Advertisement

Universal actions and representations of locally finite groups on metric spaces

  • Michal DouchaEmail author
Article
  • 9 Downloads

Abstract

We construct a universal action of a countable locally finite group (the Hall group) on a separable metric space by isometries. This single action contains all actions of all countable locally finite groups on all separable metric spaces as subactions. The main ingredient is the amalgamation of actions by isometries. We show that an equivalence class of this universal action is generic.

We show that the restriction to locally finite groups in our results is necessary as analogous results do not hold for infinite non-locally finite groups.

We discuss the problem also for actions by linear isometries on Banach spaces.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. Baumslag, On the residual finiteness of generalized free products of nilpotent groups, Transactions of the American Mathematical Society 106 (1963), 193–209.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    I. Ben Yaacov, Fraïssé limits of metric structures, Journal of Symbolic Logic 80 (2015), 100–115.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1., American Mathematical Society Colloquium Publications, Vol. 48, American Mathematical Society, Providence, RI, 2000.Google Scholar
  4. [4]
    M. Dadarlat and E. Guentner, Constructions preserving Hilbert space uniform embeddability of discrete group, Transactions of the American Mathematical Society 355 (2003), 3253–3275.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    M. Doucha, Metrically universal abelian groups, Transactions of the American Mathematical Society 369 (2017), 5981–5998.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M. Doucha and M. Malicki, Generic representations of countable groups, arXiv:1710.08170 [math.GR].Google Scholar
  7. [7]
    M. Doucha, M. Malicki and A. Valette, Property (T), finite-dimensional representations, and generic representations, Journal of Group Theory 22 (2019), 1–13.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Y. Glasner, D. Kitroser and J. Melleray, From isolated subgroups to generic permutation representations, Journal of the London Mathematical Society 94 (2016), 688–708.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    G. Godefroy and N. J. Kalton, Lipschitz-free Banach spaces, Studia Mathematica 159 (2003), 121–141.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    V. I. Gurarij, Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces, Sibirskiĭ Matematicheskiĭ Zhurnal 7 (1966), 1002–1013.MathSciNetzbMATHGoogle Scholar
  11. [11]
    P. Hall, Some constructions for locally finite groups, Journal of the London Mathematical Society 34 (1959), 305–319.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    G. Higman, A finitely generated group with an isomorphic proper factor group, Journal of the London Mathematical Society 26 (1951), 59–61.CrossRefzbMATHGoogle Scholar
  13. [13]
    W. Hodges, Model Theory, Encyclopedia of Mathematics and its Applications, Vol. 42, Cambridge University Press, Cambridge, 1993.Google Scholar
  14. [14]
    D. Kerr, H. Li and M. Pichot, Turbulence, representations, and trace-preserving actions, Proceedings of the London Mathematical Society 100 (2010), 459–484.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    W. Kubiś, Fraïssé sequences: category-theoretic approach to universal homogeneous structures, Annals of Pure and Applied Logic 165 (2014), 1755–1811.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 89, Springer, Berlin–New York, 1977.Google Scholar
  17. [17]
    J. Melleray and T. Tsankov, Generic representations of abelian groups and extreme amenability, Israel Journal of Mathematics 198 (2013), 129–167.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    J. Melleray, Polish groups and Baire category methods, Confluentes Mathematici 8 (2016), 89–164.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    B. H. Neumann, Permutational products of groups, Journal of the Australian Mathematical Society 1 (1959/1960), 299–310.Google Scholar
  20. [20]
    P. Nowak and G. Yu, Large Scale Geometry, EMS Textbooks in Mathematics, European Mathematical Society, Zürich, 2012.CrossRefzbMATHGoogle Scholar
  21. [21]
    V. Pestov, Dynamics of Infinite-Dimensional Groups, University Lecture Series, Vol. 40, American Mathematical Society, Providence, RI, 2006.Google Scholar
  22. [22]
    L. Ribes and P. A. Zalesskiĭ, On the profinite topology on a free group, Bulletin of the London Mathematical Society 25 (1993), 37–43.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    C. Rosendal, Finitely approximable groups and actions part I: The Ribes–Zalesskiĭ property, Journal of Symbolic Logic 76 (2011), 1297–1306.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    C. Rosendal, Finitely approximable groups and actions part II: Generic representations, Journal of Symbolic Logic 76 (2011), 1307–1321.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    P. S. Urysohn, Sur un espace métrique universel, Bulletin des Sciences Mathématiques 51 (1927), 43–64, 74–96.zbMATHGoogle Scholar
  26. [26]
    N. Weaver, Lipschitz Algebras, World Scientific Publishing, River Edge, NJ, 1999.CrossRefzbMATHGoogle Scholar

Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Institute of MathematicsCzech Academy of SciencesPraha 1Czech Republic

Personalised recommendations