Foliations and Diffeomorphism Groups — Question 36

Definitions and mathematical statement

Let $T^2={\mathbb{R}}^2/{\mathbb{Z}}^2$, and let $p:E\to S$ be the torus bundle in the supplied question. The source denotes its monodromy by $\rho$, but does not specify the hypotheses on $S$, the bundle's structure group, or the target group of $\rho$. A compact leaf of a foliation means a leaf that is compact in its intrinsic leaf topology.

Question. Under what hypotheses on the monodromy $\rho$ does $E$ admit a foliation $\mathcal F$ with no compact leaves that is "close to a fiber"? The no-compact-leaves requirement is \[ \text{for every leaf }L\text{ of }\mathcal F,\qquad L\text{ is noncompact}. \] The quoted closeness condition is unresolved: the supplied excerpt specifies neither a topology nor a quantitative neighborhood, and does not say whether arbitrarily close foliations are required. It also does not explicitly specify the dimension or regularity of $\mathcal F$. Consequently, the existence question cannot be given a fully quantified precise formulation from this excerpt alone.

Supplied vector-field context. For vector fields $X,Y$ on a common manifold $M$, commuting means $[X,Y]=0$, and a common zero is a point $x\in M$ with $X(x)=Y(x)=0$. The excerpt credits Elon Lima with a common-zero result for commuting vector fields in the trivial-representation case. The underlying manifold, regularity, and other hypotheses of that result are absent, so this attribution must not be read as an unrestricted common-zero theorem.

The excerpt also contains a first-person report of an example of a $C^1$ commuting pair with no common zero, equivariant with respect to $\rho$, when $\rho(\pi_1(S))$ is generated by a single hyperbolic matrix. In the usual linear terminology, a hyperbolic matrix has no eigenvalue of modulus $1$. The supplied excerpt does not identify the speaker, the vector fields' domain, the relevant group actions, or the precise equivariance condition. It describes the torus-bundle question as essentially equivalent to the preceding problem; the missing setting prevents a precise equivalence statement.

Short English statement

Determine which monodromies of a torus bundle over $S$ allow a foliation without compact leaves that is close to a fiber. The supplied excerpt leaves the assumptions on $S$, foliation regularity and dimension, and meaning of closeness unspecified. It also reports related common-zero and equivariant vector-field results whose full hypotheses are missing.

Sources

• Local UnsolvedMath registry, permanent ID $800095$, source code catalogue entry, title "Foliations and Diffeomorphism Groups --- Question 36." The supplied registry attributes the record to Nariman and Yazdi, with proposed year $2025$, and labels it open. This label is supplied source metadata, not an independently verified status assessment.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 36. Supplied source URL: https://arxiv.org/abs/2503.18274. The record reports access on 2026-10-16 and extraction from arXiv LaTeX; these are supplied provenance statements, not a claim of independent source consultation.
• The supplied excerpt credits Elon Lima with the trivial-representation common-zero result, without giving a bibliographic citation or its complete hypotheses. The first-person example and the claimed essential equivalence are preserved as reports from that excerpt.