On a geometric description of Gal(Qp/Qp) and a p-adic avatar of GT

We develop a p-adic version of the so-called Grothendieck-Teichmüller theory (which studies Gal(Q/Q) by means of its action on profinite braid groups or mapping class groups). For every place v of Q, we give some geometrico-combinatorial descriptions of the local Galois group Gal(Qv/Qv) inside Gal(Q/Q). We also show that Gal(Qp/Qp) is the automorphism group of an appropriate (pi)1-functor in p-adic geometry.