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Publications mathématiques de l'IHÉS

, Volume 116, Issue 1, pp 115–221 | Cite as

The structure of approximate groups

  • Emmanuel Breuillard
  • Ben Green
  • Terence Tao
Article

Abstract

Let K⩾1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that AA is covered by K left translates of A.

The main result of this paper is a qualitative description of approximate groups as being essentially finite-by-nilpotent, answering a conjecture of H. Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary groups.

We begin by establishing a correspondence principle between approximate groups and locally compact (local) groups that allows us to recover many results recently established in a fundamental paper of Hrushovski. In particular we establish that approximate groups can be approximately modeled by Lie groups.

To prove our main theorem we apply some additional arguments essentially due to Gleason. These arose in the solution of Hilbert’s fifth problem in the 1950s.

Applications of our main theorem include a finitary refinement of Gromov’s theorem, as well as a generalized Margulis lemma conjectured by Gromov and a result on the virtual nilpotence of the fundamental group of Ricci almost nonnegatively curved manifolds.

Keywords

Local Group Global Group Approximate Group Escape Norm Local Homomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    I. Benjamini and G. Kozma, A resistance bound via an isoperimetric inequality, Combinatorica, 25 (2005), 645–650. MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    L. Bieberbach, Über einen Satz des Herrn C. Jordan in der Theorie der endlichen Gruppen linearer Substitutionen, Sitzber. Preuss. Akad. Wiss, Berlin, 1911. Google Scholar
  3. 3.
    Y. Bilu, Addition of sets of integers of positive density, J. Number Theory, 64 (1997), 233–275. MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Y. Bilu, Structure of sets with small sumset, Astérisque, 258 (1999), 77–108. Structure theory of set addition. MathSciNetGoogle Scholar
  5. 5.
    E. Breuillard and B. Green, Approximate groups. I: the torsion-free nilpotent case, J. Inst. Math. Jussieu, 10 (2011), 37–57. MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    E. Breuillard and B. Green, Approximate groups. II: the solvable linear case, Q. J. of Math., Oxf., 62 (2011), 513–521. MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    E. Breuillard and B. Green, Approximate groups. III: the unitary case, Turk. J. Math., 36 (2012), 199–215. MathSciNetzbMATHGoogle Scholar
  8. 8.
    E. Breuillard, B. Green, and T. Tao, Approximate subgroups of linear groups, Geom. Funct. Anal., 21 (2011), 774–819. MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Y. D. Burago and V. A. Zalgaller, Geometric Inequalities, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ, Springer Series in Soviet Mathematics. zbMATHGoogle Scholar
  10. 10.
    M.-C. Chang, A polynomial bound in Freiman’s theorem, Duke Math. J., 113 (2002), 399–419. MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    J. Cheeger and T. H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. Math., 144 (1996), 189–237. MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    L. J. Corwin and F. Greenleaf, Representations of Nilpotent Lie Groups and Their Applications, CUP, Cambridge, 1990. zbMATHGoogle Scholar
  13. 13.
    E. Croot and O. Sisask, A probabilistic technique for finding almost-periods of convolutions, Geom. Funct. Anal., 20 (2010), 1367–1396. MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    D. Fisher, N. H. Katz, and I. Peng, Approximate multiplicative groups in nilpotent Lie groups, Proc. Am. Math. Soc., 138 (2010), 1575–1580. MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    G. A. Freiman, Foundations of a Structural Theory of Set Addition, American Mathematical Society, Providence, 1973. Translated from the Russian, Translations of Mathematical Monographs, vol. 37. zbMATHGoogle Scholar
  16. 16.
    K. Fukaya and T. Yamaguchi, The fundamental groups of almost non-negatively curved manifolds, Ann. Math., 136 (1992), 253–333. MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry, Universitext, Springer, Berlin, 1987. zbMATHCrossRefGoogle Scholar
  18. 18.
    N. Gill and H. Helfgott, Growth in solvable subgroups of GLr(Z/p Z), preprint (2010), arXiv:1008.5264.
  19. 19.
    N. Gill and H. Helfgott, Growth of small generating sets in SLn(Z/p Z), Int. Math. Res. Not., 18 (2011), 4226–4251. MathSciNetGoogle Scholar
  20. 20.
    A. M. Gleason, The structure of locally compact groups, Duke Math. J., 18 (1951), 85–104. MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    A. M. Gleason, Groups without small subgroups, Ann. Math., 56 (1952), 193–212. MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    K. Gödel, Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory, Proc. Natl. Acad. Sci, 24 (1938), 556–557. CrossRefGoogle Scholar
  23. 23.
    I. Goldbring, Nonstandard Methods in Lie Theory, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2009. Google Scholar
  24. 24.
    I. Goldbring, Hilbert’s fifth problem for local groups, Ann. Math., 172 (2010), 1269–1314. MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    B. Green and I. Z. Ruzsa, Freiman’s theorem in an arbitrary abelian group, J. Lond. Math. Soc., 75 (2007), 163–175. MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    B. Green and T. Tao, Compressions, convex geometry and the Freiman-Bilu theorem, Q. J. Math., 57 (2006), 495–504. MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. IHÉS, 53 (1981), 53–73. MathSciNetzbMATHGoogle Scholar
  28. 28.
    M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Modern Birkhäuser Classics, Birkhäuser, Boston, 2007. Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates. zbMATHGoogle Scholar
  29. 29.
    M. Hall Jr., The Theory of Groups, Chelsea Publishing Co., New York, 1976. Reprinting of the 1968 edition. zbMATHGoogle Scholar
  30. 30.
    H. A. Helfgott, Growth and generation in SL2(Z/p Z), Ann. Math., 167 (2008), 601–623. MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    H. A. Helfgott, Growth in SL3(Z/p Z), J. Eur. Math. Soc., 13 (2011), 761–851. MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    J. Hirschfeld, The nonstandard treatment of Hilbert’s fifth problem, Trans. Am. Math. Soc., 321 (1990), 379–400. MathSciNetzbMATHGoogle Scholar
  33. 33.
    E. Hrushovski, Stable group theory and approximate subgroups, J. Am. Math. Soc., 25 (2012), 189–243. MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    I. Kaplansky, Lie Algebras and Locally Compact Groups, The University of Chicago Press, Chicago, 1971. zbMATHGoogle Scholar
  35. 35.
    V. Kapovitch, A. Petrunin, and W. Tuschmann, Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces, Ann. Math., 171 (2010), 343–373. MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    V. Kapovitch and B. Wilking, Structure of fundamental groups of manifolds with Ricci curvature bounded below, preprint (2011), arXiv:1105.5955.
  37. 37.
    B. Kleiner, A new proof of Gromov’s theorem on groups of polynomial growth, J. Am. Math. Soc., 23 (2010), 815–829. MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    J. Lee and Y. Makarychev, Eigenvalue multiplicity and volume growth, preprint (2008), arXiv:0806.1745.
  39. 39.
    D. Montgomery and L. Zippin, Small subgroups of finite-dimensional groups, Ann. Math., 56 (1952), 213–241. MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience Publishers, New York, 1955. zbMATHGoogle Scholar
  41. 41.
    P. J. Olver, Non-associative local Lie groups, J. Lie Theory, 6 (1996), 23–51. MathSciNetzbMATHGoogle Scholar
  42. 42.
    C. Pittet and L. Saloff-Coste, A survey on the relationships between volume growth, isoperimetry, and the behavior of simple random walk on Cayley graphs, with examples, survey, preprint (2000). Google Scholar
  43. 43.
    L. Pyber and E. Szabó, Growth in finite simple groups of Lie type of bounded rank, preprint (2010), arXiv:1005.1858.
  44. 44.
    I. Z. Ruzsa, Generalized arithmetical progressions and sumsets, Acta Math. Hung., 65 (1994), 379–388. MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    I. Z. Ruzsa, An analog of Freiman’s theorem in groups, Astérisque, 258 (1999), 323–326. MathSciNetGoogle Scholar
  46. 46.
    T. Sanders, From polynomial growth to metric balls in monomial groups, preprint (2009), arXiv:0912.0305.
  47. 47.
    T. Sanders, On a non-abelian Balog-Szemerédi-type lemma, J. Aust. Math. Soc., 89 (2010), 127–132. MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    T. Sanders, On the Bogolyubov-Ruzsa lemma. Anal. Partial Differ. Equ. (2010), to appear, arXiv:1011.0107.
  49. 49.
    T. Sanders, A quantitative version of the non-abelian idempotent theorem, Geom. Funct. Anal., 21 (2011), 141–221. MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    J.-P. Serre, Lie Algebras and Lie Groups, Lecture Notes in Mathematics, vol. 1500, Springer, Berlin, 2006. 1964 lectures given at Harvard University, Corrected fifth printing of the second (1992) edition. Google Scholar
  51. 51.
    Y. Shalom and T. Tao, A finitary version of Gromov’s polynomial growth theorem, Geom. Funct. Anal., 20 (2010), 1502–1547. MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    T. Tao, Product set estimates for non-commutative groups, Combinatorica, 28 (2008), 547–594. MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    T. Tao, Freiman’s theorem for solvable groups, Contrib. Discrete Math., 5 (2010), 137–184. MathSciNetGoogle Scholar
  54. 54.
    T. Tao and V. Vu, Additive Combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006. zbMATHCrossRefGoogle Scholar
  55. 55.
    W. P. Thurston, Three-Dimensional Geometry and Topology, vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, 1997. Edited by Silvio Levy. zbMATHGoogle Scholar
  56. 56.
    L. van den Dries and I. Goldbring, Globalizing locally compact local groups, J. Lie Theory, 20 (2010), 519–524. MathSciNetzbMATHGoogle Scholar
  57. 57.
    L. van den Dries and I. Goldbring, Seminar notes on Hilbert’s 5th problem, preprint (2010). Google Scholar
  58. 58.
    L. van den Dries and A. J. Wilkie, Gromov’s theorem on groups of polynomial growth and elementary logic, J. Algebra, 89 (1984), 349–374. MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    N. T. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. Google Scholar
  60. 60.
    H. Yamabe, A generalization of a theorem of Gleason, Ann. Math., 58 (1953), 351–365. MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    H. Yamabe, On the conjecture of Iwasawa and Gleason, Ann. Math., 58 (1953), 48–54. MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© IHES and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques, Bâtiment 425Université Paris Sud 11OrsayFrance
  2. 2.Centre for Mathematical SciencesCambridgeEngland
  3. 3.Department of MathematicsUCLALos AngelesUSA

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