Geometric Group Theory and L^2 Cohomology

Davis和Januszkiewicz计划继续研究几何群论、非正曲空间和L^2上同调。他们的主要工作将集中在与Coxeter群相关的复形的“加权L^2上同调”这一新理论上。“权”取决于Coxeter群中的词长和一个正实参数q。当q是积分时，该理论与一般的建筑物的L^2上同调密切相关。当q从0变化到无穷大时，该理论在普通上同调和紧支集上同调之间进行插补。Davis和Januszkiewicz正在试图计算当Q位于某个中间范围时的这些加权上同调空间。他们还试图将这一理论推广到除Coxeter群之外的其他类型的群。由反射生成的群的理论在许多不同的数学领域中都扮演着重要的角色，例如在李理论和代数群理论中。大约在1960年，J.Tits提出了“Coxeter group”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。最近，它们在几何群论中变得重要，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。还有更多的东西有待发现。

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.