A characterization of totally disconnected compactly ruled groups

Abstract

A locally compact group G is called compactly ruled if it is a directed union of compact open subgroups. We denote by \({\mathcal {SUB}}\left( G\right) \) the space of closed subgroups of G equipped with the Chabauty topology. In this paper, we show that the subspace \({\mathcal {SUB}}_{\scriptscriptstyle \mathrm {co}}\left( G\right) \) of \({\mathcal {SUB}}\left( G\right) \) consisting of compact open subgroups is dense in \({\mathcal {SUB}}\left( G\right) \) if and only if G is totally disconnected compactly ruled.

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References

  1. 1.

    Bourbaki N, Éléments de mathématique, intégration, chapitres 7–8 (2007) (Springer-Verlag)

  2. 2.

    Cornulier Y and Harpe P, Metric geometry of locally compact groups (2016) (European Mathematical Society)

  3. 3.

    Engelking R, General topology (1989) (Heldermann)

  4. 4.

    Gartside P and Smith M, Counting the closed subgroups of profinite groups, J. Group Theory 13 (2010) 41–61

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Gartside P and Smith M, Classifying spaces of subgroups of profinite groups, J. Group Theory 13 (2010) 315–336

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Hamrouni H and Hofmann K H, Locally compact groups approximable by subgroups isomorphic to \({\mathbb{Z} }\) or \({\mathbb{R}}\), Topology Appl.  215 (2017) 58–77

  7. 7.

    Hamrouni H and Kadri B, On the compact space of closed subgroups of locally compact groups, J. Lie Theory 23 (2014) 715–723

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Herfort W, Hofmann K H and Russo F G, Periodic locally compact groups, de Gruyter Studies in Mathematics 71 (2019) (Berlin)

  9. 9.

    Hewitt E and Ross K A, Abstract harmonic analysis I, Grundlehren der Mathematischen Wissenschaften 115 (1963) (Berlin: Springer)

  10. 10.

    Hofmann K H and Morris. S A, The structure of compact groups, 2nd revised and augmented edition (2006) (W. de Gruyter)

  11. 11.

    Kelley J L, General topology, Graduate Texts in Mathematics (Book 27) (1975) (Springer)

  12. 12.

    Palmer T W, Banach algebras and the general theory of \(*\)-algebras, Encyclopedia of Mathematics and its Applications, vol. 79 (2001) (Cambridge University Press)

  13. 13.

    Schochetman I, Nets of subgroups and amenability, Proc. Amer. Math. Soc. 29 (1971) 397–403

    MathSciNet  Article  Google Scholar 

  14. 14.

    Willis G,Totally disconnected groups and proofs of conjectures of Hofmann and Mukherjea, Bull. Austral. Math. Soc. 51 (1995) 489–494

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referee for his/her detailed suggestions that helped in formulating a final version of the paper.

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Correspondence to Hatem Hamrouni.

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Communicating Editor: B Sury

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Hamrouni, H., Jlali, Z. A characterization of totally disconnected compactly ruled groups. Proc Math Sci 131, 6 (2021). https://doi.org/10.1007/s12044-020-00601-8

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Keywords

  • Totally disconnected locally compact group
  • compactly ruled
  • Chabauty topology

Mathematics Subject Classification

  • 22D05