Asymptotic Dynamics on Character Varieties over Finite Fields

TL;DR

The paper analyzes the arithmetic dynamics of the outer automorphism group $\mathrm{Out}(\mathbb{Z}^r)$ acting on the $\mathbb{F}_q$-points of the $\mathrm{SL}_n$-character variety of $\mathbb{Z}^r$ for $n=2,3$. By stratifying diagonalizable tuples and counting points through Weyl-group orbits, it derives explicit $E$-polynomials for $\mathfrak{X}_{\mathbb{Z}^r}(\mathrm{SL}_n(\mathbb{C}))$ via Katz’s polynomial-count theorem, and establishes that the action is not asymptotically transitive: the maximal orbit size grows with leading coefficient at most $\tfrac{1}{2}$ of the total number of points as $q\to\infty$. For $n=2$, the total count yields $E$-polynomial $E=\frac{(q+1)^r+(q-1)^r}{2}$; for $n=3$, one obtains $E=\frac{(q-1)^{2r}}{6}+\frac{(q^2-1)^r}{2}+\frac{(q^2+q+1)^r}{3}$. The results extend the understanding of arithmetic dynamics on character varieties and suggest analogous analyses for higher $n$ and free groups, with potential implications for relating finite-field counts to topological invariants in characteristic zero.

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.

Theorems & Definitions (54)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Definition 1
• Definition 2
• Lemma 4
• proof
• Remark 1
• Lemma 5