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$L^2$-determinant class and approximation of $L^2$-Betti numbers


Author: Thomas Schick
Journal: Trans. Amer. Math. Soc. 353 (2001), 3247-3265
MSC (2000): Primary 58G50; Secondary 55N25, 55P29, 58G52
DOI: https://doi.org/10.1090/S0002-9947-01-02699-X
Published electronically: April 10, 2001
MathSciNet review: 1828605
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Abstract:

A standing conjecture in $L^2$-cohomology says that every finite $CW$-complex $X$ is of $L^2$-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class $\mathcal G$ of groups containing, e.g., all extensions of residually finite groups with amenable quotients, all residually amenable groups, and free products of these. If, in addition, $X$ is $L^2$-acyclic, we also show that the $L^2$-determinant is a homotopy invariant -- giving a short and easy proof independent of and encompassing all known cases. Under suitable conditions we give new approximation formulas for $L^2$-Betti numbers.


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Additional Information

Thomas Schick
Affiliation: Fachbereich Mathematik, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
Email: thomas.schick@math.uni-muenster.de

DOI: https://doi.org/10.1090/S0002-9947-01-02699-X
Keywords: $L^2$-determinant, $L^2$-Betti numbers, approximation, $L^2$-torsion, homotopy invariance
Received by editor(s): July 15, 1998
Received by editor(s) in revised form: March 12, 1999
Published electronically: April 10, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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