Kirby Problem 2.19

Definitions and mathematical statement

Let $S$ be an orientable finite-type surface of negative Euler characteristic and $f:S\to S$ an orientation-preserving homeomorphism. Its mapping torus is \[ M_f=S\times[0,1]/(x,1)\sim(f(x),0). \] Restrict to pseudo-Anosov mapping classes: these preserve a pair of transverse measured foliations, expanding one and contracting the other by a factor $\lambda>1$. Their mapping tori have complete finite-volume hyperbolic structures on their interiors. Write $M_f=\mathbb H^3/\Gamma_f$ using its holonomy group $\Gamma_f\leq\operatorname{PSL}_2(\mathbb C)$.

A finite-volume hyperbolic $3$-manifold is arithmetic when its lattice is commensurable, up to conjugacy, with the image of the norm-one elements of an order in a quaternion algebra over a number field having one complex place, ramified at every real place. Groups are commensurable if their intersection has finite index in both.

Characterize exactly the mapping classes $[f]$ whose mapping tori are arithmetic. The desired criteria should be expressed through monodromy or mapping-class data, and should distinguish the arithmetic classes from other pseudo-Anosov classes. Both closed fibers and punctured finite-type fibers are included; conjugate mapping classes give equivalent bundle monodromies.

Short English statement

Which surface monodromies produce arithmetic hyperbolic mapping tori?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2767 (KP-2.19), 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 2.19. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.