Uncountability of local isomorphism classes of compactly generated simple groups

Let $\mathscr{S}$ be the class of all non-discrete, compactly generated, totally disconnected locally compact groups that are topologically simple. Is the quotient of $\mathscr{S}$ by local isomorphism uncountable; equivalently, does $|\mathscr{S}/\cong_{\mathrm{loc}}|>\aleph_0$ ? The cited preprint claims the stronger conclusion $|\mathscr{S}/\cong_{\mathrm{loc}}|=2^{\aleph_0}$ , where $G\cong_{\mathrm{loc}}H$ means that $G$ and $H$ have isomorphic open subgroups.

References

Primary source

arXiv

Additional references

Uncountably many local isomorphism types of compactly generated simple groups — arXiv — 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

geometric-group-theory locally-compact-groups

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims the strongest possible answer, producing continuum many distinct local types, but the work has not yet been peer reviewed.

The problem asks whether non-discrete compactly generated totally disconnected topologically simple groups have $2^{\aleph_0}$ distinct local isomorphism classes. A new preprint claims that they do.

Known results

Smith’s work established uncountably many isomorphism classes in this group class, but a 2021 survey said it was unknown whether they comprise only countably many local isomorphism classes.

September 2026 claimed resolution

Castellano, Fariña-Asategui, Garciarena, Marchionna, Quick, Reid, Smith, Tornier, Vannacci, and Wilson claim the existence of $2^{\aleph_0}$ local isomorphism classes. The preprint is presented as an overview of forthcoming work and is not yet peer reviewed, so the resolution remains unverified.

Current status (as of September 2026): A preprint claims $2^{\aleph_0}$ local isomorphism classes, while the claim remains unverified.

Sources

No solutions have been posted yet.