groups, one distinguishes [*geometrically finite*]{} subgroups. They form a large and flexible class of discrete subgroups which are strongly tied to the negatively curved symmetric spaces they act on. Therefore they have especially good geometric, topological and dynamical properties and one can prove many interesting results about them. The simplest are geometrically finite subgroups [*without parabolics*]{}, which are lie at the root of this paper. They can be characterized in many (not obviously) equivalent ways: As convex cocompact subgroups, as undistorted subgroups, as subgroups with conical limit set, as subgroups which are expanding at their limit set, and as intrinsically word hyperbolic subgroups with Gromov boundary equivariantly homeomorphic to their limit set, to name some.

In higher rank, a satisfying and sufficiently broad definition of geometric finiteness, with or without parabolics, remains yet to be found. Convex cocompactness turns out to be much too restrictive a condition: it was shown by Kleiner and the second author [@convcoco] that in higher rank only few subgroups are convex cocompact. Undistortion by itself, on the other hand, is way too weak: undistorted subgroups can even fail to be finitely presented. Thus, one is forced to look for suitable replacements of these notions in 