asymptotic embeddedness, we keep the word hyperbolicity of the subgroup but, instead of identifying its Gromov boundary with the limit set as in asymptotic embeddedness, we only require a boundary map embedding the Gromov boundary into the flag manifold. Compared to CEA, we require a stronger form of expansivity, now at the image of the boundary map.

We call a subgroup $\Ga<G$ [*$\taumod$-boundary embedded*]{} if $\Ga$ is word hyperbolic and there exists a $\Ga$-equivariant continuous embedding $$\beta:\geo\Ga\lra\Flagt$$ sending distinct visual boundary points to antipodal simplices. If $\Ga$ is virtually cyclic, we require in addition that it is discrete in $G$. (Otherwise, discreteness is a consequence.) We will refer to $\beta$ as a [*boundary embedding*]{}. In general, boundary embeddings are not unique.

The [*infinitesimal expansion factor*]{} of an element $g\in G$ at a simplex $\tau\in\Flagt$ is $$\eps(g, \tau)= \min_u |dg(u)|$$ where the minimum is taken over all unit tangent vectors $u\in T_\tau \Flagt$, again using the Riemannian background metric.

Now we can formulate our version of the Anosov condition:

We say that a subgroup $\Ga<G$ is [*$\taumod$-Anosov*]{} if it is $\taumod$-boundary embedded with boundary embedding $\beta$ and satisfies the following expansivity condition: For every ideal point $\zeta\in \geo \Ga$ and every 