Characterizing o-minimal groups in tame expansions of o-minimal structures

TL;DR

This paper provides the first comprehensive results on the structure and properties of groups definable in tame expansions of o-minimal structures, including characterizations and conditions for o-minimality.

Contribution

It establishes a Weil's group chunk theorem, characterizes o-minimal definable groups by dimension, and shows all groups are o-minimal in expansions by dense independent sets.

Findings

Definable groups with an o-minimal group chunk are o-minimal.

O-minimal definable groups have maximal dimension.

All groups are o-minimal in expansions by dense independent sets.

Abstract

We establish the first global results for groups definable in tame expansions of o-minimal structures. Let $\mathcal N$ be an expansion of an o-minimal structure $\mathcal M$ that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of $\mathcal M$ by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil's group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely their dimension equals the dimension of their topological closure, (3) if $\mathcal N$ expands $\mathcal M$ by a dense independent set, then every definable group is o-minimal.