Analytic L2-invariants of non-positively curved spaces

众所周知，拉普拉斯算子和狄拉克算子的谱与流形的几何形状有着有趣的联系。解析l2不变量应用冯·诺伊曼代数技术提取光谱信息，即使空间有顶点和端点。我们想研究这些不变量，并将它们与流形在有限体积局部对称空间、渐近双曲流形和有限体积Kähler双曲流形中适当紧化的组合l2不变量进行比较。在这些情况下，Gromov、chen - singer和Hopf提出的问题尚未得到解决，我们希望给出新的部分答案。有必要将行之有效的方法转移到非紧凑的环境中。特别地，我们要证明具有双曲尖端的流形的l2指数定理。另外，l2不变量的扭曲版本也成为关注的焦点。对于l2 -扭转，我们想立即开始对这些扭曲版本的研究：作为表征变体的函数。我们想把它和类似的函数联系起来比如在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).