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. These refinements allow us to prove new results about primitive elements and free factors in a free group, including 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.
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