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Wolfgang Lück, Thomas Schick

Approximating L²-Signatures by Their Compact Analogues

Let ? be a group together with a sequence of normal subgroups ? ? ?₁ ? ?₂... of finite index [? : ?_k] such that ?_k ?_k = {1}. Let (X, Y) be a (compact) 4n-dimensional Poincaré pair and p : (X?, Y?) ? (X, Y) be a ?-covering, i.e. normal covering with ? as deck transformation group. We get associated ?/?_k-coverings (X_k, Y_k). We prove that

where sign or sign⁽²⁾ is the signature or L²-signature, respectively, and the convergence of the right side for any such sequence (?_k)_k?1 is part of the statement.

If ? is amenable, we prove in a similar way an approximation theorem for sign⁽²⁾(X?, Y?) in terms of the signatures of a regular exhaustion of X?.

Our results are extensions of Lück's approximation results for L²-Betti numbers [10, Theorem 0.1].

Forum Mathematicum, Walter de Gruyter

Print ISSN: 0933-7741
Volume: 17, 01/2005
Pages: 31 - 65

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