Foliated Teichmüller theory — Question 11.1

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. 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. Consider automorphisms of a foliated pair $(M,\mathcal F)$ taking leaves to leaves, and their classes modulo isotopies through such automorphisms. This is the foliation mapping-class viewpoint; it can be richer than the ordinary mapping class group of $M$.

Statement recorded in the supplied source.

Determine what nontrivial mapping-class elements can occur for taut foliations, including their geometric realization and possible infinite-order behavior. In the slithering example of the source a transverse flow can have a time-one map preserving the foliation, even though intermediate times need not preserve leaves; such elements must not be discarded merely because the underlying ambient map is isotopic to the identity. [S2]

Short English statement

Classify nontrivial foliation-preserving mapping classes, including those produced by a transverse flow and invisible to the ordinary ambient mapping class group.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300047 (AMR-102-0047). 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 11.1; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.