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

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.