Geometric Group Theory and L^2 Cohomology

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Davis and Januszkiewicz plan to continue their research on geometricgroup theory, nonpositively curved spaces and L^2-cohomology. The major portion of their work will focus on the new theory of the ''weighted L^2-cohomology'' of the complex associated to a Coxeter group. The "weighting" depends on word length in the Coxeter group and a positive real parameter q. When q is integral, the theory is intimately tied to the ordinary L^2-cohomology of buildings. As q varies from 0 to infinity the theory interpolates between ordinary cohomology and cohomology with compact supports. Davis and Januszkiewicz are trying to calculate these weighted cohomology spaces when q lies in a certain intermediate range. They are also trying to extend this theory to other classes of groups besides Coxeter groups.The theory of groups generated by reflections plays an important role in many different areas of mathematics, for example, in Lie theory and in the theory of algebraic groups. Around 1960 J. Tits introduced the notion of a "Coxeter group." Synonymous terminology might be an "abstract reflection group." Coxeter groups form a much wider class of groups than do the classical examples of geometric reflection groups. Recently, they 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. Much more remains to be discovered.