Topology & Geometric Group Theory Seminar

Fall 2008

1:30 – 2:30, Malott 406

Tuesday, November 4

Talia Fernós, UCLA

Property (T) and friends

Property (T) has many successes in the study of a broad range of areas: von Neumann Algebras, dynamics, and geometric group theory to name a few. It is a property of groups that was introduced by Kazhdan in 1967; he showed that all higher rank lattices share this property. All property (T) groups are finitely generated. This is a key ingredient in Margulis' Arithmeticity Theorem for higher rank lattices. Simply stated, property (T) is a type of deformation rigidity for unitary representations. Namely, unitary representation which are close to containing the trivial representation actually do.

Property (T) can be seen as an analytical (versus geometric) property: it is an invariant under measure equivalence but not under quasi isometry. Property (T) groups do not admit non-trivial actions on many "simple" spaces. Such spaces include trees, the circle (with a sufficiently smooth action), and walled spaces. In this talk we will give a survey of property (T) and related properties and discuss some recent developments and open questions.

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