Periodic Topologically Simple Locally Compact Groups

Definitions and mathematical statement

Let $G$ be a Hausdorff topological group. It is locally compact if every point has a compact neighbourhood, and totally disconnected if every connected component is a singleton. It is compactly generated if there is a compact subset $K\subseteq G$ such that $G=\langle K\rangle$, where $\langle K\rangle$ denotes the subgroup algebraically generated by $K$. It is topologically simple if its only closed normal subgroups are $\{e\}$ and $G$, and nondiscrete if $\{e\}$ is not open.

The supplied question asks whether there exists a compactly generated, totally disconnected, locally compact group $G$ that is topologically simple, nondiscrete, and periodic. The word "periodic" is not defined in the supplied record, so its intended meaning requires confirmation.

Under the topological interpretation of periodicity, the precise existence question is whether such a group $G$ exists with \[ \forall g\in G,\qquad \overline{\langle g\rangle}\text{ compact in }G, \qquad \langle g\rangle=\{g^n:n\in{\mathbb{Z}}\}. \] Under the torsion interpretation of periodicity, the final condition instead reads \[ \forall g\in G\;\exists n\in{\mathbb{Z}}_{\geq 1},\qquad g^n=e. \] These are distinct formulations; this draft does not identify them or resolve which the source intends. No second countability assumption or bound on element orders is supplied. The background description as a "topological-Tarski-monster question" supplies no additional subgroup condition.

Short English statement

Does there exist a nondiscrete, compactly generated, totally disconnected locally compact group with no proper nontrivial closed normal subgroup that is periodic? The intended meaning of periodicity needs confirmation: compact cyclic closures and finite element orders give different formulations. The supplied registry labels the problem open; this is a statement-only draft requiring human mathematical review.

Sources

• Supplied local problem registry record, permanent UnsolvedMath ID $30002703$, title "Periodic Topologically Simple Locally Compact Groups," exact source problem code catalogue entry. The record labels the statement "exact" and says it was retained byte-for-byte; this is supplied source metadata, not independent verification.
• Supplied source citation: Arbeitsgemeinschaft: Totally Disconnected Groups (2015), Oberwolfach Reports, pp. 2619--2665, DOI 10.4171/owr/2014/47. Supplied source URL and literature-source link, labelled "Oberwolfach Report 47/2014": https://doi.org/10.4171/owr/2014/47. The record attributes the problem to the workshop-report collection, gives proposed year 2014, and names no individual proposer. The citation's 2015 date and the report's 2014 designation are retained as supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the existence question is still presented as a topological-Tarski-monster question, with no authoritative construction found. This is supplied dated curation derived from the Oberwolfach material, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.