Curvature and Computation

When undergraduates ask me what geometric group theorists study, I describe a theorem due to Gromov which relates the groups with an intrinsic geometry like that of the hyperbolic plane to those in which certain computations can be efficiently carried out. In short, I describe the close but surprising connection between negative curvature and efficient computation. This theorem was one of the clearest early indications that applying a metric perspective to traditional group theory problems might lead to new and important insights. The theorem I want to discuss asserts that there is a close relationship between two collections of groups: one collection is defined geometrically and the other is defined computationally. The first section describes the relevant geometric and topological ideas, the second discusses the key algebraic and computational concepts, and the short final section describes the relationship between them. An informal style, similar to the one I use when answering this question face-toface, is maintained throughout. 1. Geometry and topology The first thing to highlight is that there is a close relationship between groups and topological spaces. More specifically, to each connected topological space X there is an associated group G called its fundamental group and absolutely every group arises in this way (in the sense that for each group G one can construct a topological space X whose fundamental group is isomorphic to G). Because of this connection and because spaces with isomorphic fundamental groups share many key properties, we can use the topology of the space X to understand the algebraic structure of its fundamental group G. Fundamental groups. In order to make this discussion as accessible as possible, here is a quick sketch of the basic idea behind the notion of a fundamental group. As an initial attempt, one could try to form a group out of a space by using the paths in the space as our elements Date: November 25, 2014. Partial support by the National Science Foundation is gratefully acknowledged.