Hyperbolic space is the model space of constant curvature -1 (the other two model spaces being the sphere and Euclidean space) and its group of orientation-preserving isometries is the real Lie group SO(n,1). There is a natural correspondence between complete, hyperbolic n-manifolds of finite volume and certain discrete subgroups of SO(n,1) called lattices. With this correspondence in mind, one can ask which algebraic properties of lattices manifest geometrically (and vice-versa). Indeed, questions of this nature have driven quite a bit of research in the fields of hyperbolic geometry and algebraic group theory.
Complex hyperbolic space is a complex analogue of (real) hyperbolic space, and it is one of the simplest examples of a non-compact symmetric space with variable negative curvature. Its group of isometries is the real Lie group SU(n,1), and just as before, we have a correspondence between lattices in SU(n,1) and finite-volume complex hyperbolic n-manifolds. Although there is much similarity between the two spaces, many of the geometric techniques from real hyperbolic geometry do not carry over into the complex setting, and so many of the analogous questions are wide open. Much of my research is focused on lattices in SU(n,1), especially those "non-arithmetic" lattices (arithmetic lattices are those lattices that have similar behavior to that of SL(n,Z) as a subgroup of SL(n,R)). These non-arithmetic lattices are rather mysterious and elusive, and the few examples we do have in low dimensions require tools and techniques from many different fields of mathematics.