Kirby Problem 3.19

Definitions and mathematical statement

Let $M$ be a closed hyperbolic three-manifold and $X$ a smooth nowhere-zero vector field on $M$. Its complete flow lifts to a flow $\widetilde\Phi_t$ on the universal cover $\widetilde M\cong\mathbb H^3$. A lifted flow line is proper if the map $t\mapsto\widetilde\Phi_t(x)$ from $\mathbb R$ to $\widetilde M$ is proper: the inverse image of every compact set is compact. In particular it escapes every compact subset in both time directions.

The orbit space $L_X$ is the quotient of $\widetilde M$ by belonging to the same flow line, with its quotient topology. The deck group $\pi_1(M)$ acts on this space, since deck transformations commute with the lifted flow. The problem asks successively:

1. Can $X$ always be chosen so that every lifted flow line is proper?
2. Can it be chosen so that, additionally, $L_X$ is Hausdorff?
3. In the resulting plane orbit-space setting, can its deck-group action be extended to a closed disk compactification?

The last request means a homeomorphism $L_X\cong\operatorname{int}(D^2)$ under which every deck transformation extends to a homeomorphism of $D^2$, and these extensions form a group action. Properness of individual orbits does not itself guarantee a Hausdorff quotient or such a boundary action.

Short English statement

Can every closed hyperbolic three-manifold support a flow whose lifted trajectories have a well-behaved plane of orbits and an invariant circle boundary?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2817 (KP-3.19), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.19. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.