Kazhdan property (T)

A locally compact group $G$ has Kazhdan's property $(T)$ if every strongly continuous unitary representation with almost-invariant vectors has a nonzero invariant vector. Equivalently, there is a compact set $Q\subseteq G$ and $\varepsilon>0$ such that any representation having a unit vector $v$ with $\sup_{g\in Q}\|\pi(g)v-v\|<\varepsilon$ contains a nonzero invariant vector.

Such $(Q,\varepsilon)$ is a Kazhdan pair. For a discrete group, $Q$ is finite.

Consequence for Koopman theory

Every ergodic probability-preserving action of a discrete property- $(T)$ group has a spectral gap. Otherwise its reduced Koopman representation would have almost-invariant vectors, so property $(T)$ would give a nonzero invariant vector, contradicting ergodicity.

The property concerns all unitary representations of the group. A particular action can have a spectral gap even when its group does not have property $(T)$ .

References

B. Bekka, P. de la Harpe, and A. Valette, Kazhdan’s Property (T), Definition 1.1.3, Theorem 1.2.1, and §6.3.