Groups, measures, and the NIP

Hrushovski, Ehud; Peterzil, Ya’acov; Pillay, Anand

doi:10.1090/S0894-0347-07-00558-9

Groups, measures, and the NIP

Authors: Ehud Hrushovski, Ya'acov Peterzil and Anand Pillay
Journal: J. Amer. Math. Soc. 21 (2008), 563-596
MSC (2000): Primary 03C68, 03C45, 22C05, 28E05
DOI: https://doi.org/10.1090/S0894-0347-07-00558-9
Published electronically: February 2, 2007
MathSciNet review: 2373360
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Abstract: We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups in saturated -minimal structures to compact Lie groups. We also prove some other structural results about such , for example the existence of a left invariant finitely additive probability measure on definable subsets of . We finally introduce the new notion of ``compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the -minimal case.

1. Yerzhan Baisalov and Bruno Poizat, Paires de structures o-minimales, J. Symbolic Logic 63 (1998), no. 2, 570–578 (French, with Esperanto summary). MR 1627306, https://doi.org/10.2307/2586850
2. Alessandro Berarducci and Margarita Otero, An additive measure in o-minimal expansions of fields, Q. J. Math. 55 (2004), no. 4, 411–419. MR 2104681, https://doi.org/10.1093/qjmath/55.4.411
3. Alessandro Berarducci and Margarita Otero, Intersection theory for o-minimal manifolds, Ann. Pure Appl. Logic 107 (2001), no. 1-3, 87–119. MR 1807841, https://doi.org/10.1016/S0168-0072(00)00027-0
4. Alessandro Berarducci, Margarita Otero, Yaa’cov Peterzil, and Anand Pillay, A descending chain condition for groups definable in o-minimal structures, Ann. Pure Appl. Logic 134 (2005), no. 2-3, 303–313. MR 2139910, https://doi.org/10.1016/j.apal.2005.01.002
5. Alfred Dolich, Forking and independence in o-minimal theories, J. Symbolic Logic 69 (2004), no. 1, 215–240. MR 2039358, https://doi.org/10.2178/jsl/1080938838
6. Mário J. Edmundo, Locally definable groups in o-minimal structures, J. Algebra 301 (2006), no. 1, 194–223. MR 2230327, https://doi.org/10.1016/j.jalgebra.2005.04.016
7. Mário J. Edmundo and Margarita Otero, Definably compact abelian groups, J. Math. Log. 4 (2004), no. 2, 163–180. MR 2114966, https://doi.org/10.1142/S0219061304000358
8. P. Eleftheriou, Ph.D. thesis, U. of Notre Dame.
9. Euclid, Elements, Book V, tr. T.L. Heath.
10. Rami Grossberg, José Iovino, and Olivier Lessmann, A primer of simple theories, Arch. Math. Logic 41 (2002), no. 6, 541–580. MR 1923196, https://doi.org/10.1007/s001530100126
11. Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
12. E. Hrushovski, Valued fields, metastable groups, draft, 2004.
13. H. Jerome Keisler, Measures and forking, Ann. Pure Appl. Logic 34 (1987), no. 2, 119–169. MR 890599, https://doi.org/10.1016/0168-0072(87)90069-8
14. L. Newelski and M. Petrykowski, Weak generic types and coverings of groups, Fund. Math. 191 (2006), 201-225.
15. A. Onshuus, Groups definable in $(\mathbb{Z},+,<)$ , preprint, 2005.
16. A. Onshuus and A. Pillay, Definable groups and -adic Lie groups, preprint, 2005.
17. Y. Peterzil, A. Pillay, and S. Starchenko, Definably simple groups in o-minimal structures, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4397–4419. MR 1707202, https://doi.org/10.1090/S0002-9947-00-02593-9
18. Y. Peterzil, A. Pillay, and S. Starchenko, Linear groups definable in o-minimal structures, J. Algebra 247 (2002), no. 1, 1–23. MR 1873380, https://doi.org/10.1006/jabr.2001.8861
19. Y. Peterzil and A. Pillay, Generic sets in definably compact groups, Fund. Math. 193 (2007), 153-170.
20. Ya’acov Peterzil and Sergei Starchenko, Definable homomorphisms of abelian groups in o-minimal structures, Ann. Pure Appl. Logic 101 (2000), no. 1, 1–27. MR 1729742, https://doi.org/10.1016/S0168-0072(99)00016-0
21. Ya’acov Peterzil and Sergei Starchenko, Uniform definability of the Weierstrass ℘ functions and generalized tori of dimension one, Selecta Math. (N.S.) 10 (2004), no. 4, 525–550. MR 2134454, https://doi.org/10.1007/s00029-005-0393-y
22. Ya’acov Peterzil and Charles Steinhorn, Definable compactness and definable subgroups of o-minimal groups, J. London Math. Soc. (2) 59 (1999), no. 3, 769–786. MR 1709079, https://doi.org/10.1112/S0024610799007528
23. Anand Pillay, Type-definability, compact Lie groups, and o-minimality, J. Math. Log. 4 (2004), no. 2, 147–162. MR 2114965, https://doi.org/10.1142/S0219061304000346
24. Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1429864
25. Bruno Poizat, A course in model theory, Universitext, Springer-Verlag, New York, 2000. An introduction to contemporary mathematical logic; Translated from the French by Moses Klein and revised by the author. MR 1757487
26. Saharon Shelah, Classification theory for elementary classes with the dependence property—a modest beginning, Sci. Math. Jpn. 59 (2004), no. 2, 265–316. Special issue on set theory and algebraic model theory. MR 2062198
27. S. Shelah, Minimal bounded index subgroup for dependent theories, to appear in Proceedings AMS.