Analytic L2-invariants of non-positively curved spaces

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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).