Small dilatation pseudo-Anosov homeomorphisms and 3-manifolds

The main result of this paper is a universal finiteness theorem for the set of all small dilatation pseudo-Anosov homeomorphisms ϕ:S→S, ranging over all surfaces S. More precisely, we consider pseudo-Anosov homeomorphisms ϕ:S→S with |χ(S)|log(λ(ϕ)) bounded above by some constant, and we prove that, after puncturing the surfaces at the singular points of the stable foliations, the resulting set of mapping tori is finite. Said differently, there is a finite set of fibered hyperbolic 3-manifolds so that all small dilatation pseudo-Anosov homeomorphisms occur as the monodromy of a Dehn filling on one of the 3-manifolds in the finite list, where the filling is on the boundary slope of a fiber.