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 $\Pi A_n$ of $F_n$ consists of those automorphisms that map each $x_i$ to a palindrome. In this paper, we investigate linear representations of $\Pi A_n$, and prove that $\Pi A_2$ is linear. We obtain conjugacy classes of involutions in $\Pi A_2$, and investigate residual nilpotency of $\Pi A_n$ and some of its subgroups. Let $I{A}_n$ 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 $E\Pi A_n$ be the elementary palindromic automorphism group of $F_n$. We prove that $P{I}_n=I{A}_n\cap E\Pi A_n^{\prime }$. This result strengthens a recent result of Fullarton.

19 pages

References in corpus (1)

• A generating set for the palindromic Torelli group

Cited by in corpus (4)

• The Birman-Hilden theory
• A generating set for the palindromic Torelli group
• Palindromic widths of graph of groups
• Palindromic Automorphisms of Free Nilpotent Groups