18

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 \(SO(n,1)\)


Publications


    Team Members

    Dr. Holger Kammeyer
    Project leader
    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
    Project leader
    Karlsruher Institut für Technologie
    roman.sauer(at)kit.edu

    Prof. Dr. Thomas Schick
    Project leader
    Georg-August-Universität Göttingen
    thomas.schick(at)math.uni-goettingen.de