our setting of discrete subgroups of semisimple Lie groups. Also our methods are rather different. We give here a geometric treatment and present the material in a form suitable to serve as a basis for the further development of our theory of discrete isometry groups acting on Riemannian symmetric spaces and euclidean buildings of higher rank, such as in our papers [@coco15; @morse; @mlem; @bordif].

We now (mostly) restrict to the class of $\taumod$-regular, equivalently, $\taumod$-convergence subgroups and introduce various geometric and dynamical conditions in the spirit of geometric finiteness. We begin with three dynamical ones:

1.  We say that a subgroup $\Ga<G$ is [*$\taumod$-asymptotically embedded*]{} if it is an antipodal $\taumod$-convergence subgroup, $\Ga$ is word hyperbolic and there exists a $\Ga$-equivariant homeomorphism $$\label{eq:mapalphatauintro}
\alpha: \geo \Gamma \stackrel{\cong}{\lra}
\LatGa\subset \Flagt$$ from its Gromov boundary onto its $\taumod$-limit set.

This condition can be understood as a continuity at infinity property for the orbit maps $o_x:\Ga\to\Ga x\subset X$: By extending an orbit map $o_x$ to infinity by the boundary map $\al$, one obtains a continuous map $$o_x\sqcup\al:\Ga\sqcup\geo\Ga\lra X\sqcup\Flagt$$ from the Gromov compactification of $\Ga$ (see Proposition \[prop:asyembprp\]).

2.  Our next condition is inspired by Sullivan’s notion of [*expanding actions*]{} [@Sullivan]. Following Sullivan, 