Compactification de Chabauty de l’espace des sous-groupes de Cartan de $${\text{ SL}}_n(\mathbb{R })$$

Abstract

Let $$G$$ be a real semisimple Lie group with finite center, with a finite number of connected components and without compact factor. We are interested in the homogeneous space of Cartan subgroups of $$G$$ , which can be also seen as the space of maximal flats of the symmetric space of $$G$$ . We define its Chabauty compactification as the closure in the space of closed subgroups of $$G$$ , endowed with the Chabauty topology. We show that when the real rank of $$G$$ is 1, or when $$G={\text{ SL}}_3(\mathbb{R })$$ or $${\text{ SL}}_4(\mathbb{R })$$ , this compactification is the set of all closed connected abelian subgroups of dimension the real rank of $$G$$ , with real spectrum. And in the case of $${\text{ SL}}_3(\mathbb{R })$$ , we study its topology more closely and we show that it is simply connected.