A fine property of Whitehead's algorithm

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.