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Analytic L2-invariants of non-positively curved spaces

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 $$SO(n,1)$$

## Team Members

Dr. Holger Kammeyer
Karlsruher Institut für Technologie
holger.kammeyer(at)kit.edu

Dr. Steffen Kionke
Researcher
Karlsruher Institut für Technologie
steffen.kionke(at)kit.edu

Prof. Dr. Roman Sauer