Diskbusting elements of the free group

In 1936 Whitehead presented an algorithm for determining whether an element (or a set of elements) in a free group $F$ is part of a free generating set for $F$. We will give a more detailed discussion of Whitehead’s algorithm and a new proof that the algorithm works. We will see that in fact Whitehead’s algorithm actually determines whether or not an element (or a set of elements) is diskbusting. If an element $\omega$ is not diskbusting, then Whitehead’s algorithm produces the smallest free factor of $F$ in which $\omega$ lies, and in that free factor $\omega$ is diskbusting.

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Published 1 January 1997