The work supported by this grant is bringing to bear on the study of hyperbolic 3-manifolds methods from the analytic theory of Kleinian groups, geometric and algebraic topology, algebra and combinatorics. One group of projects aims at understanding, in a quantitative way, how the volume of a hyperbolic manifold reflects its underlying topology. Another project studies the relationship between the algebraic rank of a hyperbolic 3- manifold and its Heegaard genus. A third group of projects addresses the construction of hyperbolic manifolds by the Dehn filling construction.
A hyperbolic manifold is a space which is locally modelled on the non-euclidean geometry of Lobachevsky, Bolyai and Gauss, in which the sum of the angles of a triangle is less than pi. Besides being of fundamental importance for classical geometry and number theory, hyperbolic manifolds have long been known to play a central role in three-dimensional topology. This has been newly confirmed by Perelman's announcement of a proof of Thurston's geometrization conjecture.
