\[sec:horocycles\] and \[sec:contraction\] are Theorem \[thm:expand\] and Proposition \[prop:expand\] establishing estimates for the contraction and expansion of isometries of $X$ acting on flag manifolds. (The other results are only used only in sections \[sec:horocycles\] and \[sec:contraction\]). Theorem \[thm:expand\] and Proposition \[prop:expand\] are used in sections \[sec:expa\] and \[sec:anosov\] while discussing discrete subgroups satisfying expansion properties (CEA and Anosov).

The material of section \[sec:fins\] is used only in section \[sec:mqg\] where it is proven that Morse quasigeodesics are uniformly closed to Finsler geodesics and that $\De$-distance projections of Finsler geodesics are again Finsler geodesics.

General metric space notation {#sec:general}
-----------------------------

We will use the notation $B(p,r)$ for the open $r$-ball with center $p$ in a metric space, and $\ol B(p,r)$ for the closed $r$-ball.

A [*geodesic*]{} in a metric space $(Z,d)$ is an isometric embedding $I\to Z$ from a (possibly infinite) interval $I\subset\R$. In the context of finitely generated groups equipped with word metrics, we will also work with [*discrete geodesics*]{}; these are isometric embeddings from intervals $I\cap\Z$ in $\Z$. The notion of [*discrete quasigeodesic*]{} will be used similarly.

Spherical buildings {#sec:spherical}
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Spherical buildings occur in this paper as the visual boundaries of symmetric spaces of noncompact type, equipped with 