Finite group extensions and the Atiyah conjecture

Linnell, Peter A.; Schick, Thomas

doi:10.1090/s0894-0347-07-00561-9

Cited by 30 publications

(61 citation statements)

References 47 publications

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“…However, this would deviate from the notation used in Linnel-Schick [11]. Moreover, all examples relevant to us satisfy the condition that H =V is solvable-by-finite.…”

Section: Remarkmentioning

confidence: 97%

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Finite group extensions and the Baum–Connes conjecture

Schick

2007

Geom. Topol.

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In this note, we exhibit a method to prove the Baum-Connes conjecture (with coefficients) for extensions with finite quotients of certain groups which already satisfy the Baum-Connes conjecture. Interesting examples to which this method applies are torsion-free finite extensions of the pure braid groups, eg the full braid groups, and fundamental groups of certain link complements in S 3 .The Baum-Connes conjecture for a group G states that the Baum-Connes mapis an isomorphism for every C -algebra A with an action of G by C -algebra homomorphisms. In this note the term "Baum-Connes conjecture" will always mean "Baum-Connes conjecture with coefficients", and all group are assumed to be discrete and countable.Here, the left hand side is the equivariant K -homology with coefficients in A of the universal space EG for proper G -actions, which is homological in nature. The right hand side, the K -theory of the reduced crossed product of A and G , belongs to the world of C -algebras and-to some extent-representations of groups. If A D ‫ރ‬ with the trivial action, the right hand side becomes the K -theory of the reduced C -algebra of G . If, in addition, G is torsion-free, the left hand side is the K -homology of the classifying space of G .The Baum-Connes conjecture has many important connections to other questions and areas of mathematics. The injectivity of r implies the Novikov conjecture about homotopy invariance of higher signatures. It also implies the stable Gromov-LawsonRosenberg conjecture about the existence of metrics with positive scalar curvature on spin-manifolds. The surjectivity, on the other hand, gives information in particular about C red G . If G is torsion-free, it implies eg that this C -algebra contains no idempotents different from zero and one. Since we are only considering the BaumConnes conjecture with coefficients, all these properties follow for all subgroups of G , as well.

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“…However, this would deviate from the notation used in Linnel-Schick [11]. Moreover, all examples relevant to us satisfy the condition that H =V is solvable-by-finite.…”

Section: Remarkmentioning

confidence: 97%

“…Proof In Linnell-Schick [11,Proposition 4.30] we prove that G has the desired properties. Because LHETH is closed under extension, G 2 LHETH, in particular G fulfills the Baum-Connes conjecture.…”

Section: Definition 18mentioning

confidence: 99%

Finite group extensions and the Baum–Connes conjecture

Schick

2007

Geom. Topol.

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“…Following [LS07] we call a group G cohomologically p-complete if the natural morphism G → G p induces an isomorphism H n cont ( G p ; F p ) → H n (G; F p ) for each n ≥ 1.…”

Section: Cohomological P-completenessmentioning

confidence: 99%

$3$-Manifold Groups are Virtually Residually $p$

Aschenbrenner

Friedl²

2013

Memoirs of the AMS

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“…In this endeavor, we will employ the following simplification of the defining property for G(p). [4] Cohomology and profinite topologies for solvable groups of finite rank 257 P 2.2. Let p be a prime.…”

Section: The Classes G( P) and G * ( P)mentioning

confidence: 99%

Cohomology and Profinite Topologies for Solvable Groups of Finite Rank

Lorensen¹

2012

Bull. Aust. Math. Soc.

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Assume that G is a solvable group whose elementary abelian sections are all finite. Suppose, further, that p is a prime such that G fails to contain any subgroups isomorphic to Cp∞. We show that if G is nilpotent, then the pro-p completion map $G\to \hat {G}_p$ induces an isomorphism $H^\ast (\hat {G}_p,M)\to H^\ast (G,M)$ for any discrete $\hat {G}_p$-module M of finite p-power order. For the general case, we prove that G contains a normal subgroup N of finite index such that the map $H^\ast (\hat {N}_p,M)\to H^\ast (N,M)$ is an isomorphism for any discrete $\hat {N}_p$-module M of finite p-power order. Moreover, if G lacks any Cp∞-sections, the subgroup N enjoys some additional special properties with respect to its pro-p topology.

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Finite group extensions and the Atiyah conjecture

Cited by 30 publications

References 47 publications

Finite group extensions and the Baum–Connes conjecture

Finite group extensions and the Baum–Connes conjecture

$3$-Manifold Groups are Virtually Residually $p$

Cohomology and Profinite Topologies for Solvable Groups of Finite Rank

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