Appears in collection : 2025 IHES Summer School – Discrete Subgroups of Lie Groups : Dynamics, Actions, Rigidity

Kazhdan’s Property T is a fundamental analytic invariant of discrete or, more generally, locally compact groups that is defined in terms of unitary representations. According to Delorme-Guichardet it can be characterized by the vanishing of the group cohomology in degree 1 for arbitrary unitary coefficients. This suggests an obvious generalizations to a higher degree Property T. We will discuss the first higher property T result by Garland from the 1970s. We will then turn to the Lie group case and their lattices, which is based on joint work with Uri Bader. As time permits, we will also discuss applications of (higher) property T.