# The Baum-Connes conjecture with coefficients for word-hyperbolic groups (after Vincent Lafforgue)

@article{Puschnigg2012TheBC, title={The Baum-Connes conjecture with coefficients for word-hyperbolic groups (after Vincent Lafforgue)}, author={Michael Puschnigg}, journal={arXiv: K-Theory and Homology}, year={2012} }

Already in the early eighties, A. Connes emphasized that Kazhdan’s property (T), which means that the trivial representation of a locally compact group is separated from all other unitary representations, might be a serious obstruction to the Baum-Connes conjecture. The only previously known approach, due to Kasparov [32], demands the construction of a homotopy among unitary representations between the regular and the trivial representation, which cannot exist for non-compact groups with…

## 7 Citations

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## References

SHOWING 1-10 OF 46 REFERENCES

### E-theory and KK-theory for groups which act properly and isometrically on Hilbert space

- Mathematics
- 2001

A good deal of research in C∗-algebra K -theory in recent years has been devoted to the Baum-Connes conjecture [3], which proposes a formula for the K -theory of group C∗-algebras that blends group…

### BOTT PERIODICITY AND THE INDEX OF ELLIPTIC OPERATORS

- Mathematics
- 1968

IN an expository article (1) I have indicated the deep connection between the Bott periodicity theorem (on the homotopy of the unitary groups) and the index of elliptio operators. It ia the purpose…

### Discrete Subgroups of Semisimple Lie Groups

- Mathematics
- 1991

1. Statement of Main Results.- 2. Synopsis of the Chapters.- 3. Remarks on the Structure of the Book, References and Notation.- 1. Preliminaries.- 0. Notation, Terminology and Some Basic Facts.- 1.…

### The Connes-Kasparov conjecture for almost connected groups and for linear p-adic groups

- Mathematics
- 2001

Let G be a locally compact group with cocompact connected component. We prove that the assembly map from the topological K-theory of G to the K-theory of the reduced C*-algebra of G is an…

### Equivariant Kasparov theory and generalized homomorphisms

- Mathematics
- 2000

Let G be a locally compact group. We describe elements of KK^G (A,B) by equivariant homomorphisms, following Cuntz's treatment in the non-equivariant case. This yields another proof for the universal…

### Essays in Group Theory

- Mathematics
- 2011

"Essays in Group Theory" contains five papers on topics of current interest which were presented in a seminar at MSRI, Berkeley in June, 1985. Special mention should be given to Gromovs paper, one of…

### Classifying Space for Proper Actions and K-Theory of Group C*-algebras

- Mathematics
- 2004

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…

### La conjecture de Baum–Connes à coefficients pour les groupes hyperboliques

- Mathematics
- 2012

This paper gives a proof of the Baum-Connes conjecture with coefficients for hyperbolic groups. More precisely the injectivity of the Baum-Connes map was established by Kasparov and Skandalis and we…