Tame fundamental groups of rigid spaces

Abstract

We introduce the tame étale fundamental group $π_1^t(X/K)$ of a rigid space X over a non-archimedean field K. We show that if X is qcqs and K has topologically finitely generated tame Galois group (e.g. algebraically closed or a local field), then $π_1^t(X/K)$ is topologically finitely generated. If X is moreover the rigid generic fibre of a strictly semistable formal scheme such that the smooth locus of its special fibre admits a projective snc compactification, then $π_1^t(X/K)$ is topologically finitely presented. The proofs rely on techniques of logarithmic geometry (extended beyond its usual scope of finitely generated monoids), in particular on an analogous finiteness statement for the tame log étale fundamental group, and on the 'vertical compactification' of a map of adic spaces.