𝐿²-determinant class and approximation of 𝐿²-Betti numbers

Schick, Thomas

doi:10.1090/S0002-9947-01-02699-X

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

Trans. Amer. Math. Soc. 353 (2001), 3247-3265

DOI: https://doi.org/10.1090/S0002-9947-01-02699-X

Published electronically: April 10, 2001

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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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