to higher rank. Our main goal here is to give several new equivalent characterizations for this important class of discrete subgroups, including a considerable simplification of their original definition. For convex cocompact subgroups as well as for word hyperbolic groups, it is very fruitful to have different viewpoints and alternative definitions, as they were developed by many authors starting with Ahlfors’ work on geometric finiteness in the 60s, and later by Beardon, Maskit, Marden, Thurston, Sullivan, Bowditch and others. Besides a deeper understanding, it enables one to switch perspectives in a nontrivial way, adapted to the situation at hand. A main purpose of this paper is to demonstrate that much of this theory extends to Anosov subgroups, and we hope that the concepts and results presented here will be useful for their further study. In our related work, they lay the basis for the results on the Higher Rank Morse Lemma [@mlem], compactifications of locally symmetric spaces for Anosov subgroups [@bordif], the local-to-global principle and the construction of Morse-Schottky subgroups [@morse]. We refer to the surveys [@anosov; @manicures] for more details on these developments.

In rank one, among Kleinian groups and, more generally, among discrete subgroups of rank one Lie 