Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T14:32:47.889Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximate groups and doubling metrics

Published online by Cambridge University Press:  13 December 2011

TOM SANDERS
TOM SANDERS*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a version of Freĭman's theorem for a class of non-abelian groups, which includes finite nilpotent, supersolvable and solvable A-groups. To do this we have to replace the small doubling hypothesis with a stronger relative polynomial growth hypothesis akin to that in Gromov's theorem (although with an effective range), and the structures we find are balls in (left and right) translation invariant pseudo-metrics with certain well behaved growth estimates.

Our work complements three other recent approaches to developing non-abelian versions of Freĭman's theorem by Breuillard and Green, Fisher, Katz and Peng, and Tao.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 3 , May 2012 , pp. 385 - 404
Copyright
Copyright © Cambridge Philosophical Society 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Bas72]Bass, H.The degree of polynomial growth of finitely generated nilpotent groups. Proc. London Math. Soc. (3), 25 (1972), 603–614.CrossRefGoogle Scholar
[BDJ+82]Bray, H. G., Deskins, W. E., Johnson, D., Humphreys, J. F., Puttaswamaiah, B. M., Venzke, P. and Walls, G. L.Between Nilpotent and Solvable (Polygonal Publ. House, Washington, N. J., 1982), Edited, and with a preface, by Michael Weinstein.Google Scholar
[BG11]Breuillard, E. and Green, B. J.Approximate groups. I: the torsion-free nilpotent case. J. Inst. Math. Jussieu 10 (1) (2011), 37–57.CrossRefGoogle Scholar
[Bil99]Bilu, Y. Structure of sets with small sumset. Astérisque (258) (1999), xi, 77–108. Structure theory of set addition.Google Scholar
[BKT04]Bourgain, J., Katz, N. H. and Tao, T. C.A sum-product estimate in finite fields, and applications. Geom. Funct. Anal. 14 (1) (2004), 27–57.CrossRefGoogle Scholar
[Bou99]Bourgain, J.On triples in arithmetic progression. Geom. Funct. Anal. 9 (5) (1999), 968–984.CrossRefGoogle Scholar
[Cha02]Chang, M.-C.A polynomial bound in Freĭman's theorem. Duke Math. J. 113 (3) (2002), 399–419.CrossRefGoogle Scholar
[Cha07]Chang, M.-C.Additive and multiplicative structure in matrix spaces. Combin. Probab. Comput. 16 (2) (2007), 219–238.CrossRefGoogle Scholar
[Cha08]Chang, M.-C.Product theorems in SL2 and SL3. J. Inst. Math. Jussieu 7 (1) (2008), 1–25.CrossRefGoogle Scholar
[DSV03]Davidoff, G., Sarnak, P. and Valette, A.Elementary number theory, group theory and Ramanujan graphs> London Mathematical Society Student Texts, vol 55 (Cambridge University Press, 2003).Google Scholar
[EK01]Elekes, G. and Király, Z.On the combinatorics of projective mappings. J. Algebraic Combin. 14 (3) (2001), 183–197.CrossRefGoogle Scholar
[FKP10]Fisher, D., Katz, N. H. and Peng, I.Approximate multiplicative groups in nilpotent Lie groups. Proc. Amer. Math. Soc. 138 (5) (2010), 1575–1580.CrossRefGoogle Scholar
[Fre73]Freĭman, G. A.Foundations of a structural theory of set addition (American Mathematical Society, 1973). Translated from the Russian, Translations of Mathematical Monographs, Vol. 37.Google Scholar
[Gow98]Gowers, W. T.A new proof of Szemerédi's theorem for arithmetic progressions of length four. Geom. Funct. Anal. 8 (3) (1998), 529–551.CrossRefGoogle Scholar
[Gow08]Gowers, W. T.Quasirandom groups. Comb. Probab. Comput. 17 (3) (2008), 363–387.CrossRefGoogle Scholar
[GR07]Green, B. J. and Ruzsa, I. Z.Freĭman's theorem in an arbitrary abelian group. J. Lond. Math. Soc. (2), 75 (1) (2007), 163–175.CrossRefGoogle Scholar
[Gre05]Green, B. J.A Szemerédi-type regularity lemma in abelian groups, with applications. Geom. Funct. Anal. 15 (2) (2005), 340–376.CrossRefGoogle Scholar
[Gro81]Gromov, M.Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. (53) (1981), 53–73.CrossRefGoogle Scholar
[GT08]Green, B. J. and Tao, T. C.An inverse theorem for the Gowers U3(G) norm. Proc. Edinb. Math. Soc. (2), 51 (1) (2008), 73–153.CrossRefGoogle Scholar
[Gui71]Guivarc'h, Y.Groupes de Lie à croissance polynomiale. C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A1695–A1696.Google Scholar
[Hel08]Helfgott, H. A.Growth and generation in SL 2(ℤ/pℤ). Ann. of Math. (2), 167 (2008), 601–623.CrossRefGoogle Scholar
[Hel11]Helfgott, H. A.Growth in SL 3(ℤ/pℤ). J. Eur. Math. Soc. 13 (3) (2011), 761–851.CrossRefGoogle Scholar
[HLS98]Hamidoune, Y. O., Lladó, A. S. and Serra, O.On subsets with small product in torsion-free groups. Combinatorica 18 (4) (1998), 529–540.CrossRefGoogle Scholar
[Hup98]Huppert, B.Character theory of finite groups. de Gruyter Expositions in Mathematics, vol. 25 (Walter de Gruyter & Co., 1998).CrossRefGoogle Scholar
[Isa94]Isaacs, I. M.Character Theory of Finite Groups (Dover Publications Inc., 1994). Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)].Google Scholar
[Kir94]Kirillov, A. A., editor. Representation theory and noncommutative harmonic analysis, I. Encyclopaedia of Mathematical Sciences, vol. 22 (Springer-Verlag, 1994). Fundamental concepts. Representations of Virasoro and affine algebras, A translation of Current problems in mathematics. Fundamental directions. Vol. 22 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988 [MR0942946 (88k:22001)], Translation by V. Souček.CrossRefGoogle Scholar
[Lin01]Lindenstrauss, E.Pointwise theorems for amenable groups. Invent. Math. 146 (2) (2001), 259–295.CrossRefGoogle Scholar
[LM08]Lee, J. R. and Makarychev, Y. Eigenvalue multiplicity and volume growth. Journal of Topology and Analysis (2008), arXiv:0806.1745. to appear.Google Scholar
[LPS88]Lubotzky, A., Phillips, R. and Sarnak, P.Ramanujan graphs. Combinatorica 8 (3) (1988), 261–277.CrossRefGoogle Scholar
[NŠ82]Naĭmark, M. A. and Štern, A. I.Theory of group representations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 246. (Springer-Verlag, 1982). Translated from the Russian by Elizabeth Hewitt, translation edited by Edwin Hewitt.CrossRefGoogle Scholar
[Ruz94]Ruzsa, I. Z.Generalized arithmetical progressions and sumsets. Acta Math. Hungar. 65 (4) (1994), 379–388.CrossRefGoogle Scholar
[Ruz99]Ruzsa, I. Z.An analog of Freĭman''s theorem in groups. Astérisque. (258) (1999), xv, 323–326. Structure theory of set addition.Google Scholar
[San09]Sanders, T.A Freĭman-type theorem for locally compact abelian groups. Ann. Inst. Fourier (Grenoble). 59 (4) (2009), 1321–1335, arXiv:0710.2545.CrossRefGoogle Scholar
[Sch03]Schoen, T.Multiple set addition in ℤp. Integers. 3 (electronic), (2003), A17, 6 pp.Google Scholar
[Shk06a]Shkredov, I. D.On a generalization of Szemerédi's theorem. Proc. London Math. Soc. (3) 93 (3) (2006), 723–760.CrossRefGoogle Scholar
[Shk06b]Shkredov, I. D.On a problem of Gowers. Izv. Ross. Akad. Nauk Ser. Mat. 70 (2) (2006), 179–221.Google Scholar
[SSV05]Sudakov, B., Szemerédi, E. and Vu, V. H.On a question of Erdős and Moser. Duke Math. J. 129 (1) (2005), 129–155.CrossRefGoogle Scholar
[ST09]Shalom, Y. and Tao, T. C. A finitary version of Gromov's polynomial growth theorem (2009), arXiv:0910.4148.Google Scholar
[SV05]Szemerédi, E. and Vu, V. H.Long arithmetic progressions in sum-sets and the number of x-sum-free sets. Proc. London Math. Soc. (3) 90 (2) (2005), 273–296.CrossRefGoogle Scholar
[Tao05]Tao, T. C. Fourier analysis on finite non-abelian groups. www.math.ucla.edu/~tao! (2005).Google Scholar
[Tao08a]Tao, T. C. The correspondence principle and finitary ergodic theory. Available at terrytao.wordpress.com! (2008).Google Scholar
[Tao08b]Tao, T. C.Product set estimates for non-commutative groups. Combinatorica. 28 (5) (2008), 547–594.CrossRefGoogle Scholar
[Tao10]Tao, T. C.Freĭman's theorem for solvable groups. Contrib. Disc. Math. 5 (2) (2010), 137–184.Google Scholar
[TV06]Tao, T. C. and Vu, H. V.Additive combinatorics. Cambridge Studies in Advanced Mathematics, vol 105 (Cambridge University Press, 2006).CrossRefGoogle Scholar
[TV07]Tao, T. C. and Vu, V. H.On the singularity probability of random Bernoulli matrices. J. Amer. Math. Soc. 20 (3) (electronic), (2007), 603–628.CrossRefGoogle Scholar
[Wol68]Wolf, J. A.Growth of finitely generated solvable groups and curvature of Riemanniann manifolds. J. Differential Geom. 2 (1968), 421–446.CrossRefGoogle Scholar