Since the set of volumes of hyperbolic
–manifolds is well
ordered, for each fixed
there is a genus–
surface bundle over the circle of minimal volume. Here, we introduce an explicit family
of genus–
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
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.
