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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: t.sanders@dpmms.cam.ac.uk
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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.

Information

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

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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