Small palindromic lengths in free groups and word equations with antimorphisms

Abstract

The palindromic length of a finite word $w$ is defined as the minimal number of palindromes such that their product is $w$. Clearly, this function may take different values depending on if we consider $w$ as an element a free semigroup or of a free group: for example, in the free semigroup, the palindromic length of $abca$ is 4 (here every letter is a palindrome), and in the free group, it is 3 since $abca=(aba)(a^{-1}a^{-1})(aca)$. In free semigroups, the palindromic length can clearly be computed, and there are fast algorithms for that. In free groups, the question is trickier. In this paper, we characterize words in the free group whose palindromic length is 2 and 3.