# Negatively Curved Groups

Cardinal Scholar
 dc.contributor.author Malone, William dc.date.accessioned 2020-09-01T20:45:58Z dc.date.available 2020-09-01T20:45:58Z dc.date.issued 2005 dc.identifier.citation Malone, W. (2005). Negatively Curved Groups. Mathematics Exchange, 3(1), 10-15. en_US dc.identifier.uri http://cardinalscholar.bsu.edu/handle/123456789/202308 dc.description Article published in Mathematics Exchange, 3(1), 2005. en_US dc.description.abstract 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. en_US 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. dc.title Negatively Curved Groups en_US dc.type Article en_US
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