Skip to main content
SpringerLink
Log in

Equivariant dissipation in non-archimedean groups

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We prove that, if a topological group G has an open subgroup of infinite index, then every net of tight Borel probability measures on G UEB-converging to invariance dissipates in G in the sense of Gromov. In particular, this solves a 2006 problem by Pestov: for every left-invariant (or right-invariant) metric d on the infinite symmetric group Sym(ℕ), compatible with the topology of pointwise convergence, the sequence of the finite symmetric groups (Sym(n), d ↾Sym(n), μSym(n))n∈ℕ equipped with the restricted metrics and their normalized counting measures dissipates, thus fails to admit a subsequence being Cauchy with respect to Gromov’s observable distance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Arhangel’skii and M. Tkachenko, Topological Groups and Related Structures, Atlantis Studies in Mathematics, Vol. 1, Atlantis Press, Paris; World Scientific Publishing, Hackensack, NJ, 2008.

    Book  Google Scholar 

  2. V. I. Bogachev, Measure Theory. Vol. I, Springer, Berlin, 2007.

    Book  Google Scholar 

  3. A. Bouziad and J.-P. Troallic, A precompactness test for topological groups in the manner of Grothendieck, Topology Proceedings 31 (2007), 19–30.

    MathSciNet  MATH  Google Scholar 

  4. V. V. Buldygin and A. B. Kharazishvili, Geometric Aspects of Probability Theory and Mathematical Statistics, Mathematics and its Applications, Vol. 514, Kluwer Academic, Dordrecht, 2000.

    Book  Google Scholar 

  5. D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, Vol. 33, American Mathematical Society, Providence, RI, 2001.

    Book  Google Scholar 

  6. T. Giordano and V. G. Pestov, Some extremely amenable groups related to operator algebras and ergodic theory, Journal of the Institute of Mathematics of Jussieu 6 (2007), 279–315.

    Article  MathSciNet  Google Scholar 

  7. E. Glasner, B. Tsirelson and B. Weiss, The automorphism group of the Gaussian measure cannot act pointwise, Israel Journal of Mathematics 148 (2005), 305–329.

    Article  MathSciNet  Google Scholar 

  8. E. Glasner and B. Weiss, Minimal actions of the group S(Z) of permutations of the integers, Geometric and Functional Analysis 12 (2002), 964–988.

    Article  MathSciNet  Google Scholar 

  9. M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics, Vol. 152, Birkhäuser Boston, Boston, MA, 1999.

    MATH  Google Scholar 

  10. M. Gromov and V. D. Milman, A topological application of the isoperimetric inequality, American Journal of Mathematics 105 (1983), 843–854.

    Article  MathSciNet  Google Scholar 

  11. P. Holm, On the Bohr compactißcation, Mathematische Annalen 156 (1964), 34–46.

    Article  MathSciNet  Google Scholar 

  12. J. L. Kelley, General Topology, Graduate Texts in Mathematics, Vol. 27, Springer, New York, 1975.

    MATH  Google Scholar 

  13. M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, Vol. 89, American Mathematical Society, Providence, RI, 2001.

    MATH  Google Scholar 

  14. P. Lévy, Leçons d’analyse fonctionnelle, Gauthier-Villars, Paris, 1922.

    MATH  Google Scholar 

  15. B. Maurey, Constructions de suites symétriques, Comptes Rendus de l’Academie des Sciences. Série A-B 288 (1979), 679–681.

    MATH  Google Scholar 

  16. V. D. Milman, Infinite-dimensional geometry of the unit sphere in Banach space, Soviet Mathematics. Doklady 8 (1967), 1440–1444.

    MATH  Google Scholar 

  17. V. D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Mathematics, Vol. 1200, Springer, Berlin-Heidelberg, 1986.

    MATH  Google Scholar 

  18. J. Pachl, Uniform Spaces and Measures, Fields Institute Monographs, Vol. 30, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2013.

    Book  Google Scholar 

  19. K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, Vol. 3, Academic Press, New York-London, 1967.

    Book  Google Scholar 

  20. V. G. Pestov, Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups, Israel Journal of Mathematics 127 (2002), 317–357.

    Article  MathSciNet  Google Scholar 

  21. V. G. Pestov, Dynamics of Infinite-Dimensional Groups: The Ramsey-Dvoretzky-Milman Phenomenon, University Lecture Series, Vol. 40, American Mathematical Society, Providence, RI, 2006.

    Book  Google Scholar 

  22. V. G. Pestov, Concentration of measure and whirly actions of Polish groups, in Probabilistic Approach to Geometry, Advanced Studied in Pure Mathematics, Vol. 57, Mathematical Society of Japan, Tokyo, 2010, pp. 383–403.

    Chapter  Google Scholar 

  23. V. G. Pestov and F. M. Schneider, On amenability and groups of measurable maps, Journal of Functional Analysis 273 (2017), 3859–3874.

    Article  MathSciNet  Google Scholar 

  24. F. M. Schneider, Equivariant concentration in topological groups, Geometry & Topology 23 (2019), 925–956.

    Article  MathSciNet  Google Scholar 

  25. F. M. Schneider and A. Thom, On Følner sets in topological groups, Compositio Mathematica 154 (2018), 1333–1362.

    Article  MathSciNet  Google Scholar 

  26. T. Shioya, Metric Measure Geometry, IRMA Lectures in Mathematics and Theoretical Physics, Vol. 25, EMS Publishing House, Zürich, 2016.

    Book  Google Scholar 

  27. S. Solecki, Actions of non-compact and non-locally compact Polish groups, Journal of Symbolic Logic 65 (2000), 1881–1894.

    Article  MathSciNet  Google Scholar 

  28. V. V. Uspenskij, On subgroups of minimal topological groups, Topology and its Applications 155 (2008), 1580–1606.

    Article  MathSciNet  Google Scholar 

  29. J. de Vries, Elements of Topological Dynamics, Mathematics and Its Applications, Vol. 257, Kluwer Academic, Dordrecht, 1993.

    Book  Google Scholar 

Download references

Acknowledgments

This research has been supported by funding of the Excellence Initiative by the German Federal and State Governments as well as the Brazilian Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), processo 150929/2017-0. The author is deeply indebted to Vladimir Pestov for a number of inspiring and insightful discussions about the concentration of measured metric spaces, as well as to Tom Hanika for a helpful exchange on combinatorics of finite permutation groups. Also, the kind hospitality of CFM-UFSC (Florianópolis) during the origination of this work is gratefully acknowledged. Furthermore, the author would like to express his sincere gratitude towards the anonymous referee for their careful reading and valuable remarks, including a substantial simplification of an earlier proof of Theorem 5.1!

Author information

Authors and Affiliations

  1. Institute of Algebra, TU Dresden, 01062, Dresden, Germany

    Friedrich Martin Schneider

Authors
  1. Friedrich Martin Schneider
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Friedrich Martin Schneider.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schneider, F.M. Equivariant dissipation in non-archimedean groups. Isr. J. Math. 234, 281–307 (2019). https://doi.org/10.1007/s11856-019-1927-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-019-1927-x

Search

Navigation