Slack calculus, quasi-laminations, and the limit cone for positive representations

We study the limit cone of a positive representation of a surface group into a real-split, semi-simple Lie group. We focus on the problem of identifying which curves and geodesic currents are able to find the boundary. We show that for a typical boundary face, the curves and currents mapping to that face are supported on a sub-flow of low complexity that we call a quasi-lamination. It is analogous to the maximally stretched lamination appearing in the story of Thurston’s asymmetric metric. Joint work with Fran\c{c}ois Gu\’eritaud and Fanny Kassel.