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

Eiko Kin, Sadayoshi Kojima and Mitsuhiko Takasawa

Algebraic & Geometric Topology 13 (2013) 3537–3602

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( 3 2) or N( 1 2) by Dehn filling both cusps. These minimizers are the same ones identified by Hironaka, Aaber and Dunfield, Kin and Takasawa independently. In the case g 6(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

limsupg6(mod 12),gglogδg+ 2logδ(D 5) 1.0870,

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

limsupnnlogδ1,n 2logδ(D4) 1.6628.

mapping class group, pseudo-Anosov, dilatation, entropy, hyperbolic volume , fibered $3$–manifold, magic manifold
Mathematical Subject Classification 2010
Primary: 57M27, 37E30
Secondary: 37B40
Received: 18 October 2011
Revised: 15 February 2013
Accepted: 19 February 2013
Published: 10 October 2013
Eiko Kin
Department of Mathematics
Graduate School of Science
Osaka University
Osaka 560-0043
Sadayoshi Kojima
Department of Math. and Computing Sciences
Tokyo Institute of Technology
Ohokayama, Meguro
Tokyo 152-8552
Mitsuhiko Takasawa
Department of Mathematical and Computing Sciences
Tokyo Institute of Technology
Ohokayama, Meguro
Tokyo 152-8552