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

Kin, Eiko¹ ; Kojima, Sadayoshi² ; Takasawa, Mitsuhiko³

¹Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
²Department of Math. and Computing Sciences, Tokyo Institute of Technology, Ohokayama, Meguro, Tokyo 152-8552, Japan
³Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ohokayama, Meguro, Tokyo 152-8552, Japan

Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3537-3602

Résumé

This paper concerns the set ℳ̂ 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 ∈ ℚ. We prove that for each g (resp. g≢0(mod6)), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of ℳ̂ 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

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

Détail

DOI : 10.2140/agt.2013.13.3537

Classification : 57M27, 37E30, 37B40
Keywords: mapping class group, pseudo-Anosov, dilatation, entropy, hyperbolic volume , fibered $3$–manifold, magic manifold

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