Quantitative properties of convex representations

Let $Γ$ be a discrete subgroup of $\textrm{PGL}(d,\R)$ and fix some euclidean norm $\|\ \|$ on $\R^d.$ Let $N_Γ(t)$ be the number of elements in $Γ$ whose operator norm is $\leq t.$ In this article we prove an asymptotic for the growth of $N_Γ(t)$ when $t\to\infty$ for a class of $Γ$'s which contains, in particular, Hitchin representations of surface groups and groups dividing a convex set of $¶(\R^d).$ We also prove analogue counting theorems for the growth of the spectral radii. More precise information is given for Hitchin representations.

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