Quantitative properties of convex representations

Abstract

Let $\Gamma$ be a discrete subgroup of $\textrm{PGL}(d,\R)$ and fix some euclidean norm $\|\ \|$ on $\R^d.$ Let $N_\Gamma(t)$ be the number of elements in $\Gamma$ whose operator norm is $\leq t.$ In this article we prove an asymptotic for the growth of $N_\Gamma(t)$ when $t\to\infty$ for a class of $\Gamma$'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.