Exponential mixing and counting conjugacy classes for Anosov subgroups

The celebrated prime geodesic theorem gives the asymptotic formula for the number of primitive closed geodesics according to their length in a closed hyperbolic manifold, i.e., for uniform lattices in SO(n, 1). What’s more, the error term is power-saving. This was generalized with the error term for convex cocompact subgroups by Naud and Stoyanov. It is natural to wonder whether one can generalize this further to the higher rank setting such as SL(n, R). This was done without the error term for Anosov subgroups by Sambarino. In keeping with a principle from Margulis's thesis, we go further and establish exponential mixing of an appropriate dynamical system and use that to produce a power-saving error term.