Approximating <i>L</i><sup>2</sup>‐invariants and the Atiyah conjecture

Dodziuk, Józef; Linnell, Peter A.; Mathai, Varghese; Schick, Thomas; Yates, Stuart

doi:10.1002/cpa.10076

Cited by 81 publications

(88 citation statements)

References 15 publications

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“…Recall, the Atiyah Conjecture for torsionfree groups predicts the existence of a skew-field ZG ⊂ K ⊂ UG. Note that the Atiyah Conjecture was established for a large class of torsionfree groups, see the results and references in [17]. In particular, condition (⋆) is known to hold for all right orderable groups and all residually torsionfree elementary amenable groups.…”

Section: Free Subgroupsmentioning

confidence: 98%

Group cocycles and the ring of affiliated operators

2011

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Abstract. In this article we study cocycles of discrete countable groups with values in ℓ 2 G and the ring of affiliated operators U G. We clarify properties of the first cohomology of a group G with coefficients in ℓ 2 G and answer several questions from [14]. Moreover, we obtain strong results about the existence of free subgroups and the subgroup structure, provided the group has a positive first ℓ 2 -Betti number. We give numerous applications and examples of groups which satisfy our assumptions.

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Section: Free Subgroupsmentioning

confidence: 98%

Group cocycles and the ring of affiliated operators

2011

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“…The final goal of this article is to prove the following theorem, which in a similar form has been conjectured by Dodziuk, Linnell, Mathai, Schick and Yates, as Conjecture 4.14 in [2], to hold for all groups: Theorem 1.1. Let Γ be a sofic group and A ∈ M n (ZΓ).…”

Section: Introductionmentioning

confidence: 77%

“…We prove the algebraic eigenvalue conjecture of Dodziuk, Linnell, Mathai, Schick and Yates from [2] for sofic groups. (ii) For all λ ∈ C, the approximation formula holds: Proof.…”

Section: Applications To L 2 -Invariantsmentioning

confidence: 92%

Sofic groups and diophantine approximation

Thom

2007

Comm Pure Appl Math

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Abstract. We prove the algebraic eigenvalue conjecture of J.Dodziuk, P.Linnell, V.Mathai, T.Schick and S.Yates (see [2]) for sofic groups. Moreover, we give restrictions on the spectral measure of elements in the integral group ring. Finally, we define integer operators and prove a quantization of the operator norm below 2. To the knowledge of the author, there is no group known, which is not sofic.

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“…In this section, we recall a few more known results about the Atiyah conjecture, proved in particular in [47] and [11].…”

Section: More Positive Results About the Atiyah Conjecturementioning

confidence: 99%

“…the case K = Q) [47, Corollary 4]. A generalization to K = Q, the field of algebraic numbers over Q in C, is given in [11]. This applies 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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Approximating L²‐invariants and the Atiyah conjecture

Cited by 81 publications

References 15 publications

Group cocycles and the ring of affiliated operators

Group cocycles and the ring of affiliated operators

Sofic groups and diophantine approximation

Finite group extensions and the Atiyah conjecture

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Approximating L2‐invariants and the Atiyah conjecture

Cited by 81 publications

References 15 publications

Group cocycles and the ring of affiliated operators

Group cocycles and the ring of affiliated operators

Sofic groups and diophantine approximation

Finite group extensions and the Atiyah conjecture

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Approximating L²‐invariants and the Atiyah conjecture