The structure of approximate groups
Breuillard, Emmanuel1 ; Green, Ben2 ; Tao, Terence3
1 Laboratoire de Mathématiques, Bâtiment 425, Université Paris Sud 11 91405, Orsay France
2 Centre for Mathematical Sciences Wilberforce Road, Cambridge, CB3 0WA England
3 Department of Mathematics, UCLA 405 Hilgard Ave, Los Angeles, CA, 90095 USA
Publications Mathématiques de l'IHÉS, Tome 116 (2012), pp. 115-221.

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 A⋅A 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.

DOI : 10.1007/s10240-012-0043-9
Breuillard, Emmanuel 1 ; Green, Ben 2 ; Tao, Terence 3

1 Laboratoire de Mathématiques, Bâtiment 425, Université Paris Sud 11 91405, Orsay France
2 Centre for Mathematical Sciences Wilberforce Road, Cambridge, CB3 0WA England
3 Department of Mathematics, UCLA 405 Hilgard Ave, Los Angeles, CA, 90095 USA 
@article{PMIHES_2012__116__115_0,
     author = {Breuillard, Emmanuel and Green, Ben and Tao, Terence},
     title = {The structure of approximate groups},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {115--221},
     publisher = {Springer-Verlag},
     volume = {116},
     year = {2012},
     doi = {10.1007/s10240-012-0043-9},
     mrnumber = {3090256},
     zbl = {1260.20062},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-012-0043-9/}
}
TY  - JOUR
AU  - Breuillard, Emmanuel
AU  - Green, Ben
AU  - Tao, Terence
TI  - The structure of approximate groups
JO  - Publications Mathématiques de l'IHÉS
PY  - 2012
SP  - 115
EP  - 221
VL  - 116
PB  - Springer-Verlag
UR  - http://archive.numdam.org/articles/10.1007/s10240-012-0043-9/
DO  - 10.1007/s10240-012-0043-9
LA  - en
ID  - PMIHES_2012__116__115_0
ER  - 
%0 Journal Article
%A Breuillard, Emmanuel
%A Green, Ben
%A Tao, Terence
%T The structure of approximate groups
%J Publications Mathématiques de l'IHÉS
%D 2012
%P 115-221
%V 116
%I Springer-Verlag
%U http://archive.numdam.org/articles/10.1007/s10240-012-0043-9/
%R 10.1007/s10240-012-0043-9
%G en
%F PMIHES_2012__116__115_0
Breuillard, Emmanuel; Green, Ben; Tao, Terence. The structure of approximate groups. Publications Mathématiques de l'IHÉS, Tome 116 (2012), pp. 115-221. doi : 10.1007/s10240-012-0043-9. http://archive.numdam.org/articles/10.1007/s10240-012-0043-9/

[1.] Benjamini, I.; Kozma, G. A resistance bound via an isoperimetric inequality, Combinatorica, Volume 25 (2005), pp. 645-650 | DOI | MR | Zbl

[2.] Bieberbach, L. Über einen Satz des Herrn C. Jordan in der Theorie der endlichen Gruppen linearer Substitutionen, Sitzber. Preuss. Akad. Wiss, Berlin, 1911 | JFM

[3.] Bilu, Y. Addition of sets of integers of positive density, J. Number Theory, Volume 64 (1997), pp. 233-275 | DOI | MR | Zbl

[4.] Bilu, Y. Structure of sets with small sumset, Astérisque, Volume 258 (1999), pp. 77-108 (Structure theory of set addition.) | Numdam | MR | Zbl

[5.] Breuillard, E.; Green, B. Approximate groups. I: the torsion-free nilpotent case, J. Inst. Math. Jussieu, Volume 10 (2011), pp. 37-57 | DOI | MR | Zbl

[6.] Breuillard, E.; Green, B. Approximate groups. II: the solvable linear case, Q. J. of Math., Oxf., Volume 62 (2011), pp. 513-521 | DOI | MR | Zbl

[7.] Breuillard, E.; Green, B. Approximate groups. III: the unitary case, Turk. J. Math., Volume 36 (2012), pp. 199-215 | MR | Zbl

[8.] Breuillard, E.; Green, B.; Tao, T. Approximate subgroups of linear groups, Geom. Funct. Anal., Volume 21 (2011), pp. 774-819 | DOI | MR | Zbl

[9.] Burago, Y. D.; Zalgaller, V. A. Geometric Inequalities, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 285, Springer, Berlin, 1988 (Translated from the Russian by A. B. Sosinskiĭ, Springer Series in Soviet Mathematics.) | MR | Zbl

[10.] Chang, M.-C. A polynomial bound in Freiman’s theorem, Duke Math. J., Volume 113 (2002), pp. 399-419 | DOI | MR | Zbl

[11.] Cheeger, J.; Colding, T. H. Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. Math., Volume 144 (1996), pp. 189-237 | DOI | MR | Zbl

[12.] Corwin, L. J.; Greenleaf, F. Representations of Nilpotent Lie Groups and Their Applications, CUP, Cambridge, 1990 | MR | Zbl

[13.] Croot, E.; Sisask, O. A probabilistic technique for finding almost-periods of convolutions, Geom. Funct. Anal., Volume 20 (2010), pp. 1367-1396 | DOI | MR | Zbl

[14.] Fisher, D.; Katz, N. H.; Peng, I. Approximate multiplicative groups in nilpotent Lie groups, Proc. Am. Math. Soc., Volume 138 (2010), pp. 1575-1580 | DOI | MR | Zbl

[15.] Freiman, G. A. Foundations of a Structural Theory of Set Addition, American Mathematical Society, Providence, 1973 (Translated from the Russian, Translations of Mathematical Monographs, vol. 37) | MR | Zbl

[16.] Fukaya, K.; Yamaguchi, T. The fundamental groups of almost non-negatively curved manifolds, Ann. Math., Volume 136 (1992), pp. 253-333 | DOI | MR | Zbl

[17.] Gallot, S.; Hulin, D.; Lafontaine, J. Riemannian Geometry, Universitext, Springer, Berlin, 1987 | DOI | MR | Zbl

[18.] N. Gill and H. Helfgott, Growth in solvable subgroups of GL r (Z/p Z), preprint (2010), . | arXiv

[19.] Gill, N.; Helfgott, H. Growth of small generating sets in SL n (Z/p Z), Int. Math. Res. Not., Volume 18 (2011), pp. 4226-4251 | MR | Zbl

[20.] Gleason, A. M. The structure of locally compact groups, Duke Math. J., Volume 18 (1951), pp. 85-104 | DOI | MR | Zbl

[21.] Gleason, A. M. Groups without small subgroups, Ann. Math., Volume 56 (1952), pp. 193-212 | DOI | MR | Zbl

[22.] Gödel, K. Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory, Proc. Natl. Acad. Sci, Volume 24 (1938), pp. 556-557 | DOI | JFM

[23.] I. Goldbring, Nonstandard Methods in Lie Theory, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2009. | MR

[24.] Goldbring, I. Hilbert’s fifth problem for local groups, Ann. Math., Volume 172 (2010), pp. 1269-1314 | DOI | MR | Zbl

[25.] Green, B.; Ruzsa, I. Z. Freiman’s theorem in an arbitrary abelian group, J. Lond. Math. Soc., Volume 75 (2007), pp. 163-175 | DOI | MR | Zbl

[26.] Green, B.; Tao, T. Compressions, convex geometry and the Freiman-Bilu theorem, Q. J. Math., Volume 57 (2006), pp. 495-504 | DOI | MR | Zbl

[27.] Gromov, M. Groups of polynomial growth and expanding maps, Publ. Math. IHÉS, Volume 53 (1981), pp. 53-73 | Numdam | MR | Zbl

[28.] Gromov, M. 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) | MR | Zbl

[29.] Hall, M. The Theory of Groups, Chelsea Publishing Co., New York, 1976 (Reprinting of the 1968 edition) | MR | Zbl

[30.] Helfgott, H. A. Growth and generation in SL2(Z/p Z), Ann. Math., Volume 167 (2008), pp. 601-623 | DOI | MR | Zbl

[31.] Helfgott, H. A. Growth in SL3(Z/p Z), J. Eur. Math. Soc., Volume 13 (2011), pp. 761-851 | DOI | MR | Zbl

[32.] Hirschfeld, J. The nonstandard treatment of Hilbert’s fifth problem, Trans. Am. Math. Soc., Volume 321 (1990), pp. 379-400 | MR | Zbl

[33.] Hrushovski, E. Stable group theory and approximate subgroups, J. Am. Math. Soc., Volume 25 (2012), pp. 189-243 | DOI | MR | Zbl

[34.] Kaplansky, I. Lie Algebras and Locally Compact Groups, The University of Chicago Press, Chicago, 1971 | MR | Zbl

[35.] Kapovitch, V.; Petrunin, A.; Tuschmann, W. Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces, Ann. Math., Volume 171 (2010), pp. 343-373 | DOI | MR | Zbl

[36.] V. Kapovitch and B. Wilking, Structure of fundamental groups of manifolds with Ricci curvature bounded below, preprint (2011), . | arXiv

[37.] Kleiner, B. A new proof of Gromov’s theorem on groups of polynomial growth, J. Am. Math. Soc., Volume 23 (2010), pp. 815-829 | DOI | MR | Zbl

[38.] J. Lee and Y. Makarychev, Eigenvalue multiplicity and volume growth, preprint (2008), . | arXiv

[39.] Montgomery, D.; Zippin, L. Small subgroups of finite-dimensional groups, Ann. Math., Volume 56 (1952), pp. 213-241 | DOI | MR | Zbl

[40.] Montgomery, D.; Zippin, L. Topological Transformation Groups, Interscience Publishers, New York, 1955 | MR | Zbl

[41.] Olver, P. J. Non-associative local Lie groups, J. Lie Theory, Volume 6 (1996), pp. 23-51 | MR | Zbl

[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).

[43.] L. Pyber and E. Szabó, Growth in finite simple groups of Lie type of bounded rank, preprint (2010), . | arXiv | MR

[44.] Ruzsa, I. Z. Generalized arithmetical progressions and sumsets, Acta Math. Hung., Volume 65 (1994), pp. 379-388 | DOI | MR | Zbl

[45.] Ruzsa, I. Z. An analog of Freiman’s theorem in groups, Astérisque, Volume 258 (1999), pp. 323-326 | Numdam | MR | Zbl

[46.] T. Sanders, From polynomial growth to metric balls in monomial groups, preprint (2009), . | arXiv

[47.] Sanders, T. On a non-abelian Balog-Szemerédi-type lemma, J. Aust. Math. Soc., Volume 89 (2010), pp. 127-132 | DOI | MR | Zbl

[48.] T. Sanders, On the Bogolyubov-Ruzsa lemma. Anal. Partial Differ. Equ. (2010), to appear, . | arXiv | MR

[49.] Sanders, T. A quantitative version of the non-abelian idempotent theorem, Geom. Funct. Anal., Volume 21 (2011), pp. 141-221 | DOI | MR | Zbl

[50.] Serre, J.-P. Lie Algebras and Lie Groups, Lecture Notes in Mathematics, 1500, Springer, Berlin, 2006 1964 lectures given at Harvard University, Corrected fifth printing of the second (1992) edition | MR | Zbl

[51.] Shalom, Y.; Tao, T. A finitary version of Gromov’s polynomial growth theorem, Geom. Funct. Anal., Volume 20 (2010), pp. 1502-1547 | DOI | MR | Zbl

[52.] Tao, T. Product set estimates for non-commutative groups, Combinatorica, Volume 28 (2008), pp. 547-594 | DOI | MR | Zbl

[53.] Tao, T. Freiman’s theorem for solvable groups, Contrib. Discrete Math., Volume 5 (2010), pp. 137-184 | MR | Zbl

[54.] Tao, T.; Vu, V. Additive Combinatorics, Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, Cambridge, 2006 | DOI | MR | Zbl

[55.] Thurston, W. P. Three-Dimensional Geometry and Topology, vol. 1, Princeton Mathematical Series, 35, Princeton University Press, Princeton, 1997 (Edited by Silvio Levy) | MR | Zbl

[56.] Dries, L.; Goldbring, I. Globalizing locally compact local groups, J. Lie Theory, Volume 20 (2010), pp. 519-524 | MR | Zbl

[57.] L. van den Dries and I. Goldbring, Seminar notes on Hilbert’s 5th problem, preprint (2010). | MR

[58.] Dries, L.; Wilkie, A. J. Gromov’s theorem on groups of polynomial growth and elementary logic, J. Algebra, Volume 89 (1984), pp. 349-374 | DOI | MR | Zbl

[59.] Varopoulos, N. T.; Saloff-Coste, L.; Coulhon, T. Analysis and Geometry on Groups, Cambridge Tracts in Mathematics, 100, Cambridge University Press, Cambridge, 1992 | MR | Zbl

[60.] Yamabe, H. A generalization of a theorem of Gleason, Ann. Math., Volume 58 (1953), pp. 351-365 | DOI | MR | Zbl

[61.] Yamabe, H. On the conjecture of Iwasawa and Gleason, Ann. Math., Volume 58 (1953), pp. 48-54 | DOI | MR | Zbl

Cité par Sources :