• Corpus ID: 117137631

# 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}
}
• M. Puschnigg
• Published 26 November 2012
• Mathematics
• arXiv: K-Theory and Homology
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

## Figures from this paper

### Bivariant KK -Theory and the Baum–Connes conjecure

The extension of K-theory from topological spaces to operator algebras provides the most powerful tool for the study of C *-algebras. On one side there now exist far reaching classification results

### Kazhdan projections, random walks and ergodic theorems

• Mathematics
Journal für die reine und angewandte Mathematik (Crelles Journal)
• 2019
Abstract In this paper we investigate generalizations of Kazhdan’s property (T) to the setting of uniformly convex Banach spaces. We explain the interplay between the existence of spectral gaps and

### Towards strong Banach property (T) for SL(3,ℝ)

We prove that SL(3, ℝ) has Strong Banach property (T) in Lafforgue’s sense with respect to the Banach spaces that are θ > 0 interpolation spaces (for the complex interpolation method) between an

### Half-dimensional collapse of ends of manifolds of nonpositive curvature

• Mathematics
Geometric and Functional Analysis
• 2019
This paper accomplishes two things. First, we construct a geometric analog of the rational Tits building for general noncompact, complete, finite volume n-manifolds M of bounded nonpositive

### Slant products on the Higson–Roe exact sequence

• Mathematics
Annales de l'Institut Fourier
• 2021
We construct a slant product $/ \colon \mathrm{S}_p(X \times Y) \times \mathrm{K}_{1-q}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{S}_{p-q}(X)$ on the analytic structure group of Higson and Roe and

### WOA: Women in Operator Algebras

We think that we have achieved both of these aims by using the format from the BIRS workshops “Women in Numbers” where the focus is on research in groups on current problems. Our workshop had 8

### The Baum–Connes conjecture: an extended survey

• Mathematics
• 2019
We present a history of the Baum–Connes conjecture, the methods involved, the current status, and the mathematics it generated.

## 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

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

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

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

"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

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