Foliations and Diffeomorphism Groups — Question 47

Definitions and mathematical statement

Let $G$ be a group, let $P$ be a topological plane equipped with the two foliation structures intended by the source, and let $\rho\colon G\to\operatorname{Homeo}(P)$ be an action. Write $\mathcal A$ for the class of such actions satisfying the source's definition of an Anosov-like action, and $\mathcal F$ for the class satisfying the definition of a flowable action attributed to Baik--Wu--Zhao, reference key baik2024reconstruction.

Question (source Question 47). Is \[ \mathcal A\subseteq\mathcal F? \] Equivalently, does every Anosov-like action on a bifoliated plane satisfy the flowability axioms? If this implication fails, find counterexample actions in $\mathcal A\setminus\mathcal F$ and determine precisely how the two notions differ. The question concerns flowability of the given action, rather than merely the existence of some flowable action of the same group.

Unresolved definitions and scope. The supplied record does not include the axioms defining either class. It also does not specify the permitted foliation singularities, the exact preservation requirements for the action, restrictions on $G$, or whether orientation preservation is required in the question itself. Consequently, the displayed inclusion expresses the intended comparison but is not a self-contained axiomatic formulation. These missing conditions must be recovered from the cited definitions before mathematical review. The source describes hyperbolicity informally as topological expansion along one leaf and contraction along another when two leaves cross; it does not supply the quantifiers, fixed-point conditions, or precise expansion and contraction requirements needed to turn that description into an axiom.

Supplied context. The source relates this question to the Anosov-like action conjecture. It reports that Baik--Wu--Zhao defined flowable actions and showed that a group admitting a flowable orientation-preserving action on a bifoliated plane is the fundamental group of a $3$-manifold with a pseudo-Anosov flow, with the bifoliated plane identified as the flow's orbit space. This is retained as a report of the source, not an independently verified theorem statement. The record does not specify whether the orbit space is that of the flow on the manifold or of a lifted flow. It says that the two sets of axioms differ slightly and identifies the informal hyperbolicity condition above as their most important common condition.

Short English statement

The question asks whether every Anosov-like action on a bifoliated plane is flowable in the sense attributed to Baik--Wu--Zhao. If not, it asks for counterexamples and a precise comparison of the notions. The supplied record omits both sets of defining axioms, so their exact hypotheses remain unresolved. Its open label is supplied registry metadata, not an independently verified status assessment.

Sources

• Local UnsolvedMath registry, permanent ID $800106$, exact source code catalogue entry, title Foliations and Diffeomorphism Groups --- Question 47. The record attributes the problem to Nariman and Yazdi, with proposed year $2025$, and supplies status open.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 47. Supplied source URL: https://arxiv.org/abs/2503.18274. The provenance records access on $2026\text{-}10\text{-}16$ and extraction by arxiv_latex; these are supplied provenance statements, not a claim of independent access or verification here.
• Baik--Wu--Zhao, cited in the supplied statement by the reference key baik2024reconstruction, for the definition of flowable actions and the reported reconstruction result. No complete bibliographic entry or separate URL for this reference is supplied, and its defining axioms and theorem hypotheses cannot be verified from the record alone.