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Geometry of growth: approximation theorems for \(L^2\) invariants. (English) Zbl 0911.53026

In [Geom. Funct. Anal. 4, 455-481 (1994; Zbl 0853.57021)], W. Lück has proved a theorem stating that von Neumann Betti numbers of the universal covering of a finite polyhedron can be found as the limits of the normalized Betti numbers of finitely sheeted normal coverings. One of the goals of the present paper is to generalize Lück’s theorem in two directions.
First, instead of finitely sheeted normal coverings, the author considers flat vector bundles of finite dimension. Secondly, instead of \(L^2\)-Betti numbers, the author studies the von Neumann dimensions of the homology of infinite-dimensional flat bundles determined by unitary representations in a von Neumann category with a trace.
The other purpose of this paper is to investigate the situations, when the statement of Lück’s theorem in its original form is incorrect. The author shows that the correcting additional term has a very interesting meaning (the torsion dimension). This correcting additional term can be understood in the framework of the formalism of extended cohomology and von Neumann categories.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
55N25 Homology with local coefficients, equivariant cohomology

Citations:

Zbl 0853.57021
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