Étale fundamental groups of affinoid 𝑝-adic curves

Abstract

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