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In mathematics, the Haagerup property, named after Uffe Haagerup and also known as Gromov's a-T-menability, is a property of groups that is a strong negation of Kazhdan's property (T). Property (T) is considered a representation-theoretic form of rigidity, so the Haagerup property may be considered a form of strong nonrigidity; see below for details.

The Haagerup property is interesting to many fields of mathematics, including harmonic analysis, representation theory, operator K-theory, and geometric group theory.

Perhaps its most impressive consequence is that groups with the Haagerup Property satisfy the Baum-Connes conjecture and the related Novikov conjecture. Groups with the Haagerup property are also uniformly embeddable into a Hilbert space.

Definitions

Let $G$ be a second countable locally compact group. The following properties are all equivalent, and any of them may be taken to be definitions of the Haagerup property:

There is a proper continuous conditionally negative definite function $Ψ : G \to ℝ^{+}$ .
$G$ has the Haagerup approximation property, also known as Property $C_{0}$ : there is a sequence of normalized continuous positive-definite functions $ϕ_{n}$ which vanish at infinity on $G$ and converge to 1 uniformly on compact subsets of $G$ .
There is a strongly continuous unitary representation of $G$ which weakly contains the trivial representation and whose matrix coefficients vanish at infinity on $G$ .
There is a proper continuous affine isometric action of $G$ on a Hilbert space.

Examples

There are many examples of groups with the Haagerup property, most of which are geometric in origin. The list includes:

All compact groups (trivially). Note all compact groups also have property (T). The converse holds as well: if a group has both property (T) and the Haagerup property, then it is compact.
SO(n,1)
SU(n,1)
Groups acting properly on trees or on $ℝ$ -trees
Coxeter groups
Amenable groups
Groups acting properly on CAT(0) cubical complexes

References

Cherix, Cowling, Jolissaint, Julg, and Valette (2001). Groups with the Haagerup Property (Gromov's a-T-menability)

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+In [[mathematics]], the '''Haagerup property''', named after [[Uffe Haagerup]] and also known as [[Mikhail Gromov (mathematician)|Gromov]]'s '''a-T-menability''', is a property of [[Group (mathematics)|group]]s that is a strong negation of [[Kazhdan]]'s [[property (T)]].  Property (T) is considered a representation-theoretic form of rigidity, so the Haagerup property may be considered a form of strong nonrigidity; see below for details.
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+The Haagerup property is interesting to many fields of mathematics, including [[harmonic analysis]], [[representation theory]], [[operator K-theory]], and [[geometric group theory]].
+Perhaps its most impressive consequence is that groups with the Haagerup Property satisfy the [[Baum-Connes conjecture]] and the related [[Novikov conjecture]].  Groups with the Haagerup property are also uniformly [[embedding|embeddable]] into a [[Hilbert space]].
+==Definitions==
+Let <math>G</math> be a [[second countable]] [[locally compact]] group.  The following properties are all equivalent, and any of them may be taken to be definitions of the Haagerup property:
+#There is a [[proper function|proper]] [[continuous function|continuous]] conditionally [[negative definite]] [[Function (mathematics)|function]] <math>\Psi\colon G \to \Bbb{R}^+</math>.
+#<math>G</math> has the '''Haagerup approximation property''', also known as '''Property <math>C_0</math>''':  there is a sequence of normalized continuous [[Positive-definite function on a group|positive-definite function]]s <math>\phi_n</math> which vanish at infinity on <math>G</math> and converge to 1 [[uniform convergence|uniformly]] on [[compact subset]]s of <math>G</math>.
+#There is a [[strongly continuous]] [[unitary representation]] of <math>G</math> which [[weak containment|weakly contains]] the [[trivial representation]] and whose matrix coefficients vanish at infinity on <math>G</math>.
+#There is a proper continuous affine isometric action of <math>G</math> on a [[Hilbert space]].
+==Examples==
+There are many examples of groups with the Haagerup property, most of which are geometric in origin.  The list includes:
+*All [[compact group]]s (trivially).  Note all compact groups also have [[property (T)]].  The converse holds as well:  if a group has both property (T) and the Haagerup property, then it is compact.
+*[[SO(n,1)]]
+*[[SU(n)|SU(n,1)]]
+*Groups acting properly on trees or on <math>\Bbb{R}</math>-trees
+*[[Coxeter groups]]
+*[[Amenable]] groups
+*Groups acting properly on [[CAT(0)]] [[cubical complex]]es
+==References==
+*Cherix, Cowling, Jolissaint, Julg, and Valette (2001).  ''Groups with the Haagerup Property (Gromov's a-T-menability)''
+[[Category:Representation theory]]
+[[Category:Geometric group theory]]

Shear and moment diagram: Difference between revisions

Revision as of 03:06, 23 January 2014

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