Hyperconvex representations and exponential growth

Abstract

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