Asymptotic Dynamics on Character Varieties over Finite Fields

In this paper, we prove the lack of asymptotic transitivity of the outer automorphism group action of ${\mathbb{Z}}^r$ on ${\mathrm{S}\mathrm{L}}_n({\mathbb{F}}_q)$-character varieties of ${\mathbb{Z}}^r$ for $n=2,3$ and $r\ge 2$. Along the way, we stratify the character varieties and compute the $E$-polynomial, also known as the Hodge-Deligne polynomial or Serre polynomial, of these character varieties.