Quantitative properties of convex representations

  • Andrés Sambarino

    Université Paris-Sud, Orsay, France


Let Γ\Gamma be a discrete subgroup of PGL(d,R)\mathrm{PGL}(d,\mathbb R) . Fix a norm  \|\ \| on Rd\mathbb R^d and let NΓ(t)N_{\Gamma}(t) be the number of elements in Γ\Gamma whose operator norm is t\leq t. In this article we prove an asymptotic for the growth of NΓ(t)N_{\Gamma}(t) when tt\to\infty for a class of Γ\Gamma’s which contains, in particular, Hitchin representations of surface groups and groups dividing a convex set of P(Rd)\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