Research in geometric group theory

Ohio State University, The, Columbus OH

Investigators

Abstract

In recent years the study of Coxeter groups and their relatives (e.g. Artin groups) and related spaces (e.g. buildings) have become increasingly important in geometric group theory. The PI (Davis) and Co-PI (Januszkiewicz) plan to compute various types of cohomology for these groups and spaces. They also will study the problem of determining whether branched covers of complex manifolds are nonpositively curved. This is related to a possible attack on the K(pi, 1) Conjecture for Artin groups. Together with Boris Okun the PI plans to compute the cohomology with group ring coefficients as well as the L2 cohomology of Artin groups, Bestvina-Brady groups and graph products of infinite groups. The Co-PI, together with his collaborators in Poland, plans to work on the problem of generalizing his notion of simplicial nonpositive curvature. Nonpositive curvature relates to areas outside pure mathematics ranging from robotics (via configuration spaces) to statistical mechanics. Coxeter groups are central in many areas of pure mathematics ranging from geometry and topology to dynamical systems to number theory and the theory of Lie groups. The study of Artin groups are a generalization of Artin's braid groups. The braid groups also have been in different areas of mathematics, for example, knot theory and algebraic geometry. The proposed research will have an impact in the area of nonpositive curvature, in the areas of Coxeter groups and Artin groups and other related areas.