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arXiv · 1805.11500

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

Pantelis E. Eleftheriou
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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.

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Pantelis E. Eleftheriou. 2018-05-29. Characterizing o-minimal groups in tame expansions of o-minimal structures. https://arxiv.org/abs/1805.11500

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