A fine property of Whitehead's algorithm

TL;DR

This paper refines Whitehead's algorithm for primitive words and free factors in free groups, providing new criteria and algorithms for subgroup analysis and free factor detection.

Contribution

It introduces a strengthened Whitehead's algorithm for subgroups and primitive elements, and develops algorithms for measuring distances in the free factor complex.

Findings

A refined Whitehead's algorithm for primitive words and subgroups.

A criterion to test if a subgroup is a free factor using primitive elements.

Algorithms to determine distances in the free factor complex up to d=4.

Abstract

We develop a refinement of Whitehead's algorithm for primitive words in a free group. We generalize to subgroups, establishing a strengthened version of Whitehead's algorithm for free factors. We make use of these refinements in proving new results about primitive elements and free factors in a free group. These include a relative version of Whitehead's algorithm, and a criterion that tests whether a subgroup is a free factor just by looking at its primitive elements. We develop an algorithm to determine whether or not two vertices in the free factor complex have distance $d$ for $d=1,2,3$, as well as $d=4$ in a special case.