Topologizing interpretable groups in $p$-adically closed fields

TL;DR

This paper studies the topology of interpretable groups in p-adically closed fields, introducing admissible topologies, and characterizes definable compactness via finitely satisfiable generics, extending previous results.

Contribution

It introduces the concept of admissible topologies for interpretable groups and characterizes definable compactness through finitely satisfiable generics in p-adic contexts.

Findings

Every interpretable set has an admissible topology.

Every interpretable group has a unique admissible group topology.

An interpretable group is definably compact iff it has finitely satisfiable generics.

Abstract

We consider interpretable topological spaces and topological groups in a $p$-adically closed field $K$. We identify a special class of "admissible topologies" with topological tameness properties like generic continuity, similar to the topology on definable subsets of $K^n$. We show every interpretable set has at least one admissible topology, and every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an interpretable group is definably compact if and only if it has finitely satisfiable generics (fsg), generalizing an earlier result on definable groups. As a consequence, we see that fsg is a definable property in definable families of interpretable groups, and that any fsg interpretable group defined over $\mathbb{Q}_p$ is definably isomorphic to a definable group.