Algebraic & Geometric Topology

Minimal dilatations of pseudo-Anosovs generated by the magic $3$–manifold and their asymptotic behavior

Eiko Kin, Sadayoshi Kojima, and Mitsuhiko Takasawa

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This paper concerns the set ̂ of pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the magic 3–manifold N by Dehn filling three cusps with a mild restriction. Let N(r) be the manifold obtained from N by Dehn filling one cusp along the slope r. We prove that for each g (resp. g0(mod6)), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of ̂ defined on a closed surface Σg of genus g is achieved by the monodromy of some Σg–bundle over the circle obtained from N(32) or N(12) by Dehn filling both cusps. These minimizers are the same ones identified by Hironaka, Aaber and Dunfield, Kin and Takasawa independently. In the case g6(mod12) we find a new family of pseudo-Anosovs defined on Σg with orientable invariant foliations obtained from N(6) or N(4) by Dehn filling both cusps. We prove that if δg+ is the minimal dilatation of pseudo-Anosovs with orientable invariant foliations defined on Σg, then

limsup g 6 ( mod 1 2 ) , g g log δ g + 2 log δ ( D 5 ) 1 . 0 8 7 0 ,

where δ(Dn) is the minimal dilatation of pseudo-Anosovs on an n–punctured disk. We also study monodromies of fibrations on N(1). We prove that if δ1,n is the minimal dilatation of pseudo-Anosovs on a genus 1 surface with n punctures, then

limsup n n log δ 1 , n 2 log δ ( D 4 ) 1 . 6 6 2 8 .

Article information

Algebr. Geom. Topol., Volume 13, Number 6 (2013), 3537-3602.

Received: 18 October 2011
Revised: 15 February 2013
Accepted: 19 February 2013
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Secondary: 37B40: Topological entropy

mapping class group pseudo-Anosov dilatation entropy hyperbolic volume fibered $3$–manifold magic manifold


Kin, Eiko; Kojima, Sadayoshi; Takasawa, Mitsuhiko. Minimal dilatations of pseudo-Anosovs generated by the magic $3$–manifold and their asymptotic behavior. Algebr. Geom. Topol. 13 (2013), no. 6, 3537--3602. doi:10.2140/agt.2013.13.3537.

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