A note on fsg groups in p-adically closed fields

TL;DR

This paper establishes a precise equivalence between the existence of finitely satisfiable generics and definable compactness for groups in p-adically closed fields, extending prior results from the rational p-adic numbers.

Contribution

It generalizes the characterization of fsg groups from the p-adic numbers to all p-adically closed fields, providing a broader understanding of their structure.

Findings

G has fsg iff G is definably compact in p-adically closed fields

Extension of previous results from Q_p to general p-adically closed fields

Clarification of the relationship between fsg and definable compactness in this setting

Abstract

Let $G$ be a definable group in a $p$-adically closed field $M$. We show that $G$ has finitely satisfiable generics (fsg) if and only if $G$ is definably compact. The case $M = \mathbb{Q}_p$ was previously proved by Onshuus and Pillay.