L2-invariants

L2-不变量在几何和拓扑学中起着重要的作用。特别地，它们在(微分)几何和由拓扑甚至代数引起的问题之间提供了有用的联系，例如计算负弯曲Kahler流形的基本群的欧拉特征的符号。这些不变量通过显式计算L2-不变量来表示具有自然紧化的拓扑不变量的非紧流形的几何。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计算所有有限体积局部对称空间的自然紧化的Novikov-Shubin不变量、L2-eta不变量和L2-挠率(以新的方式将紧化的拓扑与几何联系起来)。此外，我们计划在两个方向上推广这样的计算：第一，仅渐近局部对称的空间；第二，凸余紧双曲流形。

L2-invariants play an important role in geometry and topology. In particular, they provide useful connections between (differential) geometry and questions arising from topology and even algebra, like the computation of the sign of the Euler characteristic for fundamental groups of negatively curved Kahler manifolds. The more refined of these invariants express the geometry of non-compact manifolds with topological invariants of natural compactifications, 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 the calculation of Novikov-Shubin invariants, L2-eta invariants, and L2-torsion for the natural compactifications of all locally symmetric spaces of finite volume (relating the topology of the compactification to the geometry in new ways). Moreover, we plan to extend such calculations in two directions: first to spaces which are only asymptotically locally symmetric,secondly to convex cocompact hyperbolic manifolds.