Exercise 29, § 9

A real Lie group $G$ is said to be of type (N) if it is finite-dimensional,

† For more details, cf. J. Tits, Sur une classe de groupes de Lie résolubles, Bull. Soc. Math. Belg., 11 (1959), pp. 100–115 and 14 (1962), pp. 196–209.

nilpotent, connected and simply connected. If G is such a group and g is its Lie algebra, the mapping exp: g → G is an isomorphism when g is given the group structure defined by the Hausdorff law (cf. Chapter II, § 6, no. 5, Remark 3). Let log: G → g denote the inverse isomorphism.

(a) Let V be a vector Q-subspace of g. Show the equivalence of the following conditions: (i) V is a Lie Q-subalgebra of g; (ii) exp(V) is a subgroup of G. (Use Exercise 5 of § 6 of Chapter II.)

For a subgroup H of G to be of the form exp(V), where V is a vector Q-subspace of g, it is necessary and sufficient that H be saturated in G (Chapter II, § 4, Exercise 14), i.e. that the relations x ∈ G, x^n ∈ H, n ≠ 0 imply x ∈ H (loc. cit.). If this is so, show that H is an integral subgroup of G if and only if log(H) is a Lie R-subalgebra of g.

(b) Let V be a Lie Q-subalgebra of g of finite dimension m, let (e_1, ..., e_m) be a basis of V over Q and let Λ be the subgroup of V generated by e_1, ..., e_m. Using the fact that the Hausdorff law is polynomial, show that there exists an integer d ≥ 1 such that, for every integer r which is a non-zero multiple of d, exp(rΛ) is a subgroup of G. Let d be such an integer and r a multiple of d. Show that exp(rΛ) is discrete if and only if (e_1, ..., e_m) is a free family over R, i.e. if the canonical mapping of V ⊗_Q R into g is injective. Suppose that this is the case; show that G/exp(rΛ) is compact if and only if V ⊗_Q R → g is bijective, i.e. if V is a Q-form of g. (To show that the condition is sufficient, argue by induction on the nilpotency class of g.)

(c) Conversely, let Γ be a discrete subgroup of G, let $\overline{\Gamma}$ be its saturation in G (Chapter II, loc. cit.) and let $g_{\Gamma} = \log(\overline{\Gamma}) = Q.\log(\Gamma)$ be the corresponding Lie Q-subalgebra. Suppose that R.g = g, i.e. that Γ is contained in no distinct integral subgroup of G. Let z be an element ≠ 1 of the centre of Γ and let x = log (z). Show that x belongs to the centre of g. If X = exp(Rx), show that X/(Γ ∩ X) is compact and that the image of Γ in G/X is discrete. Deduce, arguing by induction on dim G, that G/Γ is compact and that $g_{\Gamma}$ is a G-form of g. If Λ is a lattice of $g_{\Gamma}$, show that there exists an integer d ≠ 0 such that Γ contains exp(dΛ) and that the index of exp(dΛ) in Γ is then finite.

Let H be an integral subgroup of G with Lie algebra h. Show that H/(H ∩ Γ) is compact if and only if h is rational with respect to the Q-structure $g_{\Gamma}$ (in particular if h is one of the terms of the lower—or upper—central series of g).

(d) Let Γ be a discrete subgroup of G, let H be the smallest integral subgroup of G containing Γ and let h be its Lie algebra. Show that H/Γ is compact and that $g \otimes_Q R \to g$ is injective and has image h. (Apply (c) to the nilpotent group H.)

(e) Let Γ be a discrete subgroup of G. Show that there exists a basis $(x_1, ..., x_q)$ of L(G) with the following properties: (i) for all $i \in \{1, q\}$, $Rx_i + \cdots + Rx_q$ is an ideal $n_i$ of $Rx_{i-1} + \cdots + Rx_q$;

(iii) there exists $p \in \{1, q\}$ such that $\Gamma$ is the set of products

where $m_p, \ldots, m_q$ are in $\mathbf{Z}$; (iii) if $G/\Gamma$ is compact, $\{n_1, n_2, \ldots, n_q\}$ contains the lower central series of $g$. (Using (d) and Proposition 16, reduce it to the case where $G/\Gamma$ is compact. Then argue by induction on $\dim G$, using (c).)