Mathematical Sciences: Geometric Topology and the Fundamental Group

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Most (perhaps all) 3-manifolds are geometric. Geometric group theory seeks to tie together combinatorial and geometric properties of geometric isometry groups. The theory is burgeoning at the hands of Gromov, Epstein, Thurston, Holt, Bestvina, Feighn, Mess, Paterson, Gersten, Short, Shapiro, Parry, Floyd, Baumslag, Casson, Brick, Stallings, a whole school of continental mathematicians, Cannon, and others. The theory has produced the notions of automatic group, negatively curved group, complex of groups, almost convex groups, combable groups, isoperimetric and isodiametric inequalities for groups. Amid these many topics, Cannon, his students, and coworkers are currently concentrating on a cluster of projects related to the existence of hyperbolic structures on 3- manifolds. Is the generic finitely presented group negatively curved, as claimed without explicit proof by Gromov? Does every negatively curved closed 3-manifold admit a Riemannian metric of constant negative curvature? Can negative curvature be recognized algorithmically? Can the space at infinity be recognized algorithmically? Can potential constant curvature be recognized algorithmically? Cannon is using as the basis for these studies his combinatorial Riemann mapping theorem developed during the previous projects, the related work by Parry, and the computer Programs of Epstein and his collaborators. Groups are the appropriate algebraic structures for describing the notion of symmetry with precision. For this reason they play an extensive role in geometry (and topology). While the more usual interaction between group theory and geometry is the use of algebraic computations involving groups to prove theorems about geometric objects possessing symmetry, there is a very significant mathematical subdiscipline in which the roles are reversed and our intuition about geometric objects assists us in conjecturing and proving theorems about associated groups. A fascinating recent development is the influence of the computer in all this. Not only is it the ubiquitous timesaver that everyone knows, enabling one to undertake computations that would otherwise be too formidable, but it has actually influenced the questions in geometric group theory in a more fundamental way. Certain classes of groups have been defined in terms of the types of hypothetical machines that would be able to perform certain computations about them. Real hardware is not involved in this, and the same definitions could have been conceived long ago, but they were not, for the existence of computing machines has influenced the way we look at the world, the types of questions we ask about it, the kinds of properties that we find interesting.