Externally definable fsg groups in NIP theories

TL;DR

This paper proves that fsg groups externally definable in NIP theories are essentially the same as interpretable groups, using honest definitions and measure-theoretic techniques, and applies these results to real closed valued fields.

Contribution

It establishes a definable isomorphism between externally definable fsg groups and interpretable groups in NIP structures, and proves a conjecture in real closed valued fields.

Findings

Externally definable fsg groups are definably isomorphic to interpretable groups.

Confirmed Eleftheriou's conjecture on fsg groups in real closed valued fields.

Characterized externally definable, definably amenable subgroups of definable groups.

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.