On Whitehead's first free-group algorithm, cutvertices, and free-product factorizations

Abstract

Let $F$ be any finite-rank free group, and $R$ be any finite subset of $\{g, [g]: g \in F-\{1\}\}$, where $[g]:= \{fgf^{-1}:f\in F\}$. By an $R$-allocating $F$-factorization we mean a set $\mathcal{H}$ of nontrivial subgroups of $F$ such that $\ast_{H \in \mathcal{H}} H = F$ and $R \subseteq \{h, [h] : h \in H, H\in \mathcal{H}\}$. We show that Whitehead's (fast) cutvertex algorithm inputs the pair $(F,R)$ and outputs a maximum-size $R$-allocating $F$-factorization. Richard Stong showed this in the case where $R \subseteq F$ or $R \subseteq \{[g] : g \in F\}$, thereby unifying and generalizing a collection of results obtained by Berge, Bestvina, Lyon, Shenitzer, Stallings, Starr, and Whitehead. Our proof is based on the interaction between two normal forms for the elements of $F$, rather than the algebraic topology of handlebodies, trees, or graph folding.