Commentarii Mathematici Helvetici

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Volume 89, Issue 2, 2014, pp. 443–488
DOI: 10.4171/CMH/324

Published online: 2014-06-18

Quantitative properties of convex representations

Andrés Sambarino[1]

(1) Université Paris-Sud, Orsay, France

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.

Keywords: Lie groups, Hitchin representations, higher rank geometry

Sambarino Andrés: Quantitative properties of convex representations. Comment. Math. Helv. 89 (2014), 443-488. doi: 10.4171/CMH/324