A palindromization map for the free group

TL;DR

This paper introduces a new self-map on the free group on two generators, connecting automorphisms, braid groups, and palindromic structures, with properties like continuity and specific conjugacy relations.

Contribution

It defines a novel palindromization map for the free group using braid group automorphisms, linking palindromes, conjugacy, and topological properties.

Findings

Map Pal produces palindromes in F_2

Pal is continuous in the profinite topology

Values of Pal relate to conjugacy of abg and bag

Abstract

We define a self-map Pal: F_2 --> F_2 of the free group on two generators a, b, using automorphisms of F_2 that form a group isomorphic to the braid group B_3. The map Pal restricts to de Luca's right iterated palindromic closure on the submonoid generated by a, b, and is continuous for the profinite topology on F_2. The values of Pal are palindromes and coincide with the elements g of F_2 such that abg is conjugate to bag.