Hyperbolic geometry is the non-Euclidean geometry discovered by Lobachevsky, Bolyai and Gauss. In hyperbolic space, the area of a triangle is determined by the sum of its angles, and more generally the volume of a configuration is determined by its shape. This phenomenon, which contrasts with the Euclidean situation, persists in hyperbolic manifolds, which are metric spaces locally isometric to hyperbolic space, and are natural objects from the point of view of differential geometry, complex analysis and number theory. The volume of a hyperbolic manifold is determined by its topological type.
For hyperbolic manifolds of dimension at least 3, even more is true: it follows from the Mostow rigidity theorem that when n is at least 3, a compact hyperbolic n-manifold is determined up to isometry by its topological type. In dimension 3 the situation is better still: over the last 30 years, through the efforts of Thurston, Perelman and others, a complete unification between the topology of 3-manifolds and the geometry of hyperbolic 3-manifolds has been achieved.
My own recent work has been concerned with making Mostow rigidity more explicit in dimension 3 by relating such geometric features of a hyperbolic 3-manifold as its volume to classical topological invariants such as homology. In this project, much of which represents joint work with Marc Culler and others, ideas from classical 3-manifold topology, developed by such pioneers as Papakyriakopoulos, Stallings, Haken and Waldhausen, are brought to bear on the study of hyperbolic 3-manifolds. In the hyperbolic setting these techniques are seen to be even richer and more powerful than might have been imagined when the topology of 3-manifolds was a relatively self-contained and isolated field.
In the course of describing this work I will try to illustrate what a rich field of research 3-dimensional hyperbolic geometry has become, involving interactions among geometry, topology, algebra, number theory and analysis.