On the $z$-classes of Palindromic automorphisms of Free Groups

TL;DR

The paper investigates $z$-classes in the palindromic automorphism group $PiA(F_n)$ of a free group and connects them to centralizer structures via the abelianization map $ ext{psi}: Aut(F_n) o GL_n(Z)$. It proves that the image $ ext{psi}(PiA(F_n))$ is a parity-constrained subgroup of $GL_n(Z)$ and uses a short exact sequence involving the palindromic Torelli group to realize these images; this underpins the identification of $z$-classes with centralizer conjugacy types. The authors establish that $PiA(F_n)$ has infinitely many $z$-classes for all $n geq 3$, by constructing infinite families of abelianization matrices with pairwise non-conjugate centralizers and extending the argument to general $n$ via block-diagonal constructions. They also provide a partial classification of conjugacy for psi-images of reducible palindromic automorphisms of $F_3$, offering explicit invariants and non-conjugate examples. These results advance understanding of dynamical types in palindromic automorphism groups and their linear representations.

Abstract

The palindromic automorphism group is a subgroup of the automorphism group $Aut(F_3).$ We establish a necessary and sufficient condition for a matrix in $GL_n(\mathbb{Z})$ representing a palindromic automorphism of $F_n.$ We prove that the number of the $z$-classes in $ΠA(F_n)$ is infinite. We further classify the conjugacy classes of the reducible palindromic automorphisms.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Corollary 3.4
• proof