Heegaard Splittings and Pseudo-Anosov Maps

Abstract

Given two 3-dimensional handlebodies whose boundaries are identified with a surface S of genus g > 1 and with different orientations, we consider the sequence of manifolds Mn obtained by gluing the handlebodies via the iteration fn of a “generic” pseudo-Anosov homeomorphism f of S. Using the deformation theory of hyperbolic structures on open hyperbolic 3-manifolds and for n sufficiently large, we construct a negatively curved metric on Mn where the sectional curvatures are pinched in a given small interval centered at –1. The construction is concrete enough to allow us describe the geometric limits of these manifolds as n tends to infinity and the metrics get closer to being hyperbolic. Such a description allows us to prove various topological and group theoretical properties of Mn, for n sufficiently large, which would not be available knowing the mere existence of a negatively curved or even hyperbolic metric on Mn.