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Finitely generated subgroups and Chabauty topology in totally disconnected locally compact groups

  • Hatem Hamrouni EMAIL logo and Yousra Kammoun
Published/Copyright: June 23, 2021
Journal of Group Theory
From the journal Journal of Group Theory Volume 26 Issue 1

Abstract

For a locally compact group 𝐺, we write S ⁢ U ⁢ B ⁢ ( G ) for the space of closed subgroups of 𝐺 endowed with the Chabauty topology. For any positive integer 𝑛, we associate to 𝐺 the function δ G , n from G n to S ⁢ U ⁢ B ⁢ ( G ) defined by

δ G , n ⁢ ( g 1 , … , g n ) = gp ¯ ⁢ ( g 1 , … , g n ) ,

where gp ¯ ⁢ ( g 1 , … , g n ) denotes the closed subgroup topologically generated by g 1 , … , g n . It would be interesting to know for which groups 𝐺 the function δ G , n is continuous for every 𝑛. Let [ HW ] be the class of such groups. Some interesting properties of the class [ HW ] are established. In particular, we prove that [ HW ] is properly included in the class of totally disconnected locally compact groups. The class of totally disconnected locally compact locally pronilpotent groups is included in [ HW ] . Also, we give an example of a solvable totally disconnected locally compact group not contained in [ HW ] .

Acknowledgements

We would like to thank Professor Karl Heinrich Hofmann for sending us a copy of the article [18] he coauthored with Professor George Willis. We also want to thank the anonymous referee for the careful reading of the paper and for his/her valuable comments.

  1. Communicated by: George Willis

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Received: 2020-10-07
Revised: 2021-05-19
Published Online: 2021-06-23
Published in Print: 2023-01-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Some simple biset functors
  3. Explicit universal minimal constants for polynomial growth of groups
  4. Shift dynamics of the groups of Fibonacci type
  5. Finitely generated subgroups and Chabauty topology in totally disconnected locally compact groups
  6. The number of set-orbits of a solvable permutation group
  7. On finite 𝜎-tower groups
  8. Orders of inner-diagonal automorphisms of some simple groups of Lie type
  9. The Lie algebra structure of the degree one Hochschild cohomology of the blocks of the sporadic Mathieu groups
  10. M, B and Co1 are recognisable by their prime graphs
https://doi.org/10.1515/jgth-2020-0159

Articles in the same Issue

  1. Frontmatter
  2. Some simple biset functors
  3. Explicit universal minimal constants for polynomial growth of groups
  4. Shift dynamics of the groups of Fibonacci type
  5. Finitely generated subgroups and Chabauty topology in totally disconnected locally compact groups
  6. The number of set-orbits of a solvable permutation group
  7. On finite 𝜎-tower groups
  8. Orders of inner-diagonal automorphisms of some simple groups of Lie type
  9. The Lie algebra structure of the degree one Hochschild cohomology of the blocks of the sporadic Mathieu groups
  10. M, B and Co1 are recognisable by their prime graphs
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