End-periodic homeomorphisms and volumes of mapping tori

Abstract

Given an irreducible, end-periodic homeomorphism f : S → S of a surface with finitely many ends, all accumulated by genus, the mapping torus, M f , is the interior of a compact, irreducible, atoroidal 3-manifold M f with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of M f in terms of the translation length of f on the pants graph of S. This builds on work of Brock and Agol in the finite-type setting. We also construct a broad class of examples of irreducible, end-periodic homeomorphisms and use them to show that our bound is asymptotically sharp.