Rigidity and moduli — Question 3.2

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. The Teichmüller polynomial of a fibered face is an element of $\mathbb Z[H_1(M;\mathbb Z)/\mathrm{torsion}]$ whose specializations encode pseudo-Anosov monodromy dilatations and their entropy data.

Statement recorded in the supplied source.

Extend the Teichmüller polynomial construction from fibered faces to the other faces of the Thurston norm ball, allowing a suitable generalized polyhedron. One possible parameter space is a polyhedron of taut foliations whose faces encode transverse pseudo-Anosov flows and which maps to the Thurston norm ball, as sought in Question 3.1 of the source. The task is an extension of the construction and its geometric information, not merely another polynomial that computes the norm. [S2]

Short English statement

Generalize the polynomial encoding fibered-face dynamics to nonfibered faces, possibly on a broader polyhedral space of taut foliations.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300006 (AMR-102-0006). 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 3.2; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.