Asymptotic Dynamics on Character Varieties over Finite Fields

TL;DR

This paper investigates the behavior of the outer automorphism group action on character varieties over finite fields, revealing lack of asymptotic transitivity and providing detailed polynomial computations.

Contribution

It demonstrates the non-transitivity of the group action on certain character varieties and computes their E-polynomials, advancing understanding of their geometric structure.

Findings

Proves lack of asymptotic transitivity for specific character varieties.

Stratifies the character varieties to analyze their structure.

Computes the E-polynomial of these varieties.

Abstract

In this paper, we prove the lack of asymptotic transitivity of the outer automorphism group action of $\mathbb{Z}^r$ on $\mathrm{SL}_n(\mathbb{F}_q)$-character varieties of $\mathbb{Z}^r$ for $n=2,3$ and $r\geq 2$. 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.