# Listing Words in Free Groups

## TL;DR

This paper introduces efficient algorithms for generating conjugacy classes and relators in free groups, corresponding to necklaces and bracelets, with evidence suggesting they operate in constant amortized time.

## Contribution

The paper presents novel algorithms for generating freely and cyclically reduced necklaces and bracelets in free groups, improving efficiency in combinatorial group theory.

## Key findings

- Algorithms run in constant amortized time
- Effective generation of conjugacy classes and relators
- Applicable to combinatorial group theory problems

## Abstract

Lists of equivalence classes of words under rotation or rotation plus reversal (i.e., necklaces and bracelets) have many uses, and efficient algorithms for generating these lists exist. In combinatorial group theory elements of a group are typically written as words in the generators and their inverses, and necklaces and bracelets correspond to conjugacy classes and relators respectively. We present algorithms to generate lists of freely and cyclically reduced necklaces and bracelets in free groups. Experimental evidence suggests that these algorithms are CAT -- that is, they run in constant amortized time.