Exercise 14, § 3

Let $G$ be a locally compact group. We say that $G$ has Kazhdan's property ]T[^{[^1]} if a unitary representation $\pi$ of $G$ has a non-zero trivial subrepresentation^{[^2]} if and only if the trivial representation 1$_G$ of $G$ on $\mathbf{C}$ is weakly contained in $\pi$.

a) Let $G$ be a locally compact group having property ]T[. Let $\mathscr{H}$ be the set of open subgroups of $G$ which are generated by a compact subset of $G$. The group $G$ is the union of the subgroups $H\in \mathscr{H}$.

b) Let $\pi$ be the representation of $G$ which is the Hilbert sum of the quasi-regular representations of $G$ on $G/H$ for $H\in \mathscr{H}$ (cf. exercise 2 of V, p. 486). We have 1$_G\leqslant \pi$.

c) The group $G$ is generated by a compact subset of $G$. In particular, if $G$ is discrete, then $G$ is generated by a finite subset of $G$.

[^1]: cf. D. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroup, Funct. Anal. Appl. 1 (1967), 63–65. [^2]: That is, there exists $x\not = 0$ in the space of E invariant by $G$.