abstract

We consider the Grothendieck--Teichm\"uller group under a new aspect. Using real algebraic geometry and web theory we show that it preserves dihedral symmetry relations, present in the fundamental groupoids of configuration spaces of marked points on $\mathbb{C}$. The motivation of this paper is to be understood in the light of Grothendieck's initial philosophy stating that throughout hidden symmetries of the moduli spaces of curves one can shed some light on the absolute Galois group. This appears as a new development of the construction of the avatar of the Grothendieck--Teichm\"uller group and prepares as well the ground for studying further relations to the motivic Galois group.