its isometry group. Our approach to studying Anosov subgroups $\Ga<G$ begins with the observation that they satisfy a strong form of discreteness which we call [*regularity*]{} and which is primarily responsible for their extrinsic “rank one behavior” alluded to above. Discreteness of a subgroup $\Ga<G$ means that for sequences $(\ga_n)$ of distinct elements the distance $d(x,\ga_nx)$ in $X$ diverges to infinity. For higher rank symmetric spaces there is a natural vector-valued refinement $d_{\De}$ of the Riemannian distance $d$, which takes values in the euclidean Weyl chamber $\De$ of $X$. The regularity assumption on $\Ga$, in its strongest form of [*$\simod$-regularity*]{}, means that $d_{\De}(x,\ga_nx)$ diverges away from the boundary of $\De$. We will work more generally with relaxations of this condition, called $\taumod$-regularity, associated with a face $\taumod$ of the model spherical Weyl chamber $\simod$, where one only requires divergence of $d_{\De}(x,\ga_nx)$ away from some of the faces of $\De$, depending on $\taumod$. To be precise, think of $\simod$ as the visual boundary of the euclidean Weyl chamber, $\simod\cong\geo\De$. Given a face $\taumod\subseteq\simod$, we define [*$\taumod$-regularity*]{} by requiring that $d_{\De}(x,\ga_nx)$ diverges away from the faces of $\De$ whose visual boundaries do not contain $\taumod$. We will also need the stronger notion 