normalized (by $r(0)=e\in \Ga$) discrete geodesic ray $r: \N\to \Ga$ asymptotic to $\zeta$, the action $\Ga\acts\Flagt$ satisfies $$\eps(r(n)^{-1}, \beta(\zeta))\ge A e^{Cn}$$ for $n\geq 0$ with constants $A, C>0$ independent of $r$. (Here, we fix a word metric on $\Ga$.)

The uniformity of expansion in this definition can be significantly weakened:

We say that a subgroup $\Ga<G$ is [*non-uniformly $\taumod$-Anosov*]{} if it is $\taumod$-boundary embedded with boundary embedding $\beta$ and, for every ideal point $\zeta\in \geo \Ga$ and every discrete geodesic ray $r: \N\to \Ga$ asymptotic to $\zeta$, the action $\Ga\acts\Flagt$ satisfies $$\sup_{n\in\N}\, \eps(r(n)^{-1}, \beta(\zeta))=+\infty.$$

The original definition of Anosov subgroups in [@Labourie; @GW] is rather involved. It is based on geodesic flows for word hyperbolic groups and formulated in terms of expansion/contraction properties for lifted flows on associated bundles over the geodesic flow spaces (see section \[sec:onos\]). Our definition requires only an expansion property for the group action on a suitable flag manifold and avoids using the geodesic flow, whose construction is highly technical for word hyperbolic groups which do not arise as the fundamental group of a closed negatively curved Riemannian manifold. The geodesic flow is replaced by a simpler coarse geometric object, the space of quasigeodesics.

Now we 