Property (T)

Remarks

Example: every compact group $G$ has property (T), with $Q = G$ and $\varepsilon = \sqrt{2}$. If $\norm{\pi(g)\xi - \xi} < \sqrt{2}$ for all $g$, then $\mathrm{Re} \ip{\pi(g)\xi}{\xi} > 0$ for all $g$. The average $\eta = \int_G \pi(g) \xi \, dg$ over the normalised Haar measure is invariant, and $\mathrm{Re} \ip{\eta}{\xi} > 0$, so $\eta \neq 0$.

Non-example: $\Z$ does not have property (T). Here compact sets are finite. In the regular representation on $\ell^2(\Z)$, let $\xi_N = N^{-1/2} \mathbf{1}_{\set{0, \ldots, N-1}}$. Then $\norm{\pi(k)\xi_N - \xi_N} \le \sqrt{2\abs{k}/N}$, which is small on any finite $Q$ when $N$ is large. But a nonzero vector of $\ell^2(\Z)$ invariant under all shifts would have constant coordinates and infinite norm.

Equivalently, the trivial representation is an isolated point of the unitary dual of $G$ in the Fell topology. For a discrete group, $Q$ can be taken finite, and a discrete group with property (T) is finitely generated. The groups $\mathrm{SL}_n(\R)$ and $\mathrm{SL}_n(\Z)$ have property (T) for $n \ge 3$, but not for $n = 2$.