Geometric Group Theory and L^2 Cohomology

几何群论和 L^2 上同调

• 批准号: 0405825
• 负责人: Michael Davis
• 金额: --
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405825&HistoricalAwards=false
• 关键词: Geometric; Group; Theory; Cohomology

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.

Davis 和 Januszkiewicz 计划继续研究几何群理论、非正弯曲空间和 L^2-上同调。 他们工作的主要部分将集中于与 Coxeter 群相关的复形的“加权 L^2-上同调”的新理论。 “权重”取决于 Coxeter 群中的字长和正实数参数 q。 当 q 是积分时，该理论与建筑物的普通 L^2 上同调密切相关。 当 q 从 0 到无穷大变化时，该理论在普通上同调和紧支持上同调之间进行插值。 Davis 和 Januszkiewicz 试图计算当 q 位于某个中间范围时这些加权上同调空间。 他们还试图将该理论扩展到除考克塞特群之外的其他群类。反射生成群的理论在数学的许多不同领域中发挥着重要作用，例如在李理论和代数群理论中。 1960 年左右，J. Tits 引入了“Coxeter 群”的概念。 同义术语可能是“抽象反射组”。 考克塞特群形成了比几何反射群的经典例子更广泛的群。 最近，它们在几何群论中变得很重要，既作为新例子的来源，又作为预测新结果的范式。 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。 还有更多的事情有待发现。