Quantitative properties of convex representations

Abstract

Let $\Gamma$ be a discrete subgroup of $\mathrm{PGL}(d,\mathbb{R})$ . Fix a norm $\Vert \Vert$ on ${\mathbb{R}}^d$ and let $N_{\Gamma }(t)$ be the number of elements in $\Gamma$ whose operator norm is $\le t$. 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 $\mathbb{P}({\mathbb{R}}^d)$. We also prove analogue counting theorems for the growth of the spectral radii. More precise information is given for Hitchin representations.