with “diamonds” in a higher rank version of the Morse property. (This is suggested, for instance, by the geometry of free Anosov subgroups, see our examples of Morse-Schottky subgroups [@morse; @manicures].) We define diamonds as follows: If $\taumod=\simod$ and $xy$ is a $\simod$-regular segment, then the [*$\simod$-diamond*]{} with tips $x,y$ is the intersection $$\diamo(x,y) = V(x,\si)\cap V(y,\hat\si)$$ of the euclidean Weyl chambers with tips at $x$ and $y$ containing $xy$. In the case of general $\taumod$, the euclidean Weyl chambers are replaced with $\taumod$-Weyl cones (see section \[sec:diamo\]).

We say that a subgroup $\Ga<G$ is [*$\taumod$-Morse*]{} if it is $\taumod$-regular, $\Ga$ is word hyperbolic and an(y) orbit map $o_x:\Ga\to\Ga x\subset X$ satisfies the following Morse condition: The images $o_x\circ s$ of discrete geodesic segments $s:[n_-,n_+]\cap\Z\to\Ga$ are contained in uniform tubular neighborhoods of $\taumod$-diamonds with tips uniformly close to the endpoints of $o_x\circ s$ (see Definition \[def:mrs\]).

The definition does not a priori assume undistortion, but we show in this paper that Morse implies URU. That, conversely, URU implies Morse may seem unexpected at first but follows from our Higher Rank Morse Lemma for regular quasigeodesics [@mlem].

We now arrive at our main result on the equivalence of various conditions introduced 