Hyperconvex representations and exponential growth

Let G be a real algebraic semi-simple Lie group and Γ be the fundamental group of a closed negatively curved manifold. In this article we study the limit cone, introduced by Benoist [Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7 (1) (1997), 1–47], and the growth indicator function, introduced by Quint [Divergence exponentielle des sous-groupes discrets en rang supérieur. Comment. Math. Helv. 77 (2002), 503–608], for a class of representations ρ : Γ → G admitting an equivariant map from ∂ Γ to the Furstenberg boundary of the symmetric space of $G, $ together with a transversality condition. We then study how these objects vary with the representation.