Kirby Problem 2.44

Definitions and mathematical statement

Let $S$ be a connected orientable surface of infinite type: it is not homeomorphic to a compact surface with finitely many points removed. Work without boundary, or use interiors when a boundary convention is needed. Given a homeomorphism $f:S\to S$, form the mapping torus \[ M_f=(S\times[0,1])/\left((x,1)\sim(f(x),0)\right). \] It is a three-manifold fibering over the circle, with fiber $S$. No compactness or finite-volume assumption is imposed on $M_f$.

Characterize the homeomorphisms $f$ for which $M_f$ admits a complete Riemannian metric of constant sectional curvature $-1$. The metric need not be unique, and completeness is part of the hyperbolic-structure requirement. Criteria may depend on the action of $f$ on curves, subsurfaces, and the ends of $S$; finite-type pseudo-Anosov terminology alone is not a definition of the desired infinite-type class.

For mapping tori that are hyperbolic, ask also whether $M_f$ is homeomorphic to the interior of a compact hyperbolic three-manifold with totally geodesic boundary. The latter means a compact manifold carrying a curvature $-1$ metric whose boundary components have zero second fundamental form. This is a topological identification with that interior, not a requirement that its incomplete restricted metric equal the chosen complete metric on $M_f$.

The second assertion is a further finiteness and geometric condition, and must be distinguished from the first existence question.

Short English statement

Which infinite-type surface maps have hyperbolic mapping tori, and when do those tori admit compact cores with totally geodesic boundary?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2792 (KP-2.44), 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 2.44. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.