Locally definable subgroups of semialgebraic groups. (English) Zbl 1485.03130
Summary: We prove the following instance of a conjecture stated in [the second and the third author, Sel. Math., New Ser. 18, No. 4, 885–903 (2012; Zbl 1273.03130)]. Let \(G\) be an abelian semialgebraic group over a real closed field \(R\) and let \(X\) be a semialgebraic subset of \(G\). Then the group generated by \(X\) contains a generic set and, if connected, it is divisible. More generally, the same result holds when \(X\) is definable in any o-minimal expansion of \(R\) which is elementarily equivalent to \(\mathbb{R}_{\mathrm{an}, \mathrm{exp}} \). We observe that the above statement is equivalent to saying: there exists an \(m\) such that \(\Sigma_{i = 1}^m(X-X)\) is an approximate subgroup of \(G\).
MSC:
| 03C64 | Model theory of ordered structures; o-minimality |
| 03C60 | Model-theoretic algebra |
| 20A15 | Applications of logic to group theory |
| 20K99 | Abelian groups |
Citations:
Zbl 1273.03130References:
| [1] | Baily, W. L. Jr., On the theory of \(\theta \)-functions, the moduli of abelian varieties, and the moduli of curves, Ann. Math.75(2) (1962) 342-381. · Zbl 0147.39702 |
| [2] | Baro, E. and Otero, M., Locally definable homotopy, Ann. Pure Appl. Logic161(4) (2010) 488-503. · Zbl 1225.03043 |
| [3] | E. Barriga, Definably compact groups definable in real closed fields, to appear in Israel J. Math. · Zbl 1485.03131 |
| [4] | Berarducci, A., Edmundo, M. J. and Mamino, M., Discrete subgroups of locally definable groups, Selecta Math. (2012) 1-17. · Zbl 1294.03026 |
| [5] | Birkenhake, C. and Lange, H., Complex Abelian Varieties, 2nd edn., , Vol. 302 (Springer-Verlag, Berlin, 2004). · Zbl 1056.14063 |
| [6] | Breuillard, E., A Brief Introduction to Approximate Groups, Vol. 61 (MSRI publications, 2013). |
| [7] | van den Dries, L. and Miller, C., On the real exponential field with restricted analytic functions, Israel J. Math.85(1-3) (1994) 19-56. · Zbl 0823.03017 |
| [8] | Edmundo, M., Locally definable groups in o-minimal structures, J. Algebra301(1) (2006) 194-223. · Zbl 1104.03032 |
| [9] | Edmundo, M. and Eleftheriou, P. E., The universal covering homomorphism in o-minimal expansions of groups, Math. Logic Q.53(6) (2007) 571-582. · Zbl 1130.03027 |
| [10] | Eleftheriou, P. E. and Peterzil, Y., Definable quotients of locally definable groups, Selecta Math. (N.S.)18(4) (2012) 885-903. · Zbl 1273.03130 |
| [11] | Eleftheriou, P. E. and Peterzil, Y., Lattices in locally definable subgroups of \(\langle R^n,+\rangle \), Notre Dame J. Formal Logic54(3-4) (2013) 449-461. · Zbl 1345.03072 |
| [12] | Hrushovski, E., Stable group theory and approximate subgroups, J. Amer. Math. Soc.25(1) (2012) 189-243. · Zbl 1259.03049 |
| [13] | Hrushovski, E. and Pillay, A., Groups definable in local fields and pseudo-finite fields, Israel J. Math.85 (1994), 203-262. · Zbl 0804.03024 |
| [14] | Hrushovski, E., Pillay, A. and Peterzil, Y., Groups, measure and the NIP, J. Amer. Math. Soc.21 (2008) 563-596. · Zbl 1134.03024 |
| [15] | Massicot, J. C. and Wagner, F., Approximate subgroups, J. Éc. polytech. Math.2 (2015) 5-64. · Zbl 1379.03008 |
| [16] | J. S. Milne, Algebraic groups, https://www.jmilne.org/math/CourseNotes/iAG200. pdf. |
| [17] | Peterzil, Y., Pillay, A. and Starchenko, S., Linear groups in o-minimal structures, J. Algebra247 (2002) 1-23. · Zbl 0991.03039 |
| [18] | Peterzil, Y. and Starchenko, S., Definable homomorphisms of abelian groups in o-minimal structures, Ann. Pure Appl. Logic101(1) (2000) 1-27. · Zbl 0949.03033 |
| [19] | Peterzil, Y. and Starchenko, S., On torsion-free groups in o-minimal structures, Illinois J. Math.49(4) (2008) 1299-1321. · Zbl 1085.03027 |
| [20] | Peterzil, Y. and Starchenko, S., Definability of restricted theta functions and families of abelian varieties, Duke Math. J.162(4) (2013) 731-765. · Zbl 1284.03215 |
| [21] | Peterzil, Y. and Steinhorn, C., Definable compactness and definable subgroups of o-minimal groups, J. London Math. Soc.59 (1999) 769-786. · Zbl 0935.03047 |
| [22] | Scanlon, T., Algebraic differential equations from covering maps, Adv. Math.330 (2018) 1071-1100. · Zbl 1423.12008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
