collect some material from the geometry of symmetric spaces and buildings. We explain the notions which are most important for the purposes of this paper, establish notation and give proofs for some of the less standard facts. No attempt of a complete review is made. For more detailed discussions, we refer the reader to [@Eberlein], [@BGS], [@qirigid] and [@habil].

We give a brief description of where various parts of this section are used in the paper:

Sections \[sec:spherical\]-\[sec:stars\] are used essentially everywhere.

While the vector valued distance function $d_{\De}$ is used in many places in the paper, the rest of the material in sections \[sec:VVD\] and \[sec:refined\] is used primarily in section \[sec:sepnest\] on the separation of nested Weyl cones and in section \[sec:mqg\] where we analyze projections of Morse quasigeodesics to the euclidean model Weyl chamber $\De$.

The material of section \[sec:ascones-shadows\] dealing with shadows at infinity is used in section \[sec:reg-con\] when we prove the equivalence of [*regularity*]{} and [*contractivity*]{} for sequences of isometries of $X$. The main result of section \[sec:sepnest\] on the separation of nested Weyl cones is used in section \[sec:asyemb\] to prove that Morse subgroups are URU (Theorem \[thm:asymbur\]).

The main results of sections 