Topics in Geometric Group Theory

摘要奖：DMS-0706259主要研究人员：Tadeusz Januszkiewicz，Michael W.Davis在过去的几年里，有两个新的研究方向：加权L^2-Coxeter群上同调和单纯非正曲率。Davis和Januszkiewicz计划继续他们在几何群论中的这些和其他主题的研究。L^2-上同调中的“权”依赖于正参数q和Coxeter群W中的字长。尚未解决的问题是在q的“中间范围”确定这个上同调，其中r是W.Januszkiewicz增长级数收敛的半径。Januszkiewicz计划发展一个“组合非正向曲率”理论，它将同时推广单纯非正曲率和非正曲率二次复形理论。在这一新理论中，细胞将是简化的产物。其他问题涉及建筑物的紧支撑上同调，超平面补的L^2-上同调，以及某些超平面补是否为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数学以外的领域。由反射生成的群理论在数学中是普遍存在的。从几何、拓扑到动力系统到数论，反射群被广泛应用，它们在李论和代数群论中扮演着重要的角色。大约在1960年，雅克·蒂茨提出了“科克塞特小组”的概念。同义术语可能是“抽象反映组”。与几何反射群的经典例子相比，Coxeter群形成的群的类别要广泛得多。1987年，MousSong证明了每个Coxter群都是某个非正弯曲空间上的一个映射群。正因为如此，Coxeter群在几何群论中已经变得非常重要，它既是新例子的来源，也是预测新结果的范例。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。还有更多的有待发现。

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.