The most important invariant of three-dimensional spaces is the fundamental group. The investigator and his coworkers are studying the fundamental group of three- dimensional manifolds asymptotically via the recursive patterns at infinity defined by those groups. This approach transfers difficult problems about three-dimensional spaces to difficult problems about recursive tiling patterns in the two-dimensional sphere at infinity. The patterns obtained are very attractive aesthetically and seem to supply deep connections among three-manifold theory, the geometry of three-manifolds, geometric and combinatorial group theory, circle packing, classical complex variable theory, Teichmueller space theory, and the theory of Kleinian and Fuchsian groups, with potential connections as well to the theories of iterated rational maps, tiling theory, Blaschke products, Grothendieck's dessins d'enfants, etc. The initial theory was developed with essentially only one aim in mind, namely to prove the conjecture that a discrete group is Kleinian if and only if it is negatively curved in the large (in the sense of Gromov) and has the two-dimensional sphere as its space at infinity. An affirmative solution to this problem would fill an important slot in Thurston's program to show that all three-manifolds admit a geometric structure. The theory of recursive tilings of the plane and two-sphere is very rich and rewarding and is likely to have applications well beyond its intended target. Discrete group theory can be used to model physical motion, geometric symmetries, and biological cell growth. The generic discrete group is infinite and negatively curved in the large (Gromov and Olshanskii) and has a recursive self- similarity structure at infinity (Cannon). One might alternatively say that the generic group casts fractal shadows at infinity. The investigator and his coworkers are studying this fractal structure at infinity by using the theory of conformal mappings to optimize the geometric shape of these shadows. The results have application to the study of discrete groups, three-dimensional spaces, complex variables, computational group theory, and, potentially, to the theory of biological cell growth. ***
