Etale fundamental groups of affinoid $p$-adic curves

TL;DR

This paper investigates the structure of the geometric étale fundamental group of smooth $p$-adic affinoid curves, revealing it as a semi-direct factor of a profinite free group and describing its maximal pro-$p$ and prime-to-$p$ quotients.

Contribution

It establishes the semi-direct factor relationship and characterizes the maximal pro-$p$ and prime-to-$p$ quotients of the fundamental group, providing new structural insights.

Findings

The geometric étale fundamental group is a semi-direct factor of a profinite free group.

Maximal pro-$p$ quotient is pro-$p$ free of infinite rank.

Maximal prime-to-$p$ quotient is free of finite computable rank.

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).