Leaf spaces and transverse structures — Question 8.8

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 lamination of $S^1$ is a closed family of unordered pairs of distinct circle points, represented by disjoint hyperbolic geodesics in the disk. Complementary regions have finite hyperbolic area. A transverse pair $\Lambda^+,\Lambda^-$ has transverse geodesic intersections between the two families. A group preserving both acts minimally on each family when every leaf orbit is dense in that leaf space. Two groups are commensurable when they have isomorphic finite-index subgroups.

Statement recorded in the supplied source.

Given such a transverse pair of circle laminations with finite-area complementary regions, and a group $\Gamma\le\operatorname{Homeo}(S^1)$ preserving them and acting minimally on the leaves of each, characterize when $\Gamma$ is commensurable with $\pi_1(M)$ for a hyperbolic $3$-manifold $M$. [S2]

Short English statement

Recognize when a group acting minimally on a pair of transverse circle laminations is commensurable with a hyperbolic three-manifold group.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300031 (AMR-102-0031). 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 8.8; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.