Proposition 8

The canonical bijection of N onto D is a homeomorphism.

Now, if $\Gamma_d$ is the set of Haar measures on the discrete subgroups of $G$, then $D$ is canonically homeomorphic to the space of orbits of the group of homotheties in $\Gamma_d$ with ratio $> 0$ (GT, I, §5, No. 2, Prop. 4). It therefore suffices to prove that the canonical mapping $\alpha \mapsto (\alpha(\{e\}), \alpha/\alpha(\{e\}))$ of $\Gamma_d$ onto $\mathbf{R}_+^* \times \mathbf{N}$ is a homeomorphism, which will result from the following lemma: