Leaf spaces and transverse structures — Question 8.1

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 irreducible if every embedded sphere bounds a ball. A nontrivial action $\rho:\pi_1(M)\to\operatorname{Homeo}(\mathbb R)$ here has no global fixed point. A transverse $(\pi_1(M),\mathbb R)$ structure is specified by an equivariant developing map $D:\widetilde M\to\mathbb R$, locally a transverse coordinate, whose fibers give the lifted foliation; equivalently it can be obtained from a section transverse to the horizontal foliation of the associated flat $\mathbb R$-bundle.

Statement recorded in the supplied source.

For an irreducible $M$ whose fundamental group has such a nontrivial action on $\mathbb R$, determine when $M$ admits a taut foliation with a transverse $(\pi_1(M),\mathbb R)$ structure. Characterize when an action can be realized in this geometric way; existence of a group action alone is the hypothesis, not a supplied foliation. [S2]

Short English statement

Determine when a nontrivial real-line action of an irreducible manifold group can be geometrically realized by a taut foliation with real transverse coordinates.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300024 (AMR-102-0024). 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 8.1; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.