Hyperbolicity of surface bundles

The problem asks whether a surface bundle over a surface can have any of four increasingly weak forms of hyperbolicity, including compact examples.

Known results

• Kent–Leininger constructions provide infinitely many homeomorphism types of atoroidal surface bundles over surfaces.
• Lafont, Miller, and Ruffoni (October 23, 2024) obtained infinitely many such bundles with signature zero, removing that obstruction to a hyperbolic metric but not producing one.

Compact atoroidal examples

The preprint Atoroidal surface bundles claims infinitely many compact surface bundles whose fundamental groups contain no subgroup isomorphic to $\mathbb{Z}^2$, settling property $(4)$ affirmatively. It presents the existence of a hyperbolic metric, negative curvature, and a Gromov-hyperbolic fundamental group as open or conjectural; the claimed results are not independently verified here.

Current status (as of September 2026): Compact atoroidal surface bundles are claimed to exist, but properties $(1)$–$(3)$ remain open in the retrieved record.