The Baum-Connes conjecture for hyperbolic groups

  title={The Baum-Connes conjecture for hyperbolic groups},
  author={Igor Mineyev and Guoliang Yu},
  journal={Inventiones mathematicae},
Abstract.We prove the Baum-Connes conjecture for hyperbolic groups and their subgroups. 
From acyclic groups to the Bass conjecture for amenable groups
Abstract.We prove that the Bost Conjecture on the ℓ1-assembly map for countable discrete groups implies the Bass Conjecture. It follows that all amenable groups satisfy the Bass Conjecture.
Banach KK-theory and the Baum-Connes conjecture
The report below describes the applications of Banach KK-theory to a conjecture of P. Baum and A. Connes about the K-theory of group C*-algebras.
  • A. Bartels
  • Mathematics
    Proceedings of the International Congress of Mathematicians (ICM 2018)
  • 2019
This note surveys axiomatic results for the Farrell-Jones Conjecture in terms of actions on Euclidean retracts and applications of these to GL_n(Z), relative hyperbolic groups and mapping class
Spectral morphisms, K-theory, and stable ranks
We give a brief account of the interplay between spectral morphisms, K-theory, and stable ranks in the context of Banach algebras.
On the algebraic K- and L-theory of word hyperbolic groups
Abstract.In this paper, the assembly maps in algebraic K- and L-theory for the family of finite subgroups are proven to be split injections for word hyperbolic groups. This is done by analyzing the
Higher invariants in noncommutative geometry
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Some geometric groups with Rapid Decay
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Hattori-Stallings trace and Euler characteristics for groups
We discuss properties of the complete Euler characteristic of a group G of type FP over the complex numbers and we relate it to the L2-Euler characteristic of the centralizers of the elements of G.
On the Farrell–Jones Conjecture and its applications
We present the status of the Farrell–Jones Conjecture for algebraic K‐theory for a group G and arbitrary coefficient rings R. We add new groups for which the conjecture is known to be true, and we


Operator -theory for groups which act properly and isometrically on Hilbert space
Let G be a countable discrete group which acts isometrically and metrically properly on an infinite-dimensional Euclidean space. We calculate the K-theory groups of the C∗-algebras C∗ max(G) and C∗
Groups acting properly on “bolic” spaces and the Novikov conjecture
We introduce a class of metric spaces which we call “bolic”. They include hyperbolic spaces, simply connected complete manifolds of nonpositive curvature, euclidean buildings, etc. We prove the
Classifying Space for Proper Actions and K-Theory of Group C*-algebras
We announce a reformulation of the conjecture in [8,9,10]. The advantage of the new version is that it is simpler and applies more generally than the earlier statement. A key point is to use the
Straightening and bounded cohomology of hyperbolic groups
Abstract. It was stated by M. Gromov [Gr2] that, for any hyperbolic group G, the map from bounded cohomology $ H^n_b(G,{\Bbb R}) $ to $ H^n(G,{\Bbb R}) $ induced by inclusion is surjective for $ n
The Kadison-Kaplansky conjecture for word-hyperbolic groups
In this paper we prove the Kadison-Kaplansky idempotent conjecture for torsion-free word-hyperbolic groups. The conjecture asserts that the following equivalent statements hold for a torsion-free
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Let r be a group. We associate to any length-function L on r the space H'{' (r) of rapidly decreasing functions on r (with respect to L), which coincides with the space of smooth functions on the
K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes
Je suis parvenu dans ma these a construire une k-theorie bivariante pour les algebres de banach. Cela m'a permis de demontrer la conjecture de baum-connes pour les groupes de lie semi-simples et les