We give a geometric interpretation of the maximal Satake compactification of symmetric
spaces
of
noncompact type, showing that it arises by attaching the horofunction boundary for a suitable
–invariant Finsler
metric on
.
As an application, we establish the existence of natural
bordifications, as orbifolds-with-corners, of locally symmetric spaces
for arbitrary discrete subgroups
. These bordifications result
from attaching
–quotients
of suitable domains of proper discontinuity at infinity. We further prove that such
bordifications are compactifications in the case of Anosov subgroups. We show,
conversely, that Anosov subgroups are characterized by the existence of such
compactifications among uniformly regular subgroups. Along the way, we give a
positive answer, in the torsion-free case, to a question of Haïssinsky and Tukia on
convergence groups regarding the cocompactness of their actions on the domains of
discontinuity.