Spaces of Kleinian Groups and Hyperbolic 3-Manifolds

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

DMS-0234540 Yair N. Minsky The proposer is co-organizing a workshop at the Newton Institute in Cambridge, UK, on the subject of ``Spaces of Kleinian groups and hyperbolic 3-manifolds'', for the summer of 2003. The 4-week workshop, including a 1-week conference, will bring together researchers active on a number of foundational topics in the field: relationships between the analytic, combinatorial and geometric structure of hyperbolic 3-manifolds; topology of deformation spaces and the arrangement of their components; classification of hyperbolic 3-manifolds by asymptotic invariants; complex projective structures; convex hull boundaries; cone manifolds, orbifolds and knot groups; the combinatorial structure of Teichmuller spaces, mapping class groups, and spaces of curves on surfaces. Recent years have seen substantial progress on longstanding foundational conjectures such as the Bers/Sullivan/Thurston Density Conjecture, Ahlfors' Measure Conjecture, Thurston's Ending Lamination Conjecture and Marden's Tameness Conjecture. The time is therefore ripe for a conference on these topics. In the study of low-dimensional geometry, topology and dynamics, there is a remarkable depth of interconnection between fields of mathematics. Henri Poincare, who studied both celestial dynamics and complex analysis (among many other things), observed in the 19th century that the standard round sphere, the setting of classical analysis and geometry, functioned also as a "horizon at infinity" for an exotic non-Euclidean geometry that we now call Hyperbolic Space. Dynamical properties of transformations of the sphere translate to geometric properties of rigid motions of this space, and give rise to families of symmetric tilings whose structure we can study by geometric and topological methods. The properties of these tilings, and the way in which they change when the tilings are varied through families (or "parameter spaces"), are still linked to some of the motivating questions about physical systems of which Poincare was aware a hundred years ago. Issues such as classification of systems, mapping out regions of stability and instability, deformation and bifurcation of families of systems, and probabilistic properties such as ergodicity, all have significance in both pure and applied mathematics. In the setting of hyperbolic geometry, recent years have seen significant progress on these issues, with both theoretical advances and computational approaches playing a role. This conference will bring together leading researchers in this area and promising young mathematicians, to disseminate recent advances and continue work on open problems.