L2-invariants of groups

L2-不变量在几何和拓扑中起着重要的作用。特别是，他们提供了有用的连接之间出现的问题，在该地区的拓扑，代数和（微分）几何。例如，可以使用L2 -不变量来证明无挠群的群环的卡普兰斯基零因子猜想的情况或估计离散群的clericiency。这些不变量的推广和放弃已经被引入到量子群中，通过中心值版本，或者根据Lp-上同调，以及相应的同调代数。这些不变量中更精细的允许通过L2-不变量的显式计算，用自然共轭的拓扑不变量来表达非紧流形的几何。然而，事实证明，除了L2-Betti数之外，缺乏明确计算的例子。我们在这个项目中的目标是：线研究中心值L2-Betti数及其可能的值;声明（和特殊情况下的证明）相应的阿蒂亚猜想;这与群环的除法闭包的环论性质的关系量子群的L2-不变量的详细研究，包括量子群的代数和算术性质的研究，Novikov-Shubin不变量，L2-eta不变量，和有限体积局部对称空间的自然紧化的L2-挠（以新的方式将紧化的拓扑与几何联系起来）。

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).