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

Abstract

This paper concerns the set M 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 2 Q. We prove that for each g (resp. g 6 0 .mod 6/), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of M defined on a closed surface g of genus g is achieved by the monodromy of some g -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 .mod 12/ we find a new family of pseudo-Anosovs defined on g with orientable invariant foliations obtained from N . 6/ or N .4/ by Dehn filling both cusps. We prove that if C g is the minimal dilatation of pseudo-Anosovs with orientable invariant foliations defined on g , then lim sup g6 .mod 12/ g!1 g log C g 2 log .D 5 / 1:0870;