CAM Actions of Residually Finite Groups

Definitions and mathematical statement

A group $G$ is residually finite if, for every $g\in G\setminus\{e\}$, there exist a finite group $F$ and a homomorphism $\varphi:G\to F$ such that $\varphi(g)\ne e_F$. Equivalently, \[ \bigcap_{\substack{N\trianglelefteq G\\ [G:N]<\infty}}N=\{e\}. \] The supplied record uses the term chaotic almost minimal action, abbreviated CAM, without defining it. Write $\mathcal{C}(G)$ provisionally for the assertion that $G$ admits an action of the type called CAM in the cited report; this notation is a placeholder for the missing definition.

The recorded existence question is:

Does every residually finite group $G$ admit a chaotic almost minimal action?

Its logical form is \[ \forall G\;\left(G\text{ is a residually finite group}\Longrightarrow\mathcal{C}(G)\right). \] The supplied report passage also asserts the necessary implication \[ \mathcal{C}(G)\Longrightarrow G\text{ is residually finite}. \] This implication is reported source context, not a theorem verified in this draft.

Unresolved qualifications. The record does not supply the definition of CAM, the class of underlying spaces, or the requirements on the action. It also does not state whether the groups must be infinite, countable, or finitely generated. The literal universal wording includes finite groups and the trivial group; whether these are intended must be checked against the defining source. Thus the displayed question preserves the supplied quantifier, but a fully specified mathematical formulation requires the missing definition and scope restrictions.

Short English statement

The question asks whether residual finiteness is sufficient for a group to admit a chaotic almost minimal action. The supplied passage states that residual finiteness is necessary. A precise interpretation still requires the definition of CAM and clarification of the permitted groups and action spaces.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005681$, entitled CAM Actions of Residually Finite Groups; exact source problem code catalogue entry. The record lists proposed year 2023 and supplies no named proposer.
• Supplied source citation: Aspects of Aperiodic Order (2024), Oberwolfach Reports, DOI 10.4171/owr/2023/37, pp. 2107--2178. https://doi.org/10.4171/owr/2023/37. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labelled "Oberwolfach report": https://doi.org/10.4171/owr/2023/37.
• Supplied literature source labelled "Cyr-Kra-Schmieding": https://sites.math.northwestern.edu/~kra/papers/CAM.pdf. No further bibliographic details are supplied.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open, states that the report presents the existence question as open, and reports that no later primary result located by its cutoff resolves it. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied verification claims; the report and linked paper have not been independently consulted for this draft.