= Free product with factor generating set
For nontrivial groups $A,B$, the <Cayley graph> of $A*B$ with $S=(A\setminus\{1\})\cup(B\setminus\{1\})$ has the edges of the <Bass-Serre tree> as its vertices. Two such vertices are adjacent exactly when the tree edges share an endpoint. Mapping each group element to its tree-edge midpoint is an isometry on the vertex metrics and has image within distance $1/2$ of every tree point, hence gives a <quasi-isometry> of metric graphs. The <normal form theorem for a free product> identifies $|g|_S$ with reduced syllable length. The factor subgroups have diameter one in this metric.
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