Quantitative properties of convex representations

Let Γ be a discrete subgroup of PGL(d, R) . Fix a norm \|\ \| on R^d and let NΓ(t) be the number of elements in Γ whose operator norm is ≤ t . In this article we prove an asymptotic for the growth of NΓ(t) when t→∞ for a class of Γ ’s which contains, in particular, Hitchin representations of surface groups and groups dividing a convex set of P( R^d) . We also prove analogue counting theorems for the growth of the spectral radii. More precise information is given for Hitchin representations.