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CHARACTERIZING O-MINIMAL GROUPS IN TAME EXPANSIONS OF O-MINIMAL STRUCTURES

Part of: Model theory Locally compact abelian groups

Published online by Cambridge University Press:  16 July 2019

Pantelis E. Eleftheriou Open the ORCID record for Pantelis E. Eleftheriou [Opens in a new window]
Pantelis E. Eleftheriou*
Affiliation:
Department of Mathematics and Statistics, University of Konstanz, Box 216, 78457Konstanz, Germany ([email protected])
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Abstract

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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) as an application, if ${\mathcal{N}}$ expands ${\mathcal{M}}$ by a dense independent set, then every definable group is o-minimal.

Keywords

o-minimal structuretame expansiondimension functiondefinable groupindependent setgroup chunk

MSC classification

Primary: 03C64: Model theory of ordered structures; o-minimality 03C68: Other classical first-order model theory
Secondary: 22B99: None of the above, but in this section
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 2 , March 2021 , pp. 699 - 724
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2019. Published by Cambridge University Press

Footnotes

The research was supported by an Independent Research Grant from the German Research Foundation (DFG) and a Zukunftskolleg Research Fellowship.

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