Kirby Problem 3.20

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.

Let $M$ be a closed hyperbolic $3$-manifold and let $\rho:\pi_1(M)\to\operatorname{Homeo}^+(\mathbb R)$ be injective, where $\operatorname{Homeo}^+(\mathbb R)$ consists of increasing homeomorphisms. A codimension-one lamination $\lambda$ is a closed subset locally modeled on a union of parallel surface plaques; it is transversely orientable when its transverse local intervals have a consistent orientation. An essential lamination has no sphere leaves, its leaves are $\pi_1$-injective, its complementary regions are irreducible, and its complement is end-incompressible (there are no essential compressing half-planes at the ends). A complementary region is irreducible if each embedded $2$-sphere bounds a $3$-ball. Let $\widetilde\lambda$ be the lifted lamination in $\widetilde M$ and $\mathcal L(\widetilde\lambda)$ its leaf space, with the quotient topology. In the source's duality convention, $\lambda$ is dual to $\rho$ if there is a map $f:\mathcal L(\widetilde\lambda)\to\mathbb R$ satisfying $f(g\ell)=\rho(g)f(\ell)$ for every $g\in\pi_1(M)$ and leaf $\ell$.

Statement recorded in the supplied source.

For every closed hyperbolic $M$ and every faithful $\rho$ as above, does there exist a transversely orientable essential lamination $\lambda$ dual to that specified action $\rho$? The lamination must realize the equivariant leaf-space map for the given representation, not merely witness that some action exists. [S2]

Short English statement

Does every faithful action of a closed hyperbolic three-manifold group on the line have a dual essential, transversely orientable lamination?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2818 (KP-3.20). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (editors), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), Problem 3.20, p. 144 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.