Palindromic Automorphisms of Free Groups

Abstract

Let $F_n$ be the free group of rank $n$ with free basis $X=\{x_1,\dots,x_n \}$. A palindrome is a word in $X^{\pm 1}$ that reads the same backwards as forwards. The palindromic automorphism group $ΠA_n$ of $F_n$ consists of those automorphisms that map each $x_i$ to a palindrome. In this paper, we investigate linear representations of $ΠA_n$, and prove that $ΠA_2$ is linear. We obtain conjugacy classes of involutions in $ΠA_2$, and investigate residual nilpotency of $ΠA_n$ and some of its subgroups. Let $IA_n$ be the group of those automorphisms of $F_n$ that act trivially on the abelianisation, $P I_n$ be the palindromic Torelli group of $F_n$, and $E ΠA_n$ be the elementary palindromic automorphism group of $F_n$. We prove that $PI_n=IA_n \cap E ΠA_n'$. This result strengthens a recent result of Fullarton.