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Approximating L 2 -Signatures by Their Compact Analogues

Approximating L 2 -Signatures by Their Compact Analogues Let Γ be a group together with a sequence of normal subgroups Γ ⊃ Γ 1 ⊃ Γ 2 ... of finite index Γ : Γ k such that ⋂ k Γ k = {1}. Let ( X , Y ) be a (compact) 4 n -dimensional Poincaré pair and p : ( X ̄, Y ̄) → ( X , Y ) be a Γ-covering, i.e. normal covering with Γ as deck transformation group. We get associated Γ/Γ k -coverings ( X k , Y k ). We prove that where sign or sign (2) is the signature or L 2 -signature, respectively, and the convergence of the right side for any such sequence (Γ k ) k ≥1 is part of the statement. If Γ 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 L 2 -Betti numbers 10, Theorem 0.1. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Approximating L 2 -Signatures by Their Compact Analogues

Lück, Wolfgang; Schick, Thomas
Forum Mathematicum , Volume 17 (1) – Jan 1, 2005
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Publisher
de Gruyter
Copyright
© de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2005.17.1.31
Publisher site
See Article on Publisher Site

Abstract

Let Γ be a group together with a sequence of normal subgroups Γ ⊃ Γ 1 ⊃ Γ 2 ... of finite index Γ : Γ k such that ⋂ k Γ k = {1}. Let ( X , Y ) be a (compact) 4 n -dimensional Poincaré pair and p : ( X ̄, Y ̄) → ( X , Y ) be a Γ-covering, i.e. normal covering with Γ as deck transformation group. We get associated Γ/Γ k -coverings ( X k , Y k ). We prove that where sign or sign (2) is the signature or L 2 -signature, respectively, and the convergence of the right side for any such sequence (Γ k ) k ≥1 is part of the statement. If Γ 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 L 2 -Betti numbers 10, Theorem 0.1.

Journal

Forum Mathematicum – de Gruyter

Published: Jan 1, 2005

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