Leaf spaces and transverse structures — Question 8.4

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 surface lamination $\Lambda$ is a closed subset of $M$ locally homeomorphic to a product of a disk with a closed transverse set. It is essential when it has no sphere leaf or torus leaf bounding a solid torus, and its complementary regions are irreducible, have incompressible boundary, and have no compressing monogon. Such a monogon is a compressing disk with one boundary arc on a boundary leaf and the other along a transverse interval fiber. It is genuine if some complementary region is not an interval bundle. A sublamination is a closed union of leaves. The universal covering group $\widetilde{\operatorname{SL}}(2,\mathbb R)$ acts on $\mathbb R$, the universal cover of the projective line $\mathbb{RP}^1$. A transverse structure of this type gives local transverse coordinates whose transition maps are restrictions of that lifted projective action.

Statement recorded in the supplied source.

For a fixed $3$-manifold $M$, describe the structure of the entire set of essential laminations equipped with a transverse $\widetilde{\operatorname{SL}}(2,\mathbb R)$ structure, including their variation and parameterization. The lamination and its transverse structure are both part of the data. [S2]

Short English statement

Describe all essential laminations on a fixed manifold with a lifted real-projective transverse structure, and how these structures vary.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300027 (AMR-102-0027). 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 8.4; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.