During the late 70's the PI initiated a study of geometric structures as a tool for deciphering three-dimensional topology. The central principle has been the geometrization conjecture, asserting that any 3-manifold has a canonical topological decomposition into pieces that have one of eight possible kinds of locally homogeneous geometric structures. The last two or three years have seen dramatic progress in this program. The most notable progress is Perlman's proof (which has been gaining in acceptance) using Ricci flow to establish geometrization. In addition, there has also been great progress on understanding the geometry of noncompact hyperbolic manifolds through the proof of the tame end and ending lamination conjectures, and there have been some notable negative results establishing that certain hyperbolic 3-manifolds don't have taut foliations, essential laminations, or quasigeodesic flows. Geometric decompositions of 3-manifolds have given us a firm grip on individual 3-manifolds. With the acceptance of the geometrization conjecture, the subject is entering a new phase of exploration of the rich interconnections among three-manifolds and various structures on manifolds. One important topic for investigation is to further clarify the relationships among taut foliations, tight contact structures, flows with varying properties, and hyperbolic structures. Another interesting area is to investigate is the master three-manifold, a laminated set whose leaves are all possible hyperbolic 3-manifolds, compact and noncompact. A third important topic is to understand the lattices of finite-sheeted covers of 3-manifolds and in particular, the questions of whether hyperbolic 3-manifolds are virtually Haken, virtually positive betti, and virtually fibered. The final and perhaps most significant topic is to develop a bigger picture to explain geometrization, using structures such as a foliated bundles over three-manifolds to build their geometry.
A 3-manifold is a space which has 3 degrees of freedom, so it can be locally described by three variables. Through the geometrization conjecture, which is now generally accepted, the topology of three-manifolds have a beautiful description in terms of crystallographic groups in 8 flavors of 3-dimensional geometry. In fact most 3-manifolds are the identification spaces of a crystallographic group in non-Euclidean or hyperbolic geometry. Unlike for standard (Euclidean) crystallography, there are infinitely many crystallographic groups in hyperbolic geometry, and many mysteries remain about their structure, their classification, and their interrelationships. This project will seek to uncover and explain some of these interrelationships.
