  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.
