Externally definable fsg groups in NIP theories

Abstract

We show that every fsg group externally definable in an NIP structure is definably isomorphic to a group interpretable in it. Our proof relies on honest definitions and a group chunk result reconstructing a hyper-definable group from its multiplication given generically with respect to a translation invariant definable Keisler measure on it. We obtain related results on externally (type-)definable sets and groups, including a proof of a conjecture of Eleftheriou on fsg groups in real closed valued fields, and a description of externally definable, definably amenable subgroups of definable groups.