On the almost-palindromic width of free groups (Q6612173)

If \(B\) is a set, then \(F_B\) is the free group on \(B\), \(W_B\) is the free monoid on the set \(B^{\pm 1}\). There is the homomorphism of monoids \(r_B \colon W_B \to F_B\) which sends a word to its corresponding reduced word and which is left-inverse to the inclusion of \(F_B\) into \(W_B\). For a word \(w \in W_B\), let \(rev(w)\) be its reverse word, that is, the word given by the letters of \(w\) in reverse order. A palindrome is a word \(p \in W_B\) with the property that \(p = rev(p)\). An \(m\)-almost-palindrome is a word which differs from a palindrome by a change of at most \(m\) letters. Thus, the palindromes are exactly the \(0\)-almost-palindromes. If \(P_{B,m}\) is the set of all \(m\)-almost-palindromes, then the image \(r_B(P_{B,m})\) is a generating set for the free group \(F_B\), for every \(m \geq 0\).\N\NIf \(F\) is a free group, together with a basis \(B\) of \(F\), then there is the canonical isomorphism \(\phi_B \colon F_B \to F\). Furthermore, note that for all \(m\) the set \(X_{B,m} = \phi_B(r_B(P_{B,m}))\) is a generating set for \(F\). The \(m\)-almost-palindromic width of \(F\) is the width of \(F\) with respect to \(X_{B,m}\) that is a minimal natural number \(c\) such that every element in \(F\) can be presented as a concatenation of \(c\), or fewer, \(m\)-almost-palindromes in \(X_{B,m}\) If such \(c\) does not exists, then the width is infinite.\N\N\textit{V. Bardakov} et al. [J. Algebra 285, No. 2, 574--585 (2005; Zbl 1085.20011)] show that, for a non-abelian free group \(F\) the \(0\)-almost-palindromic width (that is, the palindromic width) of \(F\) is infinite. Extending on this, \textit{Bardakov} [The Kourovka Notebook, no. 19, 2018, Problem 19.8] asked whether there is an \(m \in \mathbb{N}\) for which the \(m\)-almost palindromic width of \(F\) is finite, assuming that \(F\) is a free group of rank two. Of course, the same question can be asked without the restriction on the rank of \(F\). The author of this note answers this question negatively for all non-abelian free groups. He proves that if \(F\) is a free non-abelian group, then for all \(m \geq 0\), the \(m\)-almost palindromic width of \(F\) is infinite.