Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-16T17:10:43.904Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximate groups. I The torsion-free nilpotent case

Part of: Sequences and sets

Published online by Cambridge University Press:  02 June 2010

Emmanuel Breuillard  and
Ben Green
Emmanuel Breuillard
Affiliation:
Laboratoire de Mathématiques Université Paris-Sud 11, 91405 Orsay cedex, France ([email protected])
Ben Green
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We describe the structure of ‘K-approximate subgroups’ of torsion-free nilpotent groups, paying particular attention to Lie groups.

Three other works, by Fisher et al., by Sanders and by Tao, have appeared that independently address related issues. We comment briefly on some of the connections between these papers.

Keywords

approximate groupnilpotent progressionadditive combinatoricsFreiman's theorem

MSC classification

Secondary: 11B30: Arithmetic combinatorics; higher degree uniformity
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Issue 1 , January 2011 , pp. 37 - 57
Copyright
Copyright © Cambridge University Press 2010

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

1.Bourbaki, N., Éléments de mathématique, XXVI, in Groupes et algèbres de Lie, Chapitre 1: Algèbres de Lie, Actualités Scientifiques et Industrielles, No. 1285 (Hermann, Paris, 1960).Google Scholar
2.Breuillard, E. and Green, B. J., Approximate subgroups of solvable Lie groups, Q. J. Math. (Oxford), in press.Google Scholar
3.Chang, M. C., A polynomial bound in Freĭman's theorem, Duke Math. J. 113(3) (2002), 399–419.CrossRefGoogle Scholar
4.Corwin, L. J. and Greenleaf, F. P., Representations of nilpotent Lie groups and applications, Volume I, Cambridge Studies in Advanced Mathematics, Volume 18 (Cambridge University Press, 1990).Google Scholar
5.Dixon, J. D., du Sautoy, M. P. F., Mann, A. and Segal, D., Analytic pro-p groups, 2nd edn, Cambridge Studies in Advanced Mathematics, Volume 61 (Cambridge University Press, 1999).CrossRefGoogle Scholar
6.Fisher, D., Katz, N. H. and Peng, I., On Freĭman's theorem in nilpotent groups, preprint (arXiv:0901.1409; 2009).Google Scholar
7.Freĭman, G. A., Foundations of a structural theory of set addition (transl. from Russian), Translations of Mathematical Monographs, Volume 37 (American Mathematical Society, Providence, RI, 1973).Google Scholar
8.Green, B. J. and Ruzsa, I. Z., Freiman's theorem in an arbitrary abelian group, J. Lond. Math. Soc. 75(1) (2007), 163–175.Google Scholar
9.Green, B. J. and Sanders, T., A quantitative version of the idempotent theorem in harmonic analysis, Annals Math. 168(3) (2008), 1025–1054.CrossRefGoogle Scholar
10.Hall, M., Theory of groups (American Mathematical Society/Chelsea, Providence, RI, 1999).Google Scholar
11.Lazard, M., Problémes d'extension concernant les N-groupes; inversion de la formule de Hausdorff, C. R. Acad. Sci. Paris Sér. I 237 (1953), 1377–1379.Google Scholar
12.Lazard, M., Sur les groupes nilpotents et les anneaux de Lie, Annales Scient. Éc. Norm. Sup. 71(2) (1954), 101–190.CrossRefGoogle Scholar
13.Leibman, A., Polynomial sequences in groups, J. Alg. 201(1) (1998), 189–206.CrossRefGoogle Scholar
14.Raghunathan, M. S., Discrete subgroups of Lie groups (Springer, 1972).CrossRefGoogle Scholar
15.Ruzsa, I. Z., Generalized arithmetical progressions and sumsets, Acta Math. Hungar. 65(4) (1994), 379–388.CrossRefGoogle Scholar
16.Sanders, T., From polynomial growth to metric balls in monomial groups, preprint (arXiv:0912.0305; 2009).Google Scholar
17.Stewart, I., An algebraic treatment of Mal'cev's theorems concerning nilpotent Lie groups and their Lie algebras, Compositio Math. 22 (1970), 289–312.Google Scholar
18.Tao, T. C., Product set estimates for non-commutative groups, Combinatorica 28(5) (2008), 547–594.CrossRefGoogle Scholar
19.Tao, T. C., Freiman's theorem for solvable groups, Contrib. Disc. Math., in press.Google Scholar
20.Tao, T. C. and Vu, V. H., Additive combinatorics (Cambridge University Press, 2006).CrossRefGoogle Scholar