The Classification Problem for Hyperbolic 3-Manifolds

University Of Chicago, Chicago IL

Investigators

Abstract

DMS-0204454 Jeffrey F. Brock THE CLASSIFICATION PROBLEM FOR HYPERBOLIC 3-MANIFOLDS The PI, Jeffrey Brock, will synthesize diverse techniques in the deformation theory of hyperbolic 3-manifolds to address classification problem for hyperbolic 3-manifolds. Brock will undertake joint work with K. Bromberg that employs the theory of hyperbolic cone-manifolds to show that each tame hyperbolic 3-manifold M is approximated by geometrically finite 3-manifolds. This conjecture, known as the Density Conjecture has recently been solved by Brock and Bromberg in certain cases. Brock will also work toward completing joint work with R. Canary and Y. Minsky to prove Thurston's ending lamination conjecture, which predicts that a tame hyperbolic 3-manifold is determined by its topology and its end invariants: combinatorial invariants attached to the ``ends'' of a hyperbolic 3-manifold. In new joint work with Bromberg, R. Evans, and J. Souto, Brock will study the question of whether each algebraic limit of a sequence of geometrically finite hyperbolic 3-manifolds is itself topologically tame. This joint project has implications for a conjecture of Ahlfors that the limit set of a finitely generated Kleinian group has either measure zero or full measure in the Riemann sphere. Classifying mathematical objects plays much the same scientific role as classifying biological, chemical, or physical phenomena in the development of these fields. For example, with the human genome "cracked," scientists may now isolate specific genetic causes or predispositions to diseases, greatly furthering the ability of science to address these problems. In the proposed research, Brock will endeavor to solve the classification problem for a "generic" class of 3-dimensional spaces, the "hyperbolic 3-manifolds." These non-Euclidean spaces have geometry locally like our own Euclidean space, but their large scale geometry is expanding exponentially: for example, light rays (a metaphor for geodesics) emanating from a point-source diverge exponentially rather than linearly. William P. Thurston's revolutionary and pioneering work in the 1970's and 1980's showed that almost all 3-manifolds are hyperbolic, and went on to raise as many questions about hyperbolic 3-manifolds as it answered. From his contributions, a compelling conjectural picture of the right classification of hyperbolic 3-manifolds has emerged as a lasting problem for researchers in the field of geometry and topology. Recent work of the PI and his collaborators has put the solution of this problem within reach; the PI will make use his NSF support to facilitate ongoing collaborations to solve this fundamental problem, thereby making a "database" of hyperbolic 3-manifolds available for wider use by other mathematicians and physicists alike.