Hyperbolic geometry — Question 10.6

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. For a cooriented codimension-one foliation choose a nowhere-zero $1$-form $\alpha$ with $T\mathcal F=\ker\alpha$. Integrability gives $d\alpha=\alpha\wedge\omega$. The Godbillon--Vey class is $\operatorname{gv}(\mathcal F)=[\omega\wedge d\omega]\in H^3(M;\mathbb R)$, independent of these choices, and its value on an oriented closed $M$ is the integral of this $3$-form.

Statement recorded in the supplied source.

Does there exist a universal finite-valued bound depending only on hyperbolic volume for the absolute Godbillon--Vey number of taut foliations? More precisely, seek a function $b$ such that $|\operatorname{gv}(\mathcal F)[M]|\le b(\operatorname{vol}(M))$ for every hyperbolic $3$-manifold $M$ and every sufficiently regular taut foliation $\mathcal F$ for which this number is defined. No linear form of $b$ is imposed. [S2]

Short English statement

Can hyperbolic volume alone uniformly bound the absolute Godbillon--Vey invariant of taut foliations?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300044 (AMR-102-0044). 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 10.6; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.