Topics in Geometric Group Theory

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AbstractAward: DMS-0706259Principal Investigator: Tadeusz Januszkiewicz, Michael W. DavisWithin the last few years, two new lines of research have openedup: weighted L^2 -cohomology of Coxeter groups and simplicialnonpositive curvature. Davis and Januszkiewicz plan to continuetheir research on these and other topics in geometric grouptheory. The "weight" in L^2 -cohomology depends on a positivereal parameter q and on word length in the Coxeter group W. Themajor unsolved problem is to determine this cohomology in the"intermediate range," for q between r and 1/r, where r is theradius of convergence of the growth series of W. Januszkiewiczplans to develop a theory of "combinatorial nonpositivecurvature" which will simultaneously generalize simplicialnonpositive curvature and the theory of nonpositively curvedcubical complexes. In this new theory the cells will be productsof simplices. Other problems concern the compactly supportedcohomology of buildings, the L^2 -cohomology of hyperplanecomplements and the question if certain hyperplane complementsare the classifying spaces for Artin groups.Nonpositive curvature relates to areas outside pure mathematicsranging from robotics to statistical mechanics. The theory ofgroups generated by reflections is ubiquitous in mathematics.Reflection groups are used in areas ranging from geometry andtopology to dynamical systems to number theory and they play adecisive role in Lie theory and in the theory of algebraicgroups. 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 widerclass of groups than do the classical examples of geometricreflection groups. In 1987 Moussong proved that each Coxetergroup acts as a reflection group on a certain nonpositivelycurved space. Because of this, Coxeter groups have becomeimportant in geometric group theory both as a source of newexamples and as a paradigm for predicting new results. The newresearch on weighted L^2 -cohomology has revealed some unexpectedconnections between several different topics in the theory ofCoxeter groups. More remains to be discovered.