Small palindromic lengths in free groups and word equations with antimorphisms

TL;DR

This paper investigates the minimal number of palindromes needed to express words in free groups, characterizing those with palindromic lengths of 2 and 3, and contrasts this with the simpler case in free semigroups.

Contribution

It provides a characterization of words in free groups with palindromic lengths of 2 and 3, advancing understanding of word structure in free groups.

Findings

Characterization of free group words with palindromic length 2

Characterization of free group words with palindromic length 3

Comparison with palindromic length in free semigroups

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.