Prime to p fundamental groups, and tame Galois actions

We show that for a local, discretely valued field F, with residue characteristic p, and a variety U over F, the map rho : Gal(F-sep/F) --> Out(pi ((p'))(1,geom) (U)) to the outer automorphisms of the prime to p geometric etale fundamental group of U maps the wild inertia onto a finite image. We show that under favourable conditions rho depends only on the reduction of U module a power of the maximal ideal of F. The proofs make use of the theory of logarithmic schemes.