Exercise 19, § 3

Let $n\geqslant 2$ be an integer. Consider the real Lie group $G =\mathbf{S}\mathbf{L}(n,\mathbf{R})$. Let N and H denote the closed subgroups of G whose elements are of the block form

$($Id$_{n-1}x)_{n-1}(g0)$

0 1 0 1

respectively. The group N is identified with $\mathbf{R}^{n-1}$ and the group H is identified with $\mathbf{S}\mathbf{L}(n-1,\mathbf{R})$.

a) Let $\pi$ be a unitary representation of G in a complex Hilbert space E. An element $x$ of E is invariant under G if and only if it is invariant under N. b) If $n\geqslant 3$, then the group G has property ]T[. (Combine the result of the preceding question with the preceding exercise.)

c) If $n\geqslant 3$, then the group $\widetilde{G} =\mathbf{R}^n\rtimes G$ has property ]T[. (Let $\pi$ be a unitary representation such that 1$_{\widetilde{G}}\leqslant \pi$; there exists a vector $x\not = 0$ invariant under the subgroup G of $\widetilde{G}$ by the preceding question; now prove that the corresponding matrix coefficient is constant.)

d) If $n\geqslant 3$, the group $\mathbf{S}\mathbf{L}(n,\mathbf{Z})$ has property ]T[. (Use exercise 7 of INT, VII, p. 116.)

e) Let $n\geqslant 3$. The group $\mathbf{S}\mathbf{L}(n,\mathbf{Z})$ is finitely generated. Let $\mathscr{H}$ be the family of normal subgroups of finite index of $\mathbf{S}\mathbf{L}(n,\mathbf{Z})$; it is infinite. For every finite generating subset S of $\mathbf{S}\mathbf{L}(n,\mathbf{Z})$, the family of Cayley graphs $(\mathscr{G}(\mathbf{S}\mathbf{L}(n,\mathbf{Z})/H$, SH$/H))_{H\in\mathscr{H}}$ is a family of expander graphs (cf. TA, 2, p. 221, exercise 7 and exercise 16 of IV, p. 323).