spaces for Anosov subgroups. The Morse property is used in [@morse] to prove a [*local-to-global principle*]{} for Anosov subgroups. The latter in turn leads to new proofs of openness and structural stability of Anosov representations, to a construction of free Anosov subgroups (Morse-Schottky subgroups), and to the semidecidability of Anosovness, see [@morse].

\(v) In our paper [@bordif] we establish two more characterizations of Anosov subgroups among uniformly regular subgroups, namely as [*coarse retracts*]{} and by [*$S$-cocompactness*]{}. The former property is a strengthening of undistortion. The latter means the existence of a certain kind of compactification of the corresponding locally symmetric space.

\(vi) Other characterizations of Anosov subgroups can be found in [@GGKW].

Boundary embeddedness appears to be a considerable weakening of asymptotic embeddedness, even in the regular case. Nevertheless two results in this paper establish a close relation between the two concepts:

\(i) For $\simod$-regular subgroups, boundary embeddedness, conversely, implies asymptotic embeddedness, while the boundary embedding may have to be modified (see Theorem \[thm:bemmdsi\]).

\(ii) For general $\taumod$-regular subgroups, there is the following [*dichotomy*]{} for boundary embeddings (see Theorem \[thm:bemblim\]) which is useful for verifying asymptotic embeddedness:

Either the image of the boundary embedding equals the $\taumod$-limit set and the 