L2-determinant class and approximation of L2-Betti numbers

Details L2-determinant class and approximation of L2-Betti numbers L2-determinant class and approximation of L2-Betti numbers

1998-07-07

arxiv.org/abs/math/9807032v2

Trans. Amer. Math. Soc. {\bf 353} (2001), no.~8, 3247--3265 (electronic); MR 2002f:58056

Mathematics

Geometric Topology

amsLaTeX2e, 26 pages; v2: Errata are added, rectifying some unproved statements about "amenable extension"

Scientific paper

A standing conjecture in L2-cohomology is that every finite CW-complex X is of L2-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class 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 L2-acyclic, we also prove that the L2-determinant is a homotopy invariant. Even in the known cases, our proof of homotopy invariance is much shorter and easier than the previous ones. Under suitable conditions we give new approximation formulas for L2-Betti numbers. Errata are added, rectifying some unproved statements about "amenable extension": throughout, amenable extensions should be extensions with \emph{normal} subgroups.

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