The spectrum of Laplace and Dirac operators is known to be intriguingly connected with the geometry of a manifold.
Analytic \(L^2\)-invariants apply von Neumann algebra techniques to extract the spectral information even if the space has cusps and ends. The project studies these invariants and compares them to combinatorial \(L^2\)-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 the project should lead to new partial answers. It will be necessary to transfer well-established methods to a non-compact setting, e.g., proving an \(L^2\)-index theorem for manifolds with hyperbolic cusp ends.
Also twisted versions of \(L^2\)-invariants have come into focus. For \(L^2\)-torsion, the project intends to initiate a study of these twisted versions all at once: as a function on representation varieties. This should relate to similar functions like the volume functions for representations in \(SO(n,1)\).