we call a subgroup $\Ga< G$ [*expanding at infinity*]{} if its action on the appropriate partial flag manifold is expanding at the limit set. More precisely:

We call a $\taumod$-convergence subgroup $\Ga< G$ [*$\taumod$-expanding at the limit set*]{} if for every limit flag in $\LatGa$ there exists a neighborhood $U$ in $\Flagt$ and an element $\ga\in\Ga$ which is uniformly expanding on $U$, i.e. for some constant $c>1$ and all $\tau_1,\tau_2\in U$ it holds that $$d(\ga\tau_1,\ga\tau_2)\geq c\cdot d(\tau_1,\tau_2) .$$ Here, and in what follows the distance $d$ is induced by a fixed Riemannian background metric on the flag manifold.

Now we can formulate our second condition:

We say that a subgroup $\Ga <G$ is [*$\taumod$-CEA*]{} (Convergence Expanding Antipodal) if it is an antipodal $\taumod$-convergence subgroup which is expanding at the limit set.

We note that the CEA condition does not a priori assume word hyperbolicity, not even finite generation.

3.  The next condition is motivated by the original definition of Anosov subgroups. It is a hybrid of the previous two definitions, where we weaken asymptotic embeddedness (to boundary embeddedness) and strengthen expansivity. We drop the regularity/convergence assumption and, accordingly, make no use of the limit set in our definition. Compared to 