Robust quasi-isometric embeddings and limits of Anosov representations

Anosov representations, introduced by Labourie and further generalized by Guichard-Wienhard, form a large class of discrete and stable hyperbolic subgroups of semisimple Lie groups, generalizing classical convex cocompact subgroups of rank one Lie groups. In this talk, we will provide a construction of stable, non-rigid, quasi-isometrically embedded hyperbolic subgroups of SL(n,C) (n sufficiently large) which are not limits of Anosov representations, showing that analogues of Sullivan's structural stability theorem and the density theorem of Kleinian groups fail for Anosov representations in higher rank Lie groups.