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L2-invariants of groups

群的 L2 不变量

基本信息

批准号：
144856302
负责人：
Professor Dr. Thomas Schick
金额：
--
依托单位：
Mathematisches Institut (MI)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2009
资助国家：
德国
起止时间：
2008-12-31 至 2016-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/144856302?language=en
关键词：
L2 invariants groups

项目摘要

L2-invariatits play an important role in geometry and topology. In particular, they provide useful connections between questions arising in the area of topology, algebra and (differential) geometry. For example, one can use L2 -invariants to prove cases of the Kaplansky zero-divisor conjecture for group rings of torsionfree groups or estimate the clericiency of a discrete group. Generalizations and relinements of these invariants have been introduced for quantum groups, via center valued versions, or in terms of Lp-cohomology, together with the corresponding homological algebra. The more refined of these invariants allow to express the geometry of noncompact manifolds in terms of topological invariants of natural conipactilications, via the explicit calculation of L2-invariants. It turns out, however, that except for L2-Betti numbers there is a lack of explicitly calculated examples. Our goals in this project are: line study of the center valued L2-Betti numbers and their possible values; statement (and proof in special cases) of a corresponding Atiyah conjecture; relation of this to the ring theoretic properties of the division closure of the group ring detailed investigation of L2-invariants of quantum groups, including the study of the algebraic and arithmetic properties of quantum groups the calculation of Novikov-Shubin invariants, L2-eta invariants, and L2-torsion for the natural compactifications of locally symmetric spaces of finite volume (relating the topology of the compactification to the geometry in new ways).

L2-不变性在几何和拓扑学中起着重要的作用。特别是，它们在拓扑学、代数和(微分)几何领域中出现的问题之间提供了有用的联系。例如，人们可以使用L2不变量来证明关于自由群的群环的Kaplansky零因子猜想的情况，或者估计离散群的可信度。通过中心值形式或LP上同调，以及相应的同调代数，这些不变量的推广和关系已被引入到量子群中。这些不变量的精化使得非紧流形的几何可以通过L2-不变量的显式计算，用自然压缩的拓扑不变量来表示。然而，事实证明，除了L2-Betti数之外，还缺乏显式计算的例子。我们在这个项目中的目标是：线性研究中心值L2-Betti数及其可能的值；相应的Atiyah猜想的陈述(和在特殊情况下的证明)；这与群环的除法闭包的环论性质的关系。详细研究量子群的L2-不变量，包括研究量子群的代数和算术性质，Novikov-Shubin不变量的计算，L2-eta不变量的计算，以及有限体积局部对称空间自然紧化的L2-扭转(以新的方式将紧化的拓扑与几何联系起来)。