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

TL;DR

This paper investigates the structure of palindromic automorphisms of free groups, providing conditions for their matrix representations, proving the infinitude of their $z$-classes, and classifying conjugacy classes of reducible automorphisms.

Contribution

It establishes a necessary and sufficient condition for matrices representing palindromic automorphisms and classifies conjugacy classes, revealing the infinite nature of $z$-classes.

Findings

Number of $z$-classes in $$-classes is infinite.

Provides a matrix criterion for palindromic automorphisms.

Classifies conjugacy classes of reducible automorphisms.

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 $\Pi A(F_n)$ is infinite. We further classify the conjugacy classes of the reducible palindromic automorphisms.