Palindromic Automorphisms of Free Groups

TL;DR

This paper explores the structure and properties of the palindromic automorphism group of free groups, proving linearity for rank 2, analyzing involutions, residual nilpotency, and subgroup relations.

Contribution

It establishes the linearity of the palindromic automorphism group for rank 2 and clarifies subgroup intersections, extending recent results in the field.

Findings

Proves $ ext{Pi}_A_2$ is linear.

Classifies conjugacy classes of involutions in $ ext{Pi}_A_2$.

Shows residual nilpotency of $ ext{Pi}_A_n$ and related subgroups.

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 $IA_n$ 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 $E \Pi A_n$ be the elementary palindromic automorphism group of $F_n$. We prove that $PI_n=IA_n \cap E \Pi A_n'$. This result strengthens a recent result of Fullarton.