Georgia Institute of Technology College of Sciences

I'll talk about the structure of the collection of all n-ples of eigenvalues of elements of Zariski-dense subgroups D of SL(n,R). Subgroups like this appear, for instance, as the images of holonomy representations of geometric structures. Our focus is a deep and useful result of Benoist, which states that the natural cone one is led to consider here has strong convexity and non-degeneracy properties, and a succinct, qualitative characterization of the cones that so arise from Zariski-dense subgroups. The theorem comes out of a study of the dynamics of the actions of D on spaces of flags such as RP^n and the collection of open subsemigroups of SL(n,R). Everything in this talk is from Benoist’s paper Propriétés Asymptotiques des Groupes Linéaires (GAFA, 2002), and holds in far more generality than I'll state.