Stable group theory and approximate subgroups

Hrushovski, Ehud

doi:10.1090/S0894-0347-2011-00708-X

Stable group theory and approximate subgroups

Author: Ehud Hrushovski
Journal: J. Amer. Math. Soc. 25 (2012), 189-243
MSC (2010): Primary 11P70, 03C45
DOI: https://doi.org/10.1090/S0894-0347-2011-00708-X
Published electronically: June 15, 2011
MathSciNet review: 2833482
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Abstract: We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group , we show that a finite subset with $\vert X X ^{-1}X \vert/ \vert X\vert$ bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of . We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.

1. Itaï Ben Yaacov and Alexander Usvyatsov, Continuous first order logic and local stability, Trans. Amer. Math. Soc. 362 (2010), no. 10, 5213–5259. MR 2657678, https://doi.org/10.1090/S0002-9947-10-04837-3
2. George M. Bergman and Hendrik W. Lenstra Jr., Subgroups close to normal subgroups, J. Algebra 127 (1989), no. 1, 80–97. MR 1029404, https://doi.org/10.1016/0021-8693(89)90275-5
3. Breuillard, Emmanuel; Green, Ben; Terence, Tao, Approximate subgroups of linear groups, arXiv:1005.1881.
4. Breuillard, Emmanuel; Green, Ben, Approximate groups, II: the solvable linear case, arXiv:0907.0927.
5. J. Bourgain, N. Katz, and T. Tao, A sum-product estimate in finite fields, and applications, Geom. Funct. Anal. 14 (2004), no. 1, 27–57. MR 2053599, https://doi.org/10.1007/s00039-004-0451-1
6. C. C. Chang and H. J. Keisler, Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland Publishing Co., Amsterdam, 1990. MR 1059055
7. Mei-Chu Chang, Product theorems in 𝑆𝐿₂ and 𝑆𝐿₃, J. Inst. Math. Jussieu 7 (2008), no. 1, 1–25. MR 2398145, https://doi.org/10.1017/S1474748007000126
8. Gregory Cherlin and Ehud Hrushovski, Finite structures with few types, Annals of Mathematics Studies, vol. 152, Princeton University Press, Princeton, NJ, 2003. MR 1961194
9. Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, AMS Chelsea Publishing, Providence, RI, 2006. Reprint of the 1962 original. MR 2215618
10. Dinai, Oren, Expansion properties of finite simple groups. Ph.D. thesis. The Hebrew University of Jerusalem (2009). arXiv:1001.5069.
11. György Elekes and Zoltán Király, On the combinatorics of projective mappings, J. Algebraic Combin. 14 (2001), no. 3, 183–197. MR 1869409, https://doi.org/10.1023/A:1012799318591
12. van den Dries, L., notes in http://www.math.uiuc.edu/vddries/.
13. L. van den Dries and A. J. Wilkie, Gromov’s theorem on groups of polynomial growth and elementary logic, J. Algebra 89 (1984), no. 2, 349–374. MR 751150, https://doi.org/10.1016/0021-8693(84)90223-0
14. P. Erdős and E. Szemerédi, On sums and products of integers, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 213–218. MR 820223
15. A. M. Gleason, The structure of locally compact groups, Duke Math. J. 18 (1951), 85–104. MR 39730
16. Isaac Goldbring, Hilbert’s fifth problem for local groups, Ann. of Math. (2) 172 (2010), no. 2, 1269–1314. MR 2680491, https://doi.org/10.4007/annals.2010.172.1273
17. Ben Green and Tom Sanders, A quantitative version of the idempotent theorem in harmonic analysis, Ann. of Math. (2) 168 (2008), no. 3, 1025–1054. MR 2456890, https://doi.org/10.4007/annals.2008.168.1025
18. Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
19. Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR 0033869
20. H. A. Helfgott, Growth and generation in 𝑆𝐿₂(ℤ/𝕡ℤ), Ann. of Math. (2) 167 (2008), no. 2, 601–623. MR 2415382, https://doi.org/10.4007/annals.2008.167.601
21. Helfgott, H. A., Growth in , arXiv:0807.2027.
22. Edwin Hewitt and Leonard J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc. 80 (1955), 470–501. MR 76206, https://doi.org/10.1090/S0002-9947-1955-0076206-8
23. Ehud Hrushovski, Pseudo-finite fields and related structures, Model theory and applications, Quad. Mat., vol. 11, Aracne, Rome, 2002, pp. 151–212. MR 2159717
24. Ehud Hrushovski and Frank Wagner, Counting and dimensions, Model theory with applications to algebra and analysis. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 350, Cambridge Univ. Press, Cambridge, 2008, pp. 161–176. MR 2436141, https://doi.org/10.1017/CBO9780511735219.005
25. Ehud Hrushovski, Ya’acov Peterzil, and Anand Pillay, Groups, measures, and the NIP, J. Amer. Math. Soc. 21 (2008), no. 2, 563–596. MR 2373360, https://doi.org/10.1090/S0894-0347-07-00558-9
26. Hrushovski, Ehud; Pillay, Anand, On NIP and invariant measures, arXiv:0710.2330.
27. Hrushovski, Ehud; Pillay, Anand; Simon, Pierre, Generically stable and smooth measures in NIP theories, arXiv:1002.4763.
28. Olav Kallenberg, On the representation theorem for exchangeable arrays, J. Multivariate Anal. 30 (1989), no. 1, 137–154. MR 1003713, https://doi.org/10.1016/0047-259X(89)90092-4
29. Jordan, Camille, Mémoire sur les équations différentielles linéaires intégrale algébriques, Crelle 84 (1878) pp. 89-215.
30. Irving Kaplansky, Lie algebras and locally compact groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1995. Reprint of the 1974 edition. MR 1324106
31. Byunghan Kim and Anand Pillay, Simple theories, Ann. Pure Appl. Logic 88 (1997), no. 2-3, 149–164. Joint AILA-KGS Model Theory Meeting (Florence, 1995). MR 1600895, https://doi.org/10.1016/S0168-0072(97)00019-5
32. J. Komlós and M. Simonovits, Szemerédi’s regularity lemma and its applications in graph theory, Combinatorics, Paul Erdős is eighty, Vol. 2 (Keszthely, 1993) Bolyai Soc. Math. Stud., vol. 2, János Bolyai Math. Soc., Budapest, 1996, pp. 295–352. MR 1395865
33. Peter H. Krauss, Representation of symmetric probability models, J. Symbolic Logic 34 (1969), 183–193. MR 275482, https://doi.org/10.2307/2271093
34. Aspects of inductive logic, Edited by Jaakko Hintikka and Patrick Suppes, North-Holland Publishing Co., Amsterdam, 1966. MR 0201269
35. Larsen, Michael J.; Pink, Richard, Finite subgroups of algebraic groups, 1998. Available from http://www.math.ethz.ch/ $\sim$ pink/publications.html.
36. John M. Lee, Riemannian manifolds, Graduate Texts in Mathematics, vol. 176, Springer-Verlag, New York, 1997. An introduction to curvature. MR 1468735
37. David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. An introduction. MR 1924282
38. David Marker, Semialgebraic expansions of 𝐶, Trans. Amer. Math. Soc. 320 (1990), no. 2, 581–592. MR 964900, https://doi.org/10.1090/S0002-9947-1990-0964900-4
39. Michael Morley, The Löwenheim-Skolem theorem for models with standard part, Symposia Mathematica, Vol. V (INDAM, Rome, 1969/70) Academic Press, London, 1971, pp. 43–52. MR 0284322
40. Nikolov, Nikolay; Pyber, László, Product decompositions of quasirandom groups and a Jordan type theorem, arXiv:math/0703343.
41. Pillay, Anand, http://www.amsta.leeds.ac.uk/pillay/.
42. Anand Pillay, An introduction to stability theory, Oxford Logic Guides, vol. 8, The Clarendon Press, Oxford University Press, New York, 1983. MR 719195
43. Bruno Poizat, Cours de théorie des modèles, Bruno Poizat, Lyon, 1985 (French). Une introduction à la logique mathématique contemporaine. [An introduction to contemporary mathematical logic]. MR 817208
44. Bruno Poizat, An introduction to algebraically closed fields & varieties, The model theory of groups (Notre Dame, IN, 1985–1987) Notre Dame Math. Lectures, vol. 11, Univ. Notre Dame Press, Notre Dame, IN, 1989, pp. 41–67 (French, with English summary). MR 985339
45. M. S. Raghunathan, Discrete subgroups of Lie groups, Math. Student Special Centenary Volume (2007), 59–70 (2008). MR 2527560
46. Shalom, Yehuda; Tao, Terence, A finitary version of Gromov's polynomial growth theorem, arXiv:0910.4148v2 [math.GR].
47. Saharon Shelah, The lazy model-theoretician’s guide to stability, Logique et Anal. (N.S.) 18 (1975), no. 71-72, 241–308. MR 539969
48. S. Shelah, Classification theory and the number of nonisomorphic models, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990. MR 1083551
49. Saharon Shelah, Simple unstable theories, Ann. Math. Logic 19 (1980), no. 3, 177–203. MR 595012, https://doi.org/10.1016/0003-4843(80)90009-1
50. Terence Tao, Product set estimates for non-commutative groups, Combinatorica 28 (2008), no. 5, 547–594. MR 2501249, https://doi.org/10.1007/s00493-008-2271-7
51. Terence Tao, The sum-product phenomenon in arbitrary rings, Contrib. Discrete Math. 4 (2009), no. 2, 59–82. MR 2592424
52. Tao, Terence, http://terrytao.wordpress.com/2007/03/02/open-question-noncommutative- freiman-theorem/.
53. Tao, Terence, Freiman's theorem for solvable groups, arXiv:0906.3535.
54. Terence Tao and Van Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006. MR 2289012
55. Hidehiko Yamabe, A generalization of a theorem of Gleason, Ann. of Math. (2) 58 (1953), 351–365. MR 58607, https://doi.org/10.2307/1969792
56. André Weil, Foundations of algebraic geometry, American Mathematical Society, Providence, R.I., 1962. MR 0144898
57. Robert J. Zimmer, Essential results of functional analysis, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1990. MR 1045444