one to relate it to the dynamical notions above.

5.  The last set of definitions concerns the [*coarse extrinsic geometry*]{}. We recall that a finitely generated subgroup $\Ga<G$ is [*undistorted*]{} if the orbit maps $\Ga\to X$ are quasiisometric embeddings. They then send discrete geodesics in $\Ga$ (with respect to a fixed word metric) to uniform quasigeodesics in $X$. Undistortion by itself is too weak a restriction, compared with the other notions defined previously. We will strengthen it in two ways. The first is by adding uniform regularity:

We say that a subgroup $\Ga<G$ is [*$\taumod$-URU*]{} if it is uniformly $\taumod$-regular and undistorted.

According to the classical [*Morse Lemma*]{} in negative curvature, quasigeodesic segments in rank one symmetric spaces are uniformly Hausdorff close to geodesic segments with the same endpoints. This is no longer true in higher rank because it fails already in euclidean plane. Another way of strengthening undistortion is therefore by imposing a “Morse” type property on the quasigeodesics arising as orbit map images of the discrete geodesics in $\Ga$.

As in the case of conicality above, where one replaces rays with Weyl cones when passing from rank one to higher rank, it is natural to replace geodesic segments 