On the almost-palindromic width of free groups

TL;DR

This paper proves that in non-abelian free groups, there is no finite bound on the number of almost-palindromes needed to represent every element, answering a question about the structure of free groups.

Contribution

It demonstrates that no finite pair of parameters can bound the almost-palindromic width in free groups, extending the result to all non-abelian free groups.

Findings

No such pair (c, m) exists for free groups.

The result applies to all non-abelian free groups.

Almost-palindromic width is unbounded in free groups.

Abstract

We answer a question of Bardakov (Kourovka Notebook, Problem 19.8) which asks for the existence of a pair of natural numbers $(c, m)$ with the property that every element in the free group on the two-element set $\{a, b\}$ can be represented as a concatenation of $c$, or fewer, $m$-almost-palindromes in letters $a^{\pm 1}, b^{\pm 1}$. Here, an $m$-almost-palindrome is a word which can be obtained from a palindrome by changing at most $m$ letters. We show that no such pair $(c, m)$ exists. In fact, we show that the analogous result holds for all non-abelian free groups.