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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 $$SO(n,1)$$

## Publications

We prove that the sign of the Euler characteristic of arithmetic groups with CSP is determined by the profinite completion.  In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type F.  Our methods imply similar results for L2-torsion as well as a strong profiniteness statement for Novikov--Shubin invariants.

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

The purpose of this article is to define and study new invariants of topological spaces: the p-adic Betti numbers and the p-adic torsion. These invariants take values in the p-adic numbers and are constructed from a virtual pro-p completion of the fundamental group. The key result of the article is an approximation theorem which shows that the p-adic invariants are limits of their classical analogues. This is reminiscent of Lück's approximation theorem for L2-Betti numbers.

After an investigation of basic properties and examples we discuss the p-adic analog of the Atiyah conjecture: When do the p-adic Betti numbers take integer values? We establish this property for a class of spaces and discuss applications to cohomology growth.

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

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
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