On the almost palindromic width of certain free constructions of groups

TL;DR

This paper establishes a structural criterion demonstrating that certain free group constructions, like HNN extensions and free products, generally have infinite almost palindromic width, with specific exceptions such as the infinite dihedral group.

Contribution

It introduces a general criterion for infinite almost palindromic width and applies it to HNN extensions and free products, extending previous results.

Findings

HNN extensions have infinite m-almost palindromic width

Free products have infinite m-almost palindromic width

The infinite dihedral group is the unique exception among free products

Abstract

We provide a general structural criterion implying that a group has infinite $m$-almost palindromic width. In particular, we prove that both HNN extensions and free products exhibit infinite $m$-almost palindromic width, with the unique exception of the infinite dihedral group among free products. This framework extends and strengthens the results of \cite{MS} and \cite{GK}.