higher rank. It turns out that some of the other equivalent characterizations of convex cocompactness in rank one do admit useful modifications in higher rank, which lead to the class of Anosov subgroups. The Anosov condition is not too rigid and, at the same time, it imposes enough restrictions on the subgroups making it possible to analyze their geometric and dynamical properties. One way to think of Anosov subgroups is as geometrically finite subgroups without parabolics which exhibit some [*rank one behavior*]{}. Indeed, they are intrinsically word hyperbolic and we will see that also extrinsically they display hyperbolic behavior in a variety of ways.

In this paper, we primarily consider four notions generalizing convex cocompactness to higher rank, all equivalent to the Anosov condition, see the Equivalence Theorem \[thm:eqvi\] below:

\(i) asymptotic embeddedness

\(ii) expansivity

\(iii) conicality

\(iv) Morse property

Whereas the conditions Anosov, (i) and (ii) are dynamical, (iii) is a condition on the asymptotic geometry of the subgroup, and (iv) is coarse geometric.

We now describe in more detail some of our concepts and results.

Let $X=G/K$ be a symmetric space of noncompact type and, for simplicity, let the semisimple Lie group $G$ be the connected component of 