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Shalom’s property HFD and extensions by ℤ of locally finite groups

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Abstract

We show that every finitely generated extension by ℤ of a locally normally finite group has Shalom’s property HFD. The statement is no longer true without the normality assumption. This permits to answer some questions of Shalom, Erschler–Ozawa and Kozma. We also obtain a Neumann–Neumann embedding result that any countable locally finite group embeds into a two-generated amenable group with property HFD.

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References

  1. I. Benjamini, H. Duminil-Copin, G. Kozma and A. Yadin, Minimal growth harmonic functions on lamplighter groups, New York Journal of Mathematics 23 (2017), 833–858.

    MathSciNet  MATH  Google Scholar 

  2. J. Brieussel, R. Tanaka and T. Zheng, Random walks on the discrete affine group, preprint, arXiv:1703.07741.

  3. J. Brieussel and T. Zheng, Speed of random walks, isoperimetry and compression of finitely generated groups, preprint, arXiv:1510.08040.

  4. D. I. Cartwright, V. A. Kaimanovich and W. Woess, Random walks on the affine group of local fields and of homogeneous trees, Université de Grenoble. Annales de l’Institut Fourier 44 (1994), 1243–1288.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Erschler and N. Ozawa, Finite-dimensional representations constructed from random walks, Commentarii Mathematici Helvetici 93 (2018), 555–586.

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Fernós and A. Valette, The Mayer–Vietoris sequence for graphs of groups, property (T), and the first 2-Betti number, Homology, Homotopy and Applications 19 (2017), 251–274.

    Article  MathSciNet  MATH  Google Scholar 

  7. T. Fernós, A. Valette and F. Martin, Reduced 1-cohomology and relative property (T), Mathematische Zeitschrift 270 (2012), 613–626.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Gournay, Mixing, malnormal subgroups and cohomology in degree one, Groups, Geometry, and Dynamics 12 (2018), 1371–1416.

    Article  MathSciNet  Google Scholar 

  9. M. Gromov, Entropy and isoperimetry for linear and non-linear group actions, Groups, Geometry, and Dynamics 2 (2008), 499–593.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Guichardet, Sur la cohomologie des groupes topologiques. II, Bulletin des Sciences Mathématiques 96 (1972), 305–332.

    MathSciNet  MATH  Google Scholar 

  11. P. Hall, Some constructions for locally finite groups, Journal of the London Mathematical Society 34 (1959), 305–319.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Hillman, Four-manifolds, Geometries and Knots, Geometry & Topology Monographs, Vol. 5, Geometry & Topology Publications, Coventry, 2002.

    MATH  Google Scholar 

  13. B. Kleiner, A new proof of Gromov’s theorem on groups of polynomial growth, Journal of the American Mathematical Society 23 (2010), 815–829.

    Google Scholar 

  14. R. Lyons and Y. Peres, Probability on Trees and Networks, Cambridge Series in Statistical and Probabilistic Mathematics, Vol. 42, Cambridge University Press, New York, 2016.

  15. B. Neumann and H. Neumann, Embedding theorems for groups, Journal of the London Mathematical Society 34 (1959), 465–479.

    Article  MathSciNet  MATH  Google Scholar 

  16. N. Ozawa, A functional analysis proof of Gromov’s polynomial growth theorem, Annales Scientifiques de l’École Normale Supérieure 51 (2018), 549–556.

    Google Scholar 

  17. P. Révész, Random Walk in Random and Non-random Environments, World Scientific Publishing, Hackensack, NJ, 2013.

    Book  MATH  Google Scholar 

  18. Y. Shalom, Rigidity of commensurators and irreducible lattices, Inventiones Mathematicae 141 (2000), 1–54.

    Article  MathSciNet  MATH  Google Scholar 

  19. Y. Shalom, Harmonic analysis, cohomology, and the large-scale geometry of amenable groups, Acta Mathematica 192 (2004), 119–185.

    Article  MathSciNet  MATH  Google Scholar 

  20. N. Th. Varopoulos, Théorie du potentiel sur des groupes et des variétés, Comptes Rendus des Séances de l’Académie des Sciences. Série I. Mathématique 302 (1986), 203–205, 1986.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, 34095, Montpellier, France

    Jérémie Brieussel

  2. Department of Mathematics, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA, 92093, USA

    Tianyi Zheng

Authors
  1. Jérémie Brieussel
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  2. Tianyi Zheng
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Correspondence to Jérémie Brieussel.

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Brieussel, J., Zheng, T. Shalom’s property HFD and extensions by ℤ of locally finite groups. Isr. J. Math. 230, 45–70 (2019). https://doi.org/10.1007/s11856-018-1818-6

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  • DOI: https://doi.org/10.1007/s11856-018-1818-6

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