# Quantitative properties of convex representations

### Andrés Sambarino

Université Paris-Sud, Orsay, France

## 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.

## Cite this article

Andrés Sambarino, Quantitative properties of convex representations. Comment. Math. Helv. 89 (2014), no. 2, pp. 443–488

DOI 10.4171/CMH/324