L2-Invariants of Regular Coverings of Compact Manifolds and CW-Complexes

Lück, Wolfgang

doi:10.1016/b978-044482432-5/50016-0

Cited by 32 publications

(41 citation statements)

References 217 publications

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“…We define L 2 -Betti numbers, Novikov-Shubin invariants and L 2 -torsion of finite CW-complexes. For a more detailed survey on L 2 -invariants we refer to the article [7]. For a discrete group G we define l 2 (G) as the Hilbert space completion of the complex group ring CG with respect to the inner product…”

Section: Some Basics On L 2 -Invariantsmentioning

confidence: 99%

“…The von Neumann dimension satisfies faithfulness, monotony, continuity and weak exactness (see [7,Lemma 1.4…”

Section: Definition 23 the Von Neumann Dimension Of A Finitely Genementioning

confidence: 99%

“…The main result of this paper is related to the following conjecture posed by Lück (see [7,Conjecture 9.21]). In case of L 2 -Betti numbers this conjecture goes back to a question of Gromov (see [4,Section 8A] Since any closed orientable manifold with non-trivial amenable fundamental group has vanishing simplicial volume (see [3, Corollary (C) on page 40]), it is obvious to consider finite aspherical CW-complexes with amenable or elementary amenable fundamental group.…”

Section: Introductionmentioning

confidence: 99%

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L 2-Invariants of finite aspherical CW-complexes

Wegner

2008

manuscripta math.

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Abstract. Let X be a finite aspherical CW-complex whose fundamental group π 1 (X) possesses a subnormal series π 1 (X) ⊲ Gm ⊲ . . . ⊲ G 0 with a non-trivial elementary amenable group G 0 . We investigate the L 2 -invariants of the universal covering of such a CW-complex X. We show that the NovikovShubin invariants αn(X) are positive. We further prove that the L 2 -torsion ρ (2) (X) vanishes if π 1 (X) has semi-integral determinant.

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Section: Some Basics On L 2 -Invariantsmentioning

confidence: 99%

“…The von Neumann dimension satisfies faithfulness, monotony, continuity and weak exactness (see [7,Lemma 1.4…”

Section: Definition 23 the Von Neumann Dimension Of A Finitely Genementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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L 2-Invariants of finite aspherical CW-complexes

Wegner

2008

manuscripta math.

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“…If G is torsion-free, the strong Atiyah conjecture for KG is particularly interesting: it implies that KG is a domain, i.e. has no non-trivial zero-divisors [38,Lemma 2.4]. More precisely, the ring DG which contains KG is a skew field (compare e.g.…”

Section: Introductionmentioning

confidence: 99%

Finite group extensions and the Atiyah conjecture

Linnell

Schick

2007

J. Amer. Math. Soc.

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The Atiyah conjecture for a discrete group G states that the L 2 -Betti numbers of a finite CW-complex with fundamental group G are integers if G is torsion-free, and in general that they are rational numbers with denominators determined by the finite subgroups of G.Here we establish conditions under which the Atiyah conjecture for a torsion-free group G implies the Atiyah conjecture for every finite extension of G. The most important requirement is that H * (G, Z/p) is isomorphic to the cohomology of the p-adic completion of G for every prime number p. An additional assumption is necessary, e.g. that the quotients of the lower central series or of the derived series are torsion-free.We prove that these conditions are fulfilled for a certain class of groups, which contains in particular Artin's pure braid groups (and more generally fundamental groups of fiber-type arrangements), free groups, fundamental groups of orientable compact surfaces, certain knot and link groups, certain primitive one-relator groups, and products of these. Therefore every finite, in fact every elementary amenable extension of these groups satisfies the Atiyah conjecture, provided the group does.As a consequence, if such an extension H is torsion-free then the group ring CH contains no non-trivial zero divisors, i.e. H fulfills the zero-divisor conjecture.In the course of the proof we prove that if these extensions are torsionfree, then they have plenty of non-trivial torsion-free quotients which are virtually nilpotent. All of this applies in particular to Artin's full braid group, therefore answering question B6 on www.grouptheory.info.Our methods also apply to the Baum-Connes conjecture. This is discussed in [46], where for example the Baum-Connes conjecture is proved for the full braid groups.MSC: 55N25 (homology with local coefficients), 16S34 (group rings, Laurent rings), 57M25 (knots and links)

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“…An early result of this type is a theorem of Brooks [1] stating that given a regular cover M of a compact manifold N , 0 is in the spectrum of 0 on M if and only if the group of deck transformations is amenable. The articles [14], [15] provide a comprehensive survey of this topic.The purpose of this article is to prove the zero-in-the-spectrum conjecture on certain regular covers associated to normal, amenable subgroups of the fundamental …”

mentioning

confidence: 99%

Zero-in-the-spectrum conjecture on regular covers of compact manifolds

Nowak¹

2009

Comment. Math. Helv.

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Abstract. We prove the zero-in-the-spectrum conjecture for large, regular covers associated to amenable subgroups of fundamental group of a closed manifold N , provided that 1 .N / is C -exact. Mathematics Subject Classification (2000). Primary 58J22; Secondary 58J50.Keywords. Property A, exact group, zero-in-the-spectrum conjecture.The zero-in-the-spectrum conjecture was first formulated by Gromov [6], [7] and asks if the spectrum Laplace-Beltrami operator acting on the square-integrable p-forms on the universal cover of a closed aspherical manifold contains zero. This fact is implied by the Strong Novikov Conjecture and thus the interest in finding a counterexample. A more general zero-in-the-spectrum conjecture on open complete manifolds was stated by Lott and it is true if there is a positive answer to the following question: does the spectrum of the Laplace-Beltrami operator p acting on square-integrable p-forms of a complete manifold M contain zero for some p D 0; 1; : : : ? The answer is negative in general: Farber and Weinberger [4] showed that for every n 6 there exists a manifold N such that zero is not in the spectrum of p for any p 2 0; 1; : : : acting on the universal cover of N . Later Higson, Roe and Schick [12] extended this result and gave a complete description of groups which can appear as fundamental groups of manifolds whose universal covers do not have zero in the spectrum of the Laplacian.Because of the origins of the problem, various covering spaces are a natural environment for considering zero-in-the-spectrum questions. An early result of this type is a theorem of Brooks [1] stating that given a regular cover M of a compact manifold N , 0 is in the spectrum of 0 on M if and only if the group of deck transformations is amenable. The articles [14], [15] provide a comprehensive survey of this topic.The purpose of this article is to prove the zero-in-the-spectrum conjecture on certain regular covers associated to normal, amenable subgroups of the fundamental

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L2-Invariants of Regular Coverings of Compact Manifolds and CW-Complexes

Cited by 32 publications

References 217 publications

L 2-Invariants of finite aspherical CW-complexes

L 2-Invariants of finite aspherical CW-complexes

Finite group extensions and the Atiyah conjecture

Zero-in-the-spectrum conjecture on regular covers of compact manifolds

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