Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T01:56:40.302Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Locally compact abelian groups Model theory

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])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Check for updates
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.

References

Baro, E. and Martin-Pizarro, A., Small groups in dense pairs, Preprint, 2018, arXiv:1801.08744v1.Google Scholar
Belegradek, O. and Zilber, B., The model theory of the field of reals with a subgroup of the unit circle, J. Lond. Math. Soc. (2) 78 (2008), 563–579.CrossRefGoogle Scholar
Berenstein, A., Ealy, C. and Günaydin, A., Thorn independence in the field of real numbers with a small multiplicative group, Ann. Pure Appl. Logic 150 (2007), 1–18.CrossRefGoogle Scholar
Berenstein, A. and Vassiliev, E., Geometric structures with a dense independent subset, Selecta Math. (N.S.) 22 (2016), 191–225.CrossRefGoogle Scholar
Berenstein, A. and Vassiliev, E., Fields with a dense-codense linearly independent multiplicative subgroup, Preprint, 2017, Modnet 1238. Archive for Mathematical Logic, to appear.Google Scholar
Boxall, G. and Hieronymi, P., Expansions which introduce no new open sets, J. Symbolic Logic 77(1) (2012), 111–121.CrossRefGoogle Scholar
Dolich, A., Miller, C. and Steinhorn, C., Structures having o-minimal open core, Trans. Amer. Math. Soc. 362 (2010), 1371–1411.CrossRefGoogle Scholar
Dolich, A., Miller, C. and Steinhorn, C., Expansions of o-minimal structures by dense independent sets, Ann. Pure Appl. Logic 167 (2016), 684–706.CrossRefGoogle Scholar
van den Dries, L., Remarks on Tarski’s problem concerning (R, +, -, exp), in Logic Colloquium (1982) (ed. Lolli, G., Longo, G. and Marcja, A.), pp. 97–121 (North-Holland, 1984).CrossRefGoogle Scholar
van den Dries, L., Dense pairs of o-minimal structures, Fund. Math. 157 (1988), 61–78.CrossRefGoogle Scholar
van den Dries, L., Dimension of definable sets, algebraic boundedness and henselian fields, Ann. Pure Appl. Logic 45 (1989), 189–209.CrossRefGoogle Scholar
van den Dries, L., Weil’s group chunk theorem: A topological setting, Illinois J. Math. 34 (1990), 127–139.CrossRefGoogle Scholar
van den Dries, L., Tame Topology and O-minimal Structures (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
van den Dries, L. and Günaydın, A., The fields of real and complex numbers with a small multiplicative group, Proc. Lond. Math. Soc. 93 (2006), 43–81.CrossRefGoogle Scholar
Eleftheriou, P., Non-standard lattices and o-minimal groups, Bull. Symbolic Logic 19 (2013), 56–76.CrossRefGoogle Scholar
Eleftheriou, P., Counting algebraic points in expansions of o-minimal structures by a dense set, Preprint, 2018, arXiv:1708.03936v3.Google Scholar
Eleftheriou, P., Günaydin, A. and Hieronymi, P., Structure theorems in tame expansions of o-minimal structures by dense sets, Preprint, 2017, arXiv:1510.03210v4.Google Scholar
Eleftheriou, P., Günaydin, A. and Hieronymi, P., The Choice Property in tame expansions of o-minimal structures, Preprint, 2017, arXiv:1708.03896v1.Google Scholar
Eleftheriou, P. and Peterzil, Y., Definable groups as homomorphic images of semilinear and field-definable groups, Selecta Math. (N.S.) 18 (2012), 905–940.CrossRefGoogle Scholar
Eleftheriou, P., Peterzil, Y. and Ramakrishnan, J., Interpretable groups are definable, J. Math. Log. 14 (2014), 1450002.CrossRefGoogle Scholar
Fornasiero, A., Groups and rings definable in d-minimal structures, Preprint, 2012, arXiv:1205.4177v1.Google Scholar
Günaydın, A. and Hieronymi, P., The real field with the rational points of an elliptic curve, Fund. Math. 215 (2011), 167–175.Google Scholar
Hrushovski, E., The Mordell–Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996), 667–690.CrossRefGoogle Scholar
Hrushovski, E., Peterzil, Y. and Pillay, A., Groups, measures, and the NIP, J. Amer. Math. Soc. 21 (2008), 563–596.CrossRefGoogle Scholar
Johns, J., An Open Mapping Theorem for O-Minimal Structures, J. Symbolic Logic 66 (2001), 1817–1820.CrossRefGoogle Scholar
Knight, J., Pillay, A. and Steinhorn, C., Definable sets in ordered structures II, Trans. Amer. Math. Soc. 295 (1986), 593–605.CrossRefGoogle Scholar
Miller, C. and Speissegger, P., Expansions of the real line by open sets: o-minimality and open cores, Fund. Math. 162 (1999), 193–208.Google Scholar
Miller, C. and Starchenko, S., A growth dichotomy for o-minimal expansions of ordered groups, Trans. Amer. Math. Soc. 350(9) 3505–3521.CrossRefGoogle Scholar
Peterzil, Y., Returning to semi-bounded sets, J. Symbolic Logic 74 (2009), 597–617.CrossRefGoogle Scholar
Pila, J., O-minimality and the André–Oort conjecture for ℂ n , Ann. Math. 173 (2011), 1779–1840.CrossRefGoogle Scholar
Pila, J. and Wilkie, A. J., The rational points of a definable set, Duke Math. J. 133 (2006), 591–616.CrossRefGoogle Scholar
Pillay, A. and Steinhorn, C., Definable sets in ordered structures I, Trans. Amer. Math. Soc. 295 (1986), 565–592.CrossRefGoogle Scholar
Robinson, A., Solution of a problem of Tarski, Fund. Math. 47 (1959), 79–204.CrossRefGoogle Scholar
Wencel, R., Groups, group actions and fields definable in first-order topological structures, Math. Log. Quart. 58 (2012), 449–467.CrossRefGoogle Scholar