Quantitative properties of convex representations

Abstract

Let \Gamma be a discrete subgroup of \mathrm{PGL}(d,\mathbb R) . Fix a norm \|\ \| on \mathbb R^d and 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 \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.