Topics in Geometric Group Theory

摘要奖：DMS-0706259主要研究者：Tadeusz Januszkiewicz，Michael W.戴维斯在过去的几年里，两条新的研究方向已经出现：Coxeter群的加权L^2-上同调和单纯非正曲率。戴维斯和Januszkiewicz计划继续他们的研究这些和其他议题的几何群论。 L^2-上同调的"权重"取决于正实参数q和Coxeter群W中的字长。 未解决的主要问题是确定在"中间范围"内的上同调，其中q在r和1/r之间，其中r是W的增长级数的收敛半径。 Januszkiewicz计划发展一个理论"组合nonpositivecurvature"，这将同时推广simplicialnonpositivecurvature和理论的nonpositivecurvedcubical复杂。 在这个新理论中，细胞将是单纯形的产物。 其他问题涉及建筑物的紧上同调、超平面补的L^2-上同调以及某些超平面补是否是Artin群的分类空间的问题。非正曲率涉及从机器人学到统计力学的纯代数学之外的领域。由反射生成的群的理论在数学中是普遍存在的，反射群的应用范围从几何、拓扑到动力系统再到数论，它们在李群理论和代数群理论中起着决定性的作用. 大约在1960年，Jacques Tits引入了“Coxeter群”的概念。同义术语可能是一个"抽象的反射组"。"考克斯特群形成了一个比几何反射群的经典例子更广泛的群类。 1987年，Moussong证明了每个Coxeter群都是某个非正曲空间上的反射群。正因为如此，Coxeter群在几何群论中变得重要，既作为新例子的来源，又作为预测新结果的范例。 关于加权L^2-上同调的新研究揭示了Coxeter群理论中几个不同主题之间的一些意想不到的联系。 还有更多有待发现。

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.