Leaf spaces and transverse structures — Question 8.2

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. The leaf space $L_{\mathcal F}=\widetilde M/\widetilde{\mathcal F}$ has its quotient topology, and can be non-Hausdorff. A taut foliation is $\mathbb R$-covered if this space is homeomorphic to $\mathbb R$. Its deck action is the holonomy representation $\rho_H:\pi_1(M)\longrightarrow\operatorname{Homeo}(L_{\mathcal F})$. Branching means nonseparated leaves in this space; one-sided branching occurs in only one transverse direction, and two-sided branching in both.

Statement recorded in the supplied source.

If $\mathcal F$ is $\mathbb R$-covered on atoroidal $M$, does there exist a coordinate homeomorphism $h:L_{\mathcal F}\to\mathbb R$ and a function $C:\pi_1(M)\to(0,\infty)$ such that, for $\rho^h=h\rho_Hh^{-1}$, \[\left|\,|\rho^h(g)(p)-\rho^h(g)(q)|-|p-q|\,\right|\le C(g)\qquad(g\in\pi_1(M),\ p,q\in\mathbb R)?\] The additive bound can depend on $g$, but not on $p,q$; it is not required to be uniform over all group elements. [S2]

Short English statement

Can the real-line leaf-space action be conjugated so that each group element changes every distance by a bounded additive amount?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300025 (AMR-102-0025). 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 8.2; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.