Leaf spaces and transverse structures — Question 8.10

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 a taut foliation with hyperbolic lifted leaves, a universal circle is a circle $S^1_{\mathrm{univ}}$ with a $\pi_1(M)$-action and continuously varying monotone maps $\phi_\lambda:S^1_{\mathrm{univ}}\to S^1_\infty(\lambda)$ to leaf ideal circles. Monotone means point fibers are contractible. The maps are equivariant: $\phi_{g\lambda}\circ\rho(g)=g\circ\phi_\lambda$. The core for a leaf is the complement of the open intervals collapsed by its map. For incomparable lifted leaves, each core lies in one gap of the other leaf. A minimal circle action has every orbit dense.

Statement recorded in the supplied source.

Determine the optimal analytic regularity obtainable for the $\pi_1(M)$-action on $S^1_{\mathrm{univ}}$, allowing a change of circle coordinates. In particular compare Lipschitz, differentiable and more regular realizations of these actions. The target concerns the universal-circle action together with its foliation origin, rather than arbitrary circle actions of the group. [S2]

Short English statement

Determine how regular an action on a foliation universal circle can be made by changing circle coordinates.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300033 (AMR-102-0033). 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 8.10; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.