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ON NON-COMPACT p-ADIC DEFINABLE GROUPS

Part of: Model theory

Published online by Cambridge University Press:  22 November 2021

WILL JOHNSON Open the ORCID record for WILL JOHNSON [Opens in a new window]  and
NINGYUAN YAO Open the ORCID record for NINGYUAN YAO [Opens in a new window]
WILL JOHNSON
Affiliation:
SCHOOL OF PHILOSOPHY FUDAN UNIVERSITY 220 HANDAN ROAD, GUANGHUA WEST BUILDING, ROOM 2503 SHANGHAI 20043, CHINAE-mail:willjohnson@fudan.edu.cnE-mail:yaony@fudan.edu.cn
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Abstract

In [16], Peterzil and Steinhorn proved that if a group G definable in an o-minimal structure is not definably compact, then G contains a definable torsion-free subgroup of dimension 1. We prove here a p-adic analogue of the Peterzil–Steinhorn theorem, in the special case of abelian groups. Let G be an abelian group definable in a p-adically closed field M. If G is not definably compact then there is a definable subgroup H of dimension 1 which is not definably compact. In a future paper we will generalize this to non-abelian G.

Keywords

definable groupsNIP groupsp-adic numbers

MSC classification

Primary: 03C60: Model-theoretic algebra 03C98: Applications of model theory
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 1 , March 2022 , pp. 188 - 213
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Belair, L., Panorama of p-adic model theory . Annales des Sciences mathématiques du Québec, vol. 36 (2012), no. 1, pp. 43–75.Google Scholar
Dolich, A., Goodrick, J., and Lippel, D., Dp-minimality: Basic facts and examples . Notre Dame Journal of Formal Logic, vol. 52 (2011), no. 3, pp. 267–288.CrossRefGoogle Scholar
van den Dries, L., Algebraic theories with definable Skolem functions, this Journal, vol. 49 (1984), pp. 625–629.Google Scholar
Fornasiero, A., Definable compactness for topological structures, in preparation, 2015.Google Scholar
Günaydin, A. and Hieronymi, P., Dependent pairs, this Journal, vol. 76 (2011), no. 2, pp. 377–390.Google Scholar
Halevi, Y., Hasson, A., and Jahnke, F., Definable V-topologies, Henselianity and NIP . Journal of Mathematical Logic, vol. 20 (2020), no. 2, p. 2050008.10.1142/S0219061320500087CrossRefGoogle Scholar
Hrushovski, E., Peterzil, Y., and Pillay, A., Groups, measures, and the NIP . Journal of the American Mathematical Society, vol. 21 (2008), no. 2, pp. 563–596.CrossRefGoogle Scholar
Hrushovski, E. and Pillay, A., Groups definable in local and pseudofinite fields . Israel Journal of Mathematics, vol. 85 (1994), pp. 203–262.CrossRefGoogle Scholar
Johnson, W., Interpretable sets in dense o-minimal structures, this Journal, vol. 83 (2018), pp. 1477–1500.Google Scholar
Kaplan, I., Onshuus, A., and Usvyatsov, A., Additivity of the dp-rank . Transactions of the American Mathematical Society, vol. 365 (2013), no. 11, pp. 5783–5804.10.1090/S0002-9947-2013-05782-0CrossRefGoogle Scholar
Macintyre, A., On definable subsets of p-adic fields, this Journal, vol. 41 (1976), pp. 605–610.Google Scholar
Marker, D., Model Theory: An Introduction, Springer, New York, 2002.Google Scholar
Onshuus, A. and Pillay, A., Definable groups and compact p-adic lie groups . Journal of the London Mathematical Society, vol. 78 (2008), no. 1, pp. 233–247.CrossRefGoogle Scholar
Penazzi, D., Pillay, A., and Yao, N.. Some model theory and topological dynamics of p-adic algebraic groups. Fundamenta Mathematicae , vol. 247 (2019), pp. 191–216.Google Scholar
Peterzil, Y. and Starchenko, S., Topological groups, μ-types and their stabilizers . Journal of the European Mathematical Society (JEMS), vol. 19 (2017), no. 10, pp. 2965–2995.CrossRefGoogle Scholar
Peterzil, Y. and Steinhorn, C., Definable compactness and definable subgroups of o-minimal groups . Journal of the London Mathematical Society, vol. 59 (1999), no. 3, pp. 769–786.CrossRefGoogle Scholar
Pillay, A., Groups and fields definable in o-minimal structures . Journal of Pure and Applied Algebra, vol. 53 (1988), pp. 233–255.CrossRefGoogle Scholar
Pillay, A., On fields definable in ℚp . Archive for Mathematical Logic, vol. 29 (1989), pp. 1–7.10.1007/BF01630806CrossRefGoogle Scholar
Pillay, A. and Yao, N., Definable f-generic groups over p-adic numbers, preprint, 2019, arXiv:1911.01833.Google Scholar
Pillay, A. and Yao, N., A note on groups definable in the p-adic field . Archive for Mathematical Logic, vol. 58 (2019), pp. 1029–1034.CrossRefGoogle Scholar
Poizat, B., A Course in Model Theory, Springer, New York, 2000.CrossRefGoogle Scholar
Simon, P., A Guide to NIP Theories, Lecture Notes in Logic, vol. 44, Cambridge University Press, Cambridge, 2015.CrossRefGoogle Scholar
Yao, N., On dimensions, standard part maps, and p-adically closed fields . Studies in Logic, vol. 13 (2020), no. 6, pp. 41–62.Google Scholar