Analytic L2-invariants of non-positively curved spaces

众所周知，拉普拉斯和狄拉克算子的谱与流形的几何有着有趣的联系。解析L2-不变量应用冯·诺依曼代数技术来提取谱信息，即使空间具有尖点和端点。 我们想研究这些不变量，并将它们与各种情况下流形的适当紧化的组合L2-不变量进行比较：有限体积局部对称空间，渐近双曲流形和有限体积Kähler双曲流形。 在这些情况下提出的问题格罗莫夫，陈-辛格和霍普夫尚未解决，我们希望给予新的部分答案。将有必要将已确立的方法转移到非紧凑的环境中。 特别地，我们想证明一个具有双曲尖点的流形的L2-指标定理。 L2不变量的扭曲版本也成为焦点。 对于L2-torsion，我们希望立即开始对这些扭曲版本的研究：作为表示簇上的函数。我们想把它与类似的函数联系起来，比如SO（n，1）中表示的体积函数。

The spectrum of Laplace and Dirac operators is known to be intriguingly connected with the geometry of a manifold. Analytic L2-invariants apply von Neumann algebra techniques to extract the spectral information even if the space has cusps and ends. We want to study these invariants and compare them to combinatorial L2-invariants of suitable compactifications of the manifold in a variety of situations: finite volume locally symmetric spaces, asymptotically hyperbolic manifolds and finite volume Kähler hyperbolic manifolds. In these cases questions raised by Gromov, Chern-Singer and Hopf are not yet settled and we hope to give new partial answers. It will be necessary to transfer well-established methods to a non-compact setting. In particular, we want to prove an L2-index theorem for manifolds with hyperbolic cusp ends. Also twisted versions of L2-invariants have come into focus. For L2-torsion, we want to initiate a study of these twisted versions all at once: as a function on representation varieties. We want to relate this to similar functions like the volume functions for representations in SO(n,1).