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

In this note we refine examples by Aka from arithmetic to S-arithmetic groups to show that the vanishing of the i-th ℓ²-Betti number is not a profinite invariant for all i≥2.

 

Related project(s):
18Analytic L2-invariants of non-positively curved spaces

Given an S-arithmetic group, we ask how much information on the ambient algebraic group, number field of definition, and set of places S is encoded in the commensurability class of the profinite completion. As a first step, we show that the profinite commensurability class of an S-arithmetic group with CSP determines the number field up to arithmetical equivalence and the places in S above unramified primes. We include some applications to profiniteness questions of group invariants.

 

Related project(s):
18Analytic L2-invariants of non-positively curved spaces

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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

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