Étale fundamental groups of affinoid p-adic curves

Abstract

We prove that the geometric Étale fundamental group of a (geometrically connected) rigid smooth p-adic affinoid curve is a semi-direct factor of a certain profinite free group. We prove that the maximal pro-p (resp. maximal prime-to-p) quotient of this geometric Étale fundamental group is pro-p free of infinite rank (resp. (pro-)prime-to-p free of finite computable rank).