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Closed surface bundles of least volume

John W Aaber and Nathan Dunfield

Algebraic & Geometric Topology 10 (2010) 2315–2342

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.

surface bundle, pseudo-Anosov, minimal volume, minimal dilatation
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 37E30, 37E40
Received: 10 September 2010
Accepted: 19 September 2010
Published: 26 November 2010
John W Aaber
Nathan Dunfield
Department of Mathematics MC-382
University of Illinois
1409 W Green Street
Urbana, IL 61801