Consider a surface group representation into a higher rank semi-simple Lie group G. The Benoist limit cone of such a representation is the closure of the set of rays spanned by the Jordan projections of all elements of the representation. It is a closed, (typically) convex cone of the positive Weyl chamber. For Fuchsian representations, i.e. those factoring through PSL2(R), this cone is just one ray. So in general, the thickness and shape of the Benoist limit cone is a natural indication of how far a representation is from being Fuchsian. We will focus on the basic problem of determining which elements (more generally, geodesic currents) may have Jordan projection on the boundary of the Benoist cone, or phrased another way, which directions in the group see the most extreme eigenvalue behavior. We focus on the cases of Hitchin representations into the rank two groups G = PSL2(R) x PSL2(R) and G = SL3R. This talk will contain more questions than answers.
