L2-invariants

l2不变量在几何和拓扑中起着重要的作用。特别是，它们提供了（微分）几何与拓扑甚至代数问题之间的有用联系，比如负弯曲Kahler流形基本群的欧拉特征符号的计算。通过l2不变量的显式计算，这些不变量更精细地表达了具有自然紧化拓扑不变量的非紧流形的几何形状。然而，事实证明，除了L2-Betti数之外，缺乏明确计算的例子。在这个项目中，我们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 -扭转（以新的方式将紧化的拓扑结构与几何结构联系起来）。此外，我们计划在两个方向上推广这种计算：第一个方向是仅渐近局部对称的空间，第二个方向是凸紧双曲流形。

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.