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