above. We state it for nonelementary subgroups because we use this assumption in some of our proofs.

\[thm:eqvi\] The following properties for subgroups $\Ga<G$ are equivalent in the nonelementary case:

\(i) $\taumod$-asymptotically embedded

\(ii) $\taumod$-CEA.

\(iii) $\taumod$-Anosov

\(iv) non-uniformly $\taumod$-Anosov

\(v) $\taumod$-RCA

\(vi) $\taumod$-Morse

These properties imply $\taumod$-URU.

Moreover, the boundary maps for properties (i), (iii) and (iv) coincide.

Here, “nonelementary” means $|\geo\Ga|\geq3$ in the Anosov conditions (iii) and (iv), which assume word hyperbolicity but no $\taumod$-regularity, and means $|\LatGa|\geq3$ in all other cases.

\(i) We prove in [@mlem] that, conversely, $\taumod$-URU implies $\taumod$-Morse (without assuming nonelementary).

\(ii) All implications between properties (i)-(vi) hold without assuming nonelementary, with the exception of (ii)$\Rightarrow$(v)$\Rightarrow$(i). In particular, the properties (i),(iii),(iv),(vi) and $\taumod$-URU are equivalent in general.

\(iii) The implication Anosov$\Rightarrow$URU had been known before [@GW].

\(iv) Some of the implications in the theorem can be regarded as a description of geometric and dynamical properties of Anosov subgroups. Different characterizations of Anosov subgroups are useful in different contexts. For example: Expansivity (ii) is used in [@coco13; @coco15] to establish the cocompactness of $\Ga$-actions on suitable domains of discontinuity in flag manifolds. Asymptotic embeddedness is used in [@bordif] to construct Finsler compactifications of locally symmetric 