Exercise 9, § 9

Let G be the universal covering of $\mathbf{SL}(2, \mathbf{R})$. We identify the kernel of the canonical morphism G → $\mathbf{SL}(2, \mathbf{R})$ with $\mathbf{Z}$ (§ 6, Exercise 2). Let a be an element of $\mathbf{T}^n$ whose powers are everywhere dense in $\mathbf{T}^n$ (General Topology, Chapter VII, § 1, Corollary 2 to Proposition 7). Let D be the discrete subgroup of G × $\mathbf{T}^n$ generated by (1, a). Let H = (G × $\mathbf{T}^n$)/D. Then

and the integral subgroup H' of H with Lie algebra $\mathfrak{sl}(2, \mathbf{R})$ is isomorphic to G and dense in H. D^nH' = H' for all n ≥ 0.