Small Dilatation Pseudo-Anosovs Coming from Dehn Fillings of Hyperbolic Fibered 3-Manifolds

Abstract

The dynamics of pseudo-Anosov homeomorphisms of surfaces are encapsulated largely by their dilatations. Since Penner's work in the 90's, questions about minimal dilatation pseudo-Anosovs have been an area of active research with progress being made in the past decade in particular by the likes of Hironaka, and Kin-Kojima-Takasawa, among others. An approach which has proven fruitful to the aforementioned authors is to fix a hyperbolic fibered 3-manifold and study the pseudo-Anosov monodromies of its Dehn fillings. By a theorem of Farb-Leininger-Margalit, all small dilatation pseudo-Anosovs are monodromies of Dehn fillings of finitely many such fully punctured 3-manifolds. In this thesis, we take this approach further by considering entire classes of 3-manifolds at once. In particular, we show that for any hyperbolic fibered 3-manifold M that is either closed or has second betti number less than or equal to 2, the sequence (\delta_g^g(M,F))_g of least dilatations arising as monodromies of Dehn fillings of M across all realizable genera converges. Furthermore, we conjecture and offer a proof method - and make partial progress on this proof method - that similar strategies could be used to show that the sequence (\delta_g^g)_g of least dilatations across all genera has finitely many accumulation points.