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

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.