Embedding $\sigma$-Compact TDLC Groups

Definitions and mathematical statement

A totally disconnected locally compact (TDLC) group means a Hausdorff topological group that is locally compact and whose connected components are singletons. Such a group $G$ is $\sigma$-compact if there are compact subsets $K_n\subseteq G$, indexed by positive integers, such that $G=\bigcup_{n\geq 1}K_n$. A topological group $H$ is compactly generated if some compact subset $K\subseteq H$ generates $H$ as an abstract group, so that $H=\langle K\rangle$. A topological group embedding $\iota:G\to H$ is an injective continuous homomorphism that is a homeomorphism onto its image with the subspace topology.

Under this interpretation of embedding, the question is: for every $\sigma$-compact TDLC group $G$, do there exist a compactly generated TDLC group $H$ and a topological group embedding $\iota:G\to H$? No second countability assumption is included.

The supplied clean statement explicitly supplies the $\sigma$-compact hypothesis. The original excerpt starts with an unspecified $G$ and does not itself state that hypothesis or define the meaning of embedding. Thus the formulation above follows the supplied clean statement and uses the topological embedding convention, which requires confirmation against the full source.

The original excerpt also contains questions about a TDLC version of the Atiyah conjecture, compactly generated TDLC groups admitting lattices of irrational covolume, equality of rational cohomological dimensions for quasi-isometric compactly generated TDLC groups $G,H$ both having finite cohomological dimension over ${\mathbb{Q}}$, and a suitable TDLC analogue of right-angled Artin groups. These are neighboring questions in the supplied excerpt; they are not additional assertions or requirements of the selected embedding question.

Short English statement

Does every $\sigma$-compact totally disconnected locally compact group occur, with its given topology, as a subgroup of a compactly generated totally disconnected locally compact group? This formulation interprets embedding topologically; the supplied excerpt leaves that convention implicit.

Sources

• Local UnsolvedMath registry, permanent ID $30005706$, source problem code catalogue entry, title: Embedding $\sigma$-Compact TDLC Groups. The supplied record labels the problem open and its statement corrected_verified; its verification note reports checking against the official Oberwolfach report. These are supplied registry assessments, not independent verification in this draft.
• Supplied source citation: Mini-Workshop: Homological aspects of TDLC-groups (2024), Oberwolfach Reports, pp. 3103--3138, DOI 10.4171/owr/2023/55. https://doi.org/10.4171/owr/2023/55. The record gives proposed year 2023; its citation title gives 2024. No proposer is specified for the selected embedding question.
• Supplied literature source, labeled "Oberwolfach report": https://ems.press/content/serial-article-files/48166.
• Supplied literature source, labeled "Castellano--Marchionna--Weigel": https://doi.org/10.1007/s00208-025-03116-7. No article title or specific supporting result is supplied.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem open and reports that no general solution was located to the embedding, TDLC Atiyah, or irrational-covolume lattice questions. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.