Growing words in the free group on two generators

TL;DR

This paper investigates minimal-length representatives of words in the free group on two generators, introducing the concept of root words and characterizing their properties related to minimality and growth.

Contribution

It provides a simple inequality for minimal words, introduces root words as minimal words that cannot be grown from others, and analyzes their properties.

Findings

Root words have lengths divisible by 4.

A simple inequality characterizes minimal-length words.

Root words are boundary cases of minimality.

Abstract

This paper is concerned with minimal-length representatives of equivalence classes of words in F_2 under Aut F_2. We give a simple inequality characterizing words of minimal length in their equivalence class. We consider an operation that "grows" words from other words, increasing the length, and we study root words -- minimal words that cannot be grown from other minimal words. Root words are "as minimal as possible" in the sense that their characterization is the boundary case of the minimality inequality. The property of being a root word is respected by equivalence classes, and the length of each root word is divisible by 4.