Closed surface bundles of least volume

Abstract

Since the set of volumes of hyperbolic $3$–manifolds is well ordered, for each fixed $g$ there is a genus–$g$ surface bundle over the circle of minimal volume. Here, we introduce an explicit family of genus–$g$ bundles which we conjecture are the unique such manifolds of minimal volume. Conditional on a very plausible assumption, we prove that this is indeed the case when $g$ is large. The proof combines a soft geometric limit argument with a detailed Neumann–Zagier asymptotic formula for the volumes of Dehn fillings.

Our examples are all Dehn fillings on the sibling of the Whitehead manifold, and we also analyze the dilatations of all closed surface bundles obtained in this way, identifying those with minimal dilatation. This gives new families of pseudo-Anosovs with low dilatation, including a genus 7 example which minimizes dilatation among all those with orientable invariant foliations.