Negatively Curved Groups

The notion of a negatively curved group is at first highly non-intuitive because it links two areas of mathematics that are not usually associated with one another. Curvature is generally a property that we associate with geometric objects like curves or surfaces in R, while a group is an algebraic structure that we associate with objects like integers or matrices. However, there is a way to define a group structure on paths in a geometric object like a manifold, the so-called fundamental group of the manifold, such that certain aspects of negative curvature are reflected in the group. Negatively curved groups are interesting not only because of their algebraic properties but also because of their applications in both computer science and art. They make up the vast majority of all fundamental groups of three-dimensional manifolds, which are spaces that look locally like the threedimensional world we live in. A famous example of the use of negatively curved reflection groups in art is M.C. Escher’s woodcut Circle Limit IV (1960), which illustrates the overall structure of such a group. The reason that these groups are important to computer scientists is that they are what is called “automatic” and consequently have “solvable word problem.” [2] This article is a brief exploration of negatively curved groups. In order to connect geometry to group theory we begin by describing the procedure of forming a fundamental group. Next, we review the concept of curvature for two-dimensional manifolds and develop an intuitive notion of what a negatively curved group should look like. We then turn this intuitive notion into an exact criterion that distinguishes negatively curved groups in general. Finally, we analyze three examples.