Algebraic & Geometric Topology

Closed surface bundles of least volume

John W Aaber and Nathan Dunfield

Full-text: Open access

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.

Article information

Source
Algebr. Geom. Topol., Volume 10, Number 4 (2010), 2315-2342.

Dates
Received: 10 September 2010
Accepted: 19 September 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715172

Digital Object Identifier
doi:10.2140/agt.2010.10.2315

Mathematical Reviews number (MathSciNet)
MR2745673

Zentralblatt MATH identifier
1205.57018

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces 37E40: Twist maps

Keywords
surface bundle pseudo-Anosov minimal volume minimal dilatation

Citation

Aaber, John W; Dunfield, Nathan. Closed surface bundles of least volume. Algebr. Geom. Topol. 10 (2010), no. 4, 2315--2342. doi:10.2140/agt.2010.10.2315. https://projecteuclid.org/euclid.agt/1513715172


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