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PreprintarXiv licencearXiv

Uncountably many local isomorphism types of compactly generated simple groups

  • Ilaria Castellano
  • Jorge Fariña-Asategui
  • Mikel Eguzki Garciarena
  • Bianca Marchionna
  • Martyn Quick
  • Colin D. Reid
  • Simon M. Smith
  • Stephan Tornier
  • Matteo Vannacci
  • John S. Wilson
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Abstract

A major open question in the theory of locally compact groups is the following. Let 𝒮 be the class of non-discrete compactly generated totally disconnected locally compact groups that are topologically simple. Is the number of local isomorphism classes of groups in 𝒮 uncountable? We have answered this question, showing that there are 2ℵ0 local isomorphism classes in 𝒮. Our result was obtained without the use of artificial intelligence; it arose from a problem session that ran over several days at the workshop "Branch groups: subgroups, rigidity, topologies" at the Universidad Complutense de Madrid, organised by Dominik Francoeur, Alejandra Garrido and Tatiana Nagnibeda.

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Cite

BibTeXRIS
BibTeX
@misc{castellano2026uncountably,
  title = {{Uncountably many local isomorphism types of compactly generated simple groups}},
  author = {Ilaria Castellano and Jorge Fariña-Asategui and Mikel Eguzki Garciarena and Bianca Marchionna and Martyn Quick and Colin D. Reid and Simon M. Smith and Stephan Tornier and Matteo Vannacci and John S. Wilson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17410},
  eprint = {2609.17410},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17410},
}
RIS
TY  - UNPB
TI  - Uncountably many local isomorphism types of compactly generated simple groups
AU  - Ilaria Castellano
AU  - Jorge Fariña-Asategui
AU  - Mikel Eguzki Garciarena
AU  - Bianca Marchionna
AU  - Martyn Quick
AU  - Colin D. Reid
AU  - Simon M. Smith
AU  - Stephan Tornier
AU  - Matteo Vannacci
AU  - John S. Wilson
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.17410
UR  - https://arxiv.org/abs/2609.17410
AB  - A major open question in the theory of locally compact groups is the following. Let 𝒮 be the class of non-discrete compactly generated totally disconnected locally compact groups that are topologically simple. Is the number of local isomorphism classes of groups in 𝒮 uncountable? We have answered this question, showing that there are 2^ℵ_0 local isomorphism classes in 𝒮. Our result was obtained without the use of artificial intelligence; it arose from a problem session that ran over several days at the workshop "Branch groups: subgroups, rigidity, topologies" at the Universidad Complutense de Madrid, organised by Dominik Francoeur, Alejandra Garrido and Tatiana Nagnibeda.
ER  -