On partial hyperconvex surface group representations

Convex cocompact groups in rank one Lie groups are geometrically significant and well studied discrete subgroups, including the special case of representations in Teichmueller space. In 2006 Labourie introduced Anosov representations as a generalization of convex cocompactness to higher rank Lie groups, with many of its properties
still valid; leading also to analogues of Teichmueller space in higher rank, so called higher Teichmueller spaces.
When restricting to surface groups in rank one groups one can distinguish representations in Teichmueller space and general convex cocompact ones via several properties, e.g. regularity of the boundary map or behavior of length functions. This however does not directly generalize to higher rank. Namely in this talk I want to discuss a class
of Anosov representations that are not in higher Teichmueller spaces, but in many ways are very similar to representations in the latter.
Based on joint work with B. Pozzetti.