come to the geometric notions.

4.  The first geometric condition concerns the [*orbit asymptotics*]{}. The notion of conicality of limit simplices, due to Albuquerque [@Albuquerque Def. 5.2], generalizes a well-known condition from the theory of Kleinian groups: In the case $\taumod=\simod$, a limit chamber $\si\in\LasGa$ of a $\simod$-regular subgroup $\Ga<G$ is called [*conical*]{} if there exists a sequence $\ga_n\to\infty$ in $\Ga$ such that for a(ny) point $x\in X$ the sequence of orbit points $\ga_nx$ is contained in a tubular neighborhood of the euclidean Weyl chamber $V(x,\si)$ with tip $x$ and asymptotic to $\si$. For general $\taumod$ and limit simplices $\tau\in\LatGa$ of $\taumod$-regular subgroups $\Ga<G$, one replaces the euclidean Weyl chamber with the Weyl cone $V(x,\st(\tau))$ over the star of $\tau$, that is, by the union of the euclidean Weyl chambers $V(x,\si)$ for all spherical Weyl chambers $\si\supset\tau$. A $\taumod$-regular subgroup $\Ga<G$ is called [*conical*]{} if all limit simplices are conical. Here is our forth condition:

We say that a subgroup $\Ga<G$ is [*$\taumod$-RCA*]{} if it is $\taumod$-regular, conical and antipodal.

For nonelementary $\taumod$-regular antipodal subgroups, this extrinsic notion of conicality is equivalent to an intrinsic one defined in terms of the dynamics on the $\taumod$-limit set (Proposition \[prop:concon\]), which enables 