Leaf spaces and transverse structures — Question 8.9

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A codimension-one foliation $\mathcal F$ decomposes a $3$-manifold $M$ into surface leaves, locally as the horizontal slices of a product chart. In the terminology of the source, it is taut if a closed curve transverse to $\mathcal F$ meets every leaf. A minimal foliation means that every leaf is dense; a minimal surface means a surface with vanishing mean curvature for the specified Riemannian metric. These two uses of minimal are distinct. A manifold is atoroidal if it has no incompressible embedded torus other than a boundary-parallel one. Write $\widetilde M$ for its universal cover, and put a tilde on a foliation or lamination for its lift. The leaf space $L_{\mathcal F}=\widetilde M/\widetilde{\mathcal F}$ has its quotient topology, and can be non-Hausdorff. A taut foliation is $\mathbb R$-covered if this space is homeomorphic to $\mathbb R$. Its deck action is the holonomy representation $\rho_H:\pi_1(M)\longrightarrow\operatorname{Homeo}(L_{\mathcal F})$. Branching means nonseparated leaves in this space; one-sided branching occurs in only one transverse direction, and two-sided branching in both. For a taut foliation with hyperbolic lifted leaves, a universal circle is a circle $S^1_{\mathrm{univ}}$ with a $\pi_1(M)$-action and continuously varying monotone maps $\phi_\lambda:S^1_{\mathrm{univ}}\to S^1_\infty(\lambda)$ to leaf ideal circles. Monotone means point fibers are contractible. The maps are equivariant: $\phi_{g\lambda}\circ\rho(g)=g\circ\phi_\lambda$. The core for a leaf is the complement of the open intervals collapsed by its map. For incomparable lifted leaves, each core lies in one gap of the other leaf. A minimal circle action has every orbit dense.

Statement recorded in the supplied source.

Classify possible universal circles and their group actions for a fixed $3$-manifold as the taut foliation varies, and for a fixed taut foliation. Determine exactly which taut foliations have a unique minimal universal circle, with uniqueness understood up to the structure-preserving equivalence of universal-circle actions and leaf maps. [S2]

Short English statement

Classify universal circles for a manifold or a chosen foliation, and characterize when the minimal one is unique.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300032 (AMR-102-0032). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Danny Calegari, Problems in foliations and laminations of 3-manifolds, version 0.78 (2002), Question 8.9; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.