Kirby Problem 3.7

Definitions and mathematical statement

Let $M$ be a connected complete finite-volume hyperbolic three-manifold. A finite-sheeted cover $\widetilde M\to M$ fibers over the circle if $\widetilde M$ is the mapping torus of a homeomorphism $\phi:F\to F$ of a finite-type surface: \[ M_\phi=(F\times[0,1])/((x,1)\sim(\phi(x),0)). \] For a cusped manifold the fiber can be punctured, or equivalently one can use compact cores with boundary. A pseudo-Anosov monodromy has invariant transverse measured singular foliations $\mathcal F^s,\mathcal F^u$, whose transverse measures are multiplied by $\lambda^{-1}$ and $\lambda$ for some $\lambda>1$.

In this question orientable pseudo-Anosov means that both invariant foliations are transversely orientable: away from their singularities, normal directions can be chosen continuously and consistently around every loop. It does not merely mean that $\phi$ preserves the orientation of $F$.

Does every finite-volume hyperbolic three-manifold have a finite-sheeted cover admitting such a fibration with orientable pseudo-Anosov monodromy? Both the cover and the fibered structure may be chosen. Virtual fibering by itself provides pseudo-Anosov monodromy but does not impose the transverse orientability required here.

Short English statement

After passing to a finite cover, can every hyperbolic three-manifold fiber over the circle with transversely orientable invariant foliations for its monodromy?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2805 (KP-3.7), 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 3.7. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.