Topics in Geometric Group Theory

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

Abstract Award: DMS-0706259 Principal Investigator: Tadeusz Januszkiewicz, Michael W. Davis Within the last few years, two new lines of research have opened up: weighted L^2 -cohomology of Coxeter groups and simplicial nonpositive curvature. Davis and Januszkiewicz plan to continue their research on these and other topics in geometric group theory. The "weight" in L^2 -cohomology depends on a positive real parameter q and on word length in the Coxeter group W. The major unsolved problem is to determine this cohomology in the "intermediate range," for q between r and 1/r, where r is the radius of convergence of the growth series of W. Januszkiewicz plans to develop a theory of "combinatorial nonpositive curvature" which will simultaneously generalize simplicial nonpositive curvature and the theory of nonpositively curved cubical complexes. In this new theory the cells will be products of simplices. Other problems concern the compactly supported cohomology of buildings, the L^2 -cohomology of hyperplane complements and the question if certain hyperplane complements are the classifying spaces for Artin groups. Nonpositive curvature relates to areas outside pure mathematics ranging from robotics to statistical mechanics. The theory of groups generated by reflections is ubiquitous in mathematics. Reflection groups are used in areas ranging from geometry and topology to dynamical systems to number theory and they play a decisive role in Lie theory and in the theory of algebraic groups. Around 1960 Jacques Tits introduced the notion of a "Coxeter group." Synonymous terminology could have been an "abstract reflection group." Coxeter groups form a much wider class of groups than do the classical examples of geometric reflection groups. In 1987 Moussong proved that each Coxeter group acts as a reflection group on a certain nonpositively curved space. Because of this, Coxeter groups have become important in geometric group theory both as a source of new examples and as a paradigm for predicting new results. The new research on weighted L^2 -cohomology has revealed some unexpected connections between several different topics in the theory of Coxeter groups. More remains to be discovered.