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arXiv · 2609.17410

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 $\mathscr{S}$ 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 $\mathscr{S}$ uncountable? We have answered this question, showing that there are $2^{\aleph_0}$ local isomorphism classes in $\mathscr{S}$. This preprint is an overview of our forthcoming paper. 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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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. 2026-09-15. Uncountably many local isomorphism types of compactly generated simple groups. https://arxiv.org/abs/2609.17410

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