Mathematical Sciences: Research in Geometric Group Theory

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New directions in geometric group theory have emerged in the past few years in the areas of word hyperbolic groups, automatic groups, and spaces of nonpositive curvature. Professors Charney and Davis will continue research on a number of problems in these areas. Charney (together with M. Shapiro) plans research on several questions related to the automaticity of Artin groups. She also plans to study the problem of putting a metric of nonpositive curvature on several building-like complexes. Davis (also with M. Shapiro) plans to work on the conjecture that all Coxeter groups are automatic. Related to this is his project to study properties of the boundary of a Coxeter group. This might also be useful in finding a group of cohomological dimension two but geometric dimension three (the Eilenberg-Ganea Question). Finally, jointly, Charney and Davis intend to work on strict hyperbolization procedures and relative hyperbolization procedures. "Strict" refers to the possibility of converting a complex into a space of strictly negative curvature (rather than just nonpositive curvature). "Relative" refers to the problem of keeping the metric on a subcomplex fixed. Charney and Davis also plan to continue their research on nonpositively curved orbifolds. The issue here is to decide when certain piecewise spherical metrics on spheres are "large." 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.