Lafforgue's strong property (T) (Part I)

Mikael De La Salle
École Normale Supérieure de Lyon

Strong property (T) is a variant of Kazhdan's property (T) which deals with representations with small exponential growth rate of the operator norms, where the original version dealt with unitary representations only. It was introduced by Vincent Lafforgue as a natural obstruction for his approach to the Baum-Connes conjecture to apply, and has then proven very influencial, especially its Banach space version. More recently it was one of the ingredients in the spectacular progresses on the Zimmer program.

My aim in this series of two talks is to present strong property (T), the way it is used in applications, and the main ideas of the proof that higher rank algebraic groups and their lattices have strong property (T). I will do my best to make the talk accessible to an audience with no or limited knowledge in operator algebras and representation theory.

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