subgroup is asymptotically embedded. Or the image is disjoint from the limit set, and the limit set is not Zariski dense. The latter cannot happen for Zariski dense subgroups.

While the main results in this paper concern discrete subgroups of Lie groups, in section \[sec:mqg\], motivated by the Morse property, we discuss [*Morse quasigeodesics*]{} and [*Finsler geodesics*]{}. We characterize Morse subgroups as word hyperbolic subgroups whose intrinsic geodesics are extrinsically uniform Morse quasigeodesics. Furthermore, we characterize Morse quasigeodesics as bounded perturbations of Finsler geodesics. Lastly, we analyze the $\De$-distance along Finsler geodesics and Morse quasigeodesics. We show that, via the $\De$-distance function, they project to Finsler geodesics and Morse quasigeodesics in $\De$.

Most of the results in this paper were already contained in chapters 1-6 of the preprint [@morse], however the presentation in this paper is more efficient. The further material on the Morse property in [@morse §7] will appear elsewhere.

[**Acknowledgements.**]{} The first author was partly supported by the NSF grants DMS-12-05312 and DMS-16-04241. He is also thankful to KIAS (the Korea Institute for Advanced Study) for its hospitality. The third author was partially supported by grant FEDER/Mineco MTM2015-66165-P.

Geometry of symmetric spaces {#sec:prelim}
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In this section, we 