Submission date: 29 May 2018. Revised 20 July 2018

Abstract:

We establish the first global results for groups definable in tame expansions of o-minimal structures. Let N be an expansion of an o-minimal structure M that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of 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 N expands M by a dense independent set, then every definable group is o-minimal.

Mathematics Subject Classification: 03C64

Keywords and phrases:

Full text arXiv 1805.11500: pdf, ps.