Hyperbolic geometry — Question 10.1

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. 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.

Statement recorded in the supplied source.

Let $M$ be hyperbolic and $\mathcal F$ a taut foliation with two-sided branching. Identify $\widetilde M$ with $\mathbb H^3$. Must there exist a lifted leaf $\lambda$ such that each side of $\mathbb H^3\setminus\lambda$ contains an open hyperbolic halfspace? A halfspace is a component of the complement of a totally geodesic plane; a leaf with this property is asymptotically separated. [S2]

Short English statement

Must a taut foliation with branching in both directions have a lifted leaf leaving an open hyperbolic halfspace on each side?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300039 (AMR-102-0039). 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 10.1; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.