Etale fundamental groups of affinoid $p$-adic curves

Abstract

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