Rigidity and moduli — Question 3.1

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. For an integral $2$-homology class $u$, the Thurston norm $x(u)$ is the minimum of $\sum_j\max\{0,-\chi(S_j)\}$ over embedded oriented representatives $S=\bigcup_j S_j$; extend it by homogeneity and continuity to real homology. Its unit ball has a polyhedral structure. A fibered face is a face whose cone contains classes represented by fibers of fibrations over $S^1$. A pseudo-Anosov flow has stable and unstable hyperbolic directions off finitely many closed prong-singular orbits. Monotone equivalence allows collapsing foliated product intervals. Almost transversality to a pseudo-Anosov flow permits the usual modifications at singular orbits; geometric limits refer to limits of the foliated geometric structures.

Statement recorded in the supplied source.

For atoroidal $M$, construct a polyhedron $\mathscr P_{\mathcal F}$ refining the Thurston norm ball so as to parameterize taut foliations with all of the following properties. Each open face $c$ determines a pseudo-Anosov flow $X_c$, or an appropriate equivalence class of such flows; associated foliations can be isotoped transverse or almost transverse to $X_c$, or have a sublamination monotonically equivalent into its stable or unstable singular foliation. A geometric limit of foliations associated to $c$ is associated to a face contained in $\overline c$. Finally there is a natural polyhedral map from $\mathscr P_{\mathcal F}$ to a subdivision of the Thurston norm unit ball. The construction includes a choice of the appropriate parameterization and flow equivalence relation, rather than presupposing a particular one. [S2]

Short English statement

Seek a polyhedral parameterization of all taut foliations, with flows assigned to faces, compatible limits on their boundaries, and a polyhedral map to the Thurston norm ball.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300005 (AMR-102-0005). 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 3.1; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.