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