The Baum–Connes Conjecture and Discrete Group Actions on Trees

Tu, Jean-Louis

doi:10.1023/a:1007751625568

Cited by 23 publications

(19 citation statements)

References 5 publications

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“…Since all maps in the above diagram are R(Ĝ)-linear this finishes the proof. We shall now show that the T I-strong Baum-Connes property is inherited by free products, compare [22], [25]. Theorem 6.6.…”

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confidence: 92%

The $$K$$ -theory of free quantum groups

Vergnioux¹,

Voigt²

2013

Math. Ann.

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Abstract. In this paper we study the K-theory of free quantum groups in the sense of Wang and Van Daele, more precisely, of free products of free unitary and free orthogonal quantum groups. We show that these quantum groups are K-amenable and establish an analogue of the Pimsner-Voiculescu exact sequence. As a consequence, we obtain in particular an explicit computation of the K-theory of free quantum groups. Our approach relies on a generalization of methods from the Baum-Connes conjecture to the framework of discrete quantum groups. This is based on the categorical reformulation of the Baum-Connes conjecture developed by Meyer and Nest. As a main result we show that free quantum groups have a γ-element and that γ = 1. As an important ingredient in the proof we adapt the Dirac-dual Dirac method for groups acting on trees to the quantum case. We use this to extend some permanence properties of the Baum-Connes conjecture to our setting.

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confidence: 92%

The $$K$$ -theory of free quantum groups

Vergnioux¹,

Voigt²

2013

Math. Ann.

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“…(i) If N is Haken, then, by [45], or [8], or [51], its fundamental group satisfies the Baum-Connes Conjecture With Coefficients. (ii) If N is hyperbolizable, then, as recalled earlier, π 1 (N ) is a discrete subgroup of SO(3, 1).…”

Section: The Proofsmentioning

confidence: 99%

“…Definition 3.2 (see [51]). -A group G verifies property (BC ) if it is contained in a discrete group G such that there exists a proper G -compact space X, a G X-algebra A and elements α ∈ KK…”

Section: Matthey (M) Oyono-oyono (H) and Pitsch (W)mentioning

confidence: 99%

“…Theorem 3.3 (see [51]). -If a group G satifies property (BC ), then 1) G satisfies the Baum-Connes conjecture; 2) G is K-amenable.…”

Section: Matthey (M) Oyono-oyono (H) and Pitsch (W)mentioning

confidence: 99%

“…Theorem 3.4 (see [51]). -Property (BC ) is hereditary under finite fundamental domain action on trees.…”

Section: Matthey (M) Oyono-oyono (H) and Pitsch (W)mentioning

confidence: 99%

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Homotopy invariance of higher signatures and $3$-manifold groups

Matthey¹,

Oyono-Oyono²,

Pitsch³

2008

Bul. Soc. Math. France

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Abstract. -For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable 3-manifolds, including the "piecewise geometric" ones in the sense of Thurston. In particular, this class, that will be carefully described, is the class of all orientable 3-manifolds if the Thurston Geometrization Conjecture is true. In fact, for this type of groups, we show that the Baum-Connes Conjecture With Coefficients holds. The non-oriented case is also discussed. Nous démontrons que pour des variétés fermées et orientées les signatures qui proviennent des groupes fondamentaux d'une large classe de variétés orientables de dimension 3 sont des invariants homotopiques. Cette classe, que nous décrivons soigneusement, contient en particulier les variétés géométriques par morceaux au sens de Thurston. Si la conjecture de géométrisation de Thurston s'avère vraie cette classe coïncide alors avec celle des groupes fondamentaux de variétés de dimension 3 orientables. Plus précisément nous démontrons que tous les groupes dans cette classe satisfont la conjecture de Baum-Connes avec coefficients. Nous discutons également le cas non-orientable.

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The Gamma Element for Groups which Admit a Uniform Embedding into Hilbert Space

2004

Recent Advances in Operator Theory, Operator Algebras, and Their Applications

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The Baum–Connes Conjecture and Discrete Group Actions on Trees

Cited by 23 publications

References 5 publications

The $$K$$ -theory of free quantum groups

The $$K$$ -theory of free quantum groups

Homotopy invariance of higher signatures and $3$-manifold groups

The Gamma Element for Groups which Admit a Uniform Embedding into Hilbert Space

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