Abstract

During the past few years, J. Kahn and I developed a theory (based among other things on deep statistical properties of geometric flows) that led to the solution of the Ehrenpreis conjecture and (in some sense more importantly) the proof of the Surface Subgroup Theorem in 3-manifold topology. In particular, this theorem implies that every hyperbolic 3-manifold is cubulated. It then follows from the Agol-Wise theory of cube complexes that every hyperbolic 3-manifold is virtually Haken and virtually fibered. I will also discuss my latest result that aims to classify hyperbolic groups whose boundary is the 2-spehere (the Cannon Conjecture), the main motivation being that this gives a fundamentally new approach toward proving the Perelman Hyperbolization Theorem for 3-manifolds of negative curvature.