Sublaminations and superlaminations — Question 6.3

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.

Statement recorded in the supplied source.

Call an essential lamination very full if every complementary region is an ideal polygon bundle over $S^1$. For a genuine lamination $\Lambda\subset M$, characterize when there exists a very full essential lamination $\Lambda'\supseteq\Lambda$, obtained by adding leaves in the complementary regions. Determine whether requiring $M$ to be hyperbolic improves existence or its characterization. Coorientation of $\Lambda'$ is not an imposed hypothesis. [S2]

Short English statement

Determine when a genuine lamination can be enlarged so that every complementary region is an ideal polygon bundle, and how hyperbolicity affects this.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300013 (AMR-102-0013). 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 6.3; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.