A geometric group is a group acting properly discontinuously and cocompactly on a standard geometry. This principal investigator is attempting to understand the combinatorics of geometric groups: What are the implications of the geometry for the combinatorics? What combinatorial properties characterize a geometric group? The investigator has just established a combinatorial characterization of cocompact and finite-volume hyperbolic groups in dimension 3. He is also in the midst of the creation of computer programs to aid in studying the recursive computational structures associated with hyperbolic and Euclidean groups. The next steps in the investigator's plan are the following: (1) apply his characterization in as many different ways as possible (Does it imply that negatively curved groups in dimensions 3 can be realized as conformal groups? Can Thurston's hyperbolization theorem be deduced from the characterization? Can space-filling curves be so studied?); (2) use the computer programs to study interesting infinite classes of groups given by generators and relators (e.g., Coxeter's groups G(3,7,n) and Conway's Fibonacci groups). The work of this project uses the subtle interplay between geometry and group theory to advance the state of each field.
