Property (T)

Summary

A locally compact group has property (T), or Kazhdan's property (T), under the following condition. Every unitary representation that almost has invariant vectors actually has a nonzero invariant vector. Kazhdan introduced it to show that lattices in groups such as $\mathrm{SL}_n(\R)$ with $n \ge 3$ are finitely generated. It is a strong rigidity property, opposite to amenability: a locally compact group that is both amenable and has property (T) is compact. It has applications to expander graphs, to ergodic theory and to the theory of operator algebras.