Compactification de Chabauty de l'espace des sous-groupes de Cartan de SLn(R)

TL;DR

This paper studies the Chabauty compactification of the space of Cartan subgroups in real semisimple Lie groups, characterizing it explicitly for certain groups and analyzing its topological properties.

Contribution

It provides a detailed description of the Chabauty compactification for specific groups like SL3(R) and SL4(R), including topological insights.

Findings

For groups with real rank 1, the compactification equals the set of all closed connected abelian subgroups of the same dimension.

In the case of SL3(R), the compactification is shown to be simply connected.

The paper characterizes the compactification explicitly for SL3(R) and SL4(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=SL3(R) or SL4(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 SL3(R), we study its topology more closely and we show that it is simply connected.