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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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$L^2$-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


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
  • MR Author ID: 635784
  • Email:
  • 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:
  • MathSciNet review: 1828605