Immersed objects — Question 14.5

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. 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. An immersed surface has amenable growth here when it admits a Følner exhaustion: compact regions $D_i$ with boundary length divided by area tending to zero, for the induced intrinsic geometry. A quasigeodesic immersed surface has quasi-isometrically embedded universal lift in the ambient universal cover.

Statement recorded in the supplied source.

Find the weakest useful kind of $2$-dimensional geometric object guaranteed in every atoroidal $3$-manifold. In particular, does every hyperbolic $3$-manifold $M$ contain an immersed quasigeodesic surface of amenable growth? The general search for a suitably weak object is part of the question, and the displayed surface criterion is one concrete candidate. [S2]

Short English statement

Seek geometric surfaces or weaker two-dimensional objects guaranteed in every atoroidal manifold; test quasigeodesic immersed surfaces with a Folner exhaustion in every hyperbolic manifold.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300065 (AMR-102-0065). 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 14.5; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.