On a geometric description of $Gal(\bar{\bf Q}_p/{\bf Q}_p)$ and a p-adic avatar of $\hat{GT}$

Abstract

We develop a $p$-adic version of the so-called Grothendieck-Teichm\"uller theory (which studies $Gal(\bar{\bf Q}/{\bf Q})$ by means of its action on profinite braid groups or mapping class groups). For every place $v$ of $\bar{\bf Q}$, we give some geometrico-combinatorial descriptions of the local Galois group $Gal(\bar{\bf Q}_v/{\bf Q}_v)$ inside $Gal(\bar{\bf Q}/{\bf Q})$. We also show that $Gal(\bar{\bf Q}_p/{\bf Q}_p)$ is the automorphism group of an appropriate $\pi_1$-functor in $p$-adic geometry.Comment: version to appear in Duke Math.