Negatively Curved Groups

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Malone, William
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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 R3, 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 alge- braic properties but also because of their applications in both computer sci- ence and art. They make up the vast majority of all fundamental groups of three-dimensional manifolds, which are spaces that look locally like the three- dimensional 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.