Palindromic automorphisms of free groups

Abstract

Abstract. Let Fn be the free group of rank n with free basis X = {x1, . . . , xn}. A palindrome is a word in X ±1 that reads the same backwards as forwards. The palindromic automorphism group ΠAn of Fn consists of those automorphisms that map each xi to a palindrome. In this paper, we investigate linear representations of ΠAn, and prove that ΠA2 is linear. We obtain conjugacy classes of involutions in ΠA2, and investigate residual nilpotency of ΠAn and some of its subgroups. Let IAn be the group of those automorphisms of Fn that act trivially on the abelianisation, P In be the palindromic Torelli group of Fn, and EΠAn be the elementary palindromic automorphism group of Fn. We prove that P In = IAn ∩ EΠA ′ n . This result strengthens a recent result of Fullarton [3].

Palindromic automorphisms of free nilpotent groups

“…It can be seen that, if a central automorphism ϕ of N n,3 given by 3 and 1 ≤ i ≤ n is tame, then ∂ 1 w 1 + · · · + ∂ n w n ≡ 0 (mod R) ,…”

“…The papers [11] and [10] indicates a deep connection between palindromic automorphisms of free groups and geometry. Recently, Bardakov, Gongopadhyay and Singh [3] investigated many algebraic properties of ΠA(F n ). In particular, they obtained conjugacy classes of involutions in ΠA(F 2 ) and investigated residual nilpotency of ΠA(F n ).…”