Quantitative properties of convex representations

TL;DR

This paper establishes asymptotic growth rates for the number of elements in certain discrete subgroups of projective linear groups, including Hitchin representations, based on their operator norms and spectral radii.

Contribution

It provides new asymptotic formulas for counting elements in classes of convex representations, especially Hitchin representations, expanding understanding of their geometric and spectral properties.

Findings

Asymptotic growth formulas for $N_ ext{Gamma}(t)$ as $t o obreak \infty$

Counting theorems for spectral radii of group elements

More detailed results for Hitchin representations

Abstract

Let $\Gamma$ be a discrete subgroup of $\textrm{PGL}(d,\R)$ and fix some euclidean norm $\|\ \|$ on $\R^d.$ 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 $\P(\R^d).$ We also prove analogue counting theorems for the growth of the spectral radii. More precise information is given for Hitchin representations.