Clopen Strict Comparison for Amenable Cantor Actions

Definitions and mathematical statement

Let $G$ be a countably infinite discrete amenable group. Here amenability means that for every finite $K\subseteq G$ and every $\varepsilon>0$, there is a nonempty finite $F\subseteq G$ such that $|kF\mathbin{\triangle}F|<\varepsilon|F|$ for every $k\in K$. Let $X$ be a Cantor space, meaning a nonempty compact metrizable totally disconnected space without isolated points. Consider an action $G\curvearrowright X$ by homeomorphisms, written $(g,x)\mapsto gx$. The action is minimal if $\overline{Gx}=X$ for every $x\in X$. No freeness assumption is included in the supplied statement.

Write $M_G(X)$ for the set of Borel probability measures $\mu$ on $X$ satisfying $\mu(gE)=\mu(E)$ for every $g\in G$ and every Borel set $E\subseteq X$. A subset of $X$ is clopen if it is both open and closed. The action has clopen strict comparison if, for all clopen $A,B\subseteq X$, \[ \left(\forall\mu\in M_G(X),\ \mu(A)<\mu(B)\right) \ \Longrightarrow\ \begin{gathered} \exists n\in{\mathbb{Z}}_{\geq 0},\ \exists A_1,\ldots,A_n\subseteq X\text{ clopen},\ \exists g_1,\ldots,g_n\in G:\\ A=\bigsqcup_{i=1}^{n}A_i,\qquad g_iA_i\subseteq B\quad(1\leq i\leq n),\\ g_iA_i\cap g_jA_j=\varnothing\quad(1\leq i<j\leq n). \end{gathered} \] The empty partition is allowed when $A=\varnothing$. The measure inequality is required separately for every invariant probability measure; no numerical gap is specified in the hypothesis.

Question. Does every minimal action by homeomorphisms of every countably infinite discrete amenable group on a Cantor space have clopen strict comparison?

The topological meaning of action used here is the conventional interpretation of the supplied Cantor action formulation. The supplied record labels the problem partially solved, but its literature assessment does not specify the precise hypotheses or full scope of the established subcases. Those claims require separate review and do not restrict the universal question above.

Short English statement

For every minimal action of a countably infinite amenable group on a Cantor space, must a clopen set that is smaller than another for every invariant probability measure split into finitely many clopen pieces whose group translates fit disjointly inside the other set? The supplied partially solved label is dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003202$, title Clopen Strict Comparison for Amenable Cantor Actions, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2016 and supplies no individual proposer.
• Supplied source citation, with mathematical title formatting normalized: $C^*$-Algebras (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/40, pp. 2269--2345. https://doi.org/10.4171/owr/2016/40. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The relationship between the supplied title year 2017 and the proposed year and DOI year 2016 is unresolved.
• Downarowicz and Zhang, The comparison property of amenable groups, as attributed in the supplied literature sources. https://arxiv.org/abs/1712.05129.
• OWR Operator Algebras report, as labeled in the supplied literature sources. https://ems.press/content/serial-article-files/46645?nt=1.
• Literature assessment checked at 2026-08-21: this is supplied dated curation, not independently verified. It labels the problem partially solved and states that comparison is established for ${\mathbb{Z}}$, for the class written as "Z d", and for broad subexponential growth classes, while the question for arbitrary countably infinite amenable groups remains open. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry assigns statement status corrected_verified and reports that the official report was checked and that the clean statement restores the group action, space, invariant measures, and partition-and-translation definition of clopen strict comparison. These are supplied verification claims; no source visit or independent mathematical verification is asserted here.