---
title: "Baum–Connes conjecture"
chunk: 2/2
source: "https://en.wikipedia.org/wiki/Baum–Connes_conjecture"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:02:45.104924+00:00"
instance: "kb-cron"
---

Discrete subgroups of 
  
    
      
        S
        O
        (
        n
        ,
        1
        )
      
    
    {\displaystyle SO(n,1)}
  
 and 
  
    
      
        S
        U
        (
        n
        ,
        1
        )
      
    
    {\displaystyle SU(n,1)}
  
.
Groups with the Haagerup property, sometimes called a-T-menable groups. These are groups that admit an isometric action on an affine Hilbert space 
  
    
      
        H
      
    
    {\displaystyle H}
  
 which is proper in the sense that 
  
    
      
        
          lim
          
            n
            →
            ∞
          
        
        
          g
          
            n
          
        
        ξ
        →
        ∞
      
    
    {\displaystyle \lim _{n\to \infty }g_{n}\xi \to \infty }
  
 for all 
  
    
      
        ξ
        ∈
        H
      
    
    {\displaystyle \xi \in H}
  
 and all sequences of group elements 
  
    
      
        
          g
          
            n
          
        
      
    
    {\displaystyle g_{n}}
  
 with 
  
    
      
        
          lim
          
            n
            →
            ∞
          
        
        
          g
          
            n
          
        
        →
        ∞
      
    
    {\displaystyle \lim _{n\to \infty }g_{n}\to \infty }
  
. Examples of a-T-menable groups are amenable groups, Coxeter groups, groups acting properly on trees, and groups acting properly on simply connected 
  
    
      
        C
        A
        T
        (
        0
        )
      
    
    {\displaystyle CAT(0)}
  
 cubical complexes.
Groups that admit a finite presentation with only one relation.
Discrete cocompact subgroups of real Lie groups of real rank 1.
Cocompact lattices in 
  
    
      
        S
        L
        (
        3
        ,
        
          R
        
        )
        ,
        S
        L
        (
        3
        ,
        
          C
        
        )
      
    
    {\displaystyle SL(3,\mathbb {R} ),SL(3,\mathbb {C} )}
  
 or 
  
    
      
        S
        L
        (
        3
        ,
        
          
            Q
          
          
            p
          
        
        )
      
    
    {\displaystyle SL(3,\mathbb {Q} _{p})}
  
. It was a long-standing problem since the first days of the conjecture to expose a single infinite property T-group that satisfies it. However, such a group was given by V. Lafforgue in 1998 as he showed that cocompact lattices in 
  
    
      
        S
        L
        (
        3
        ,
        
          R
        
        )
      
    
    {\displaystyle SL(3,\mathbb {R} )}
  
 have the property of rapid decay and thus satisfy the conjecture.
Gromov hyperbolic groups and their subgroups.
Among non-discrete groups, the conjecture has been shown in 2003 by J. Chabert, S. Echterhoff and R. Nest for the vast class of all almost connected groups (i. e. groups having a cocompact connected component), and all groups of 
  
    
      
        k
      
    
    {\displaystyle k}
  
-rational points of a linear algebraic group over a local field 
  
    
      
        k
      
    
    {\displaystyle k}
  
 of characteristic zero (e.g. 
  
    
      
        k
        =
        
          
            Q
          
          
            p
          
        
      
    
    {\displaystyle k=\mathbb {Q} _{p}}
  
). For the important subclass of real reductive groups, the conjecture had already been shown in 1987 by Antony Wassermann.
Injectivity is known for a much larger class of groups thanks to the Dirac-dual-Dirac method. This goes back to ideas of Michael Atiyah and was developed in great generality by Gennadi Kasparov in 1987.
Injectivity is known for the following classes:

Discrete subgroups of connected Lie groups or virtually connected Lie groups.
Discrete subgroups of p-adic groups.
Bolic groups (a certain generalization of hyperbolic groups).
Groups which admit an amenable action on some compact space.
The simplest example of a group for which it is not known whether it satisfies the conjecture is 
  
    
      
        S
        
          L
          
            3
          
        
        (
        
          Z
        
        )
      
    
    {\displaystyle SL_{3}(\mathbb {Z} )}
  
.

== References ==

Mislin, Guido & Valette, Alain (2003), Proper Group Actions and the Baum–Connes Conjecture, Basel: Birkhäuser, ISBN 0-8176-0408-1.
Valette, Alain (2002), Introduction to the Baum-Connes Conjecture, Basel: Birkhäuser, ISBN 978-3-7643-6706-0.

== External links ==
On the Baum-Connes conjecture by Dmitry Matsnev.