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Lectures on approximate groups and Hilbert’s 5th problem

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Recent Trends in Combinatorics

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 159))

Abstract

This paper gathers four lectures, based on a mini-course at IMA in 2014, whose aim was to discuss the structure of approximate subgroups of an arbitrary group, following the works of Hrushovski and of Green, Tao and the author. Along the way we discuss the proof of the Gleason–Yamabe theorem on Hilbert’s 5th problem about the structure of locally compact groups and explain its relevance to approximate groups. We also present several applications in particular to uniform diameter bounds for finite groups and to the determination of scaling limits of vertex transitive graphs with large diameter.

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Notes

  1. 1.

    They can be avoided if one only shoots for Theorem 1.11, see the proof in [29], but seem necessary for the finer structure described in Remark d) after Theorem 1.11, and hence for the strong form Theorem 2.14.

  2. 2.

    Ulam stablility may fail for certain target groups other that U(H), for example it fails for G the p-adic integers, see [57][Prop. 1.]

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Acknowledgements

These lectures were prepared for the workshop on Additive and Analytic Combinatorics held at the IMA in Minneapolis from September 29 to October 3, 2014. Due to a health problem, I was unfortunately not able to give the lectures. I am very grateful to Terry Tao, who replaced me on the spot. I also thank T. Tao, R. Tessera and M. Tointon for their valuable comments on the text. The author is partially supported by ERC grant no. 617129.

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  1. Laboratoire de Mathématiques, Bâtiment 425, Université Paris Sud 11, 91405, Orsay, France

    Emmanuel Breuillard

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  1. Emmanuel Breuillard
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Correspondence to Emmanuel Breuillard .

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Editors and Affiliations

  1. Dept of Math, Statistics & Comp.Sc, Macalester College, St. Paul, Minnesota, USA

    Andrew Beveridge

  2. Dept. of Mathematics, University of South Carolina, Columbia, South Carolina, USA

    Jerrold R. Griggs

  3. Dept. of Mathematics, Iowa State University, Ames, Iowa, USA

    Leslie Hogben

  4. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Gregg Musiker

  5. School of Mathematics & Computer Sc, Georgia Institute of Technology, Atlanta, Georgia, USA

    Prasad Tetali

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Breuillard, E. (2016). Lectures on approximate groups and Hilbert’s 5th problem. In: Beveridge, A., Griggs, J., Hogben, L., Musiker, G., Tetali, P. (eds) Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-24298-9_16

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