Hyperbolic geometry has been a powerful tool in the study of manifold
topology. Beyond the classical theory of surfaces, Thurston showed
that the family of surface bundles over the circle is a rich source of
hyperbolic 3-manifolds. In dimension 4, the correct analogue is given
surface bundles over surfaces. In order for such a bundle to admit a
hyperbolic metric, it needs to satisfy some conditions, such as being
atoroidal and having zero signature. Surprisingly enough, the first
examples of atoroidal surface bundles over surfaces were constructed
only a few months ago by Kent-Leininger. In this talk I’ll explain why
these examples also have signature zero, meaning that they could admit
hyperbolic metrics. This is joint work in progress with J-F. Lafont
and N. Miller.