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.
and to study aspherical manifolds. Geometric group theory deals with symmetry groups of geometric objects. One place such symmetry groups occur is in the theory of aspherical manifolds. Other examples are reflection groups, which occur as symmetries of tilings of Euclidean and non-Euclidean spaces. Coxeter groups are abstract reflection groups. Artin groups and graph products of groups are two families of groups, which are close relatives to Coxeter groups. During the grant period the PI wrote several papers in which certain cohomology groups of Artin groups and graph products of groups were computed. For graph products of groups, in the case of Erdos-Renyi random graphs, the answer is surprising. In joint work, the PI and Co-PI also calculated certain cohomology groups of complements of hyperplane arrangements in complex space. The PI proved a number of new results concerning aspherical manifolds. Notably, he showed that, in each dimension greater than five, there are aspherical manifolds that cannot be triangulated. Intellectual merit. Coxeter groups are ubiquitous in mathematics. They play central roles in quadratic forms, dynamical systems, singularity theory, Lie groups, algebraic groups and in areas of differential and algebraic geometry. Recently, there have been some important applications of right-angled Artin groups in geometric group theory and topology, e.g., Agol's recent proof of Thurston's remaining conjectures about 3-dimensional hyperbolic manifolds. Broader impact. The PI is a leader of an active research group in geometric group theory at Ohio State University. He is involved in training graduate students and postdoctoral instructors in this area. Two of his students completed their PhD dissertations in August of 2013. A third student has substantial results. During the grant period the PI co-organized some major conferences on geometric group theory at OSU. There were five such conferences in 2010-11 and another in June of 2014. The PI lectured on his research both nationally and internationally. For example, over the last three years, he has given lecture series and mini-courses about his work on Coxeter groups and Artin groups in Korea, China and Brazil.
