On a geometric description of Gal( Q ¯ p / Q p ) and a p-adic avatar of GT ^

Abstract

We develop a p-adic version of the so-called Grothendieck-Teichmüller theory (which studies $\text{Gal}\left(\overline{Q}/Q\right)$ by means of its action on profinite braid groups or mapping class groups). For every place v of $\overline{Q}$, we give some geometrico-combinatorial descriptions of the local Galois group $\text{Gal}({\overline{Q}}_v/Q_v)$ inside $\text{Gal}\left(\overline{Q}/Q\right)$. We also show that $\text{Gal}({\overline{Q}}_p/Q_p)$ is the automorphism group of an appropriate ${\pi }_1$-functor in p-adic geometry.