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

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- by Thomas Schick PDF
- Trans. Amer. Math. Soc.
**353**(2001), 3247-3265 Request permission

## 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.## References

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## Additional Information

**Thomas Schick**- Affiliation: Fachbereich Mathematik, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
- MR Author ID: 635784
- Email: thomas.schick@math.uni-muenster.de
- Received by editor(s): July 15, 1998
- Received by editor(s) in revised form: March 12, 1999
- Published electronically: April 10, 2001
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1828605