The Classification Problem for Hyperbolic 3-Manifolds

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DMS-0204454Jeffrey F. BrockTHE CLASSIFICATION PROBLEM FOR HYPERBOLIC 3-MANIFOLDSThe PI, Jeffrey Brock, will synthesize diverse techniques in thedeformation theory of hyperbolic 3-manifolds to address classificationproblem for hyperbolic 3-manifolds. Brock will undertake joint workwith K. Bromberg that employs the theory of hyperbolic cone-manifoldsto show that each tame hyperbolic 3-manifold M is approximated bygeometrically finite 3-manifolds. This conjecture, known as theDensity Conjecture has recently been solved by Brock and Bromberg incertain cases. Brock will also work toward completing joint workwith R. Canary and Y. Minsky to prove Thurston's ending laminationconjecture, which predicts that a tame hyperbolic 3-manifold isdetermined by its topology and its end invariants: combinatorialinvariants attached to the ``ends'' of a hyperbolic 3-manifold. Innew joint work with Bromberg, R. Evans, and J. Souto, Brock will studythe question of whether each algebraic limit of a sequence ofgeometrically finite hyperbolic 3-manifolds is itself topologicallytame. This joint project has implications for a conjecture of Ahlforsthat the limit set of a finitely generated Kleinian group has eithermeasure zero or full measure in the Riemann sphere. Classifying mathematical objects plays much the same scientific roleas classifying biological, chemical, or physical phenomena in thedevelopment of these fields. For example, with the human genome"cracked," scientists may now isolate specific genetic causes orpredispositions to diseases, greatly furthering the ability of scienceto address these problems. In the proposed research, Brock willendeavor to solve the classification problem for a "generic" class of3-dimensional spaces, the "hyperbolic 3-manifolds." Thesenon-Euclidean spaces have geometry locally like our own Euclideanspace, but their large scale geometry is expanding exponentially:for example, light rays (a metaphor for geodesics) emanating from apoint-source diverge exponentially rather than linearly. WilliamP. Thurston's revolutionary and pioneering work in the 1970's and1980's showed that almost all 3-manifolds are hyperbolic, and went onto raise as many questions about hyperbolic 3-manifolds as itanswered. From his contributions, a compelling conjectural picture ofthe right classification of hyperbolic 3-manifolds has emerged as alasting problem for researchers in the field of geometry and topology.Recent work of the PI and his collaborators has put the solution ofthis problem within reach; the PI will make use his NSF support tofacilitate ongoing collaborations to solve this fundamental problem,thereby making a "database" of hyperbolic 3-manifolds available forwider use by other mathematicians and physicists alike.