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Hidden symmetries of the Grothendieck--Teichmüller group

We consider the Grothendieck--Teichmüller 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--Teichmüller group and prepares as well the ground for studying further relations to the motivic Galois group.