Free products of free-group automorphism groups: a finite-rank embedding by separating cocycles

Let \(I\) be finite and let \(G_i\leq\operatorname{Aut}(F_{n_i})\), where the ranks \(n_i\) may vary with \(i\). We construct an embedding \(\mathop{\ast}_{i\in I}G_i\hookrightarrow\operatorname{Aut}(F_{\sum_i n_i+|I|+1})\). For two factors this gives \(G*H\hookrightarrow\operatorname{Aut}(F_{n+m+3})\) and answers Problem 18.11 of the Kourovka Notebook affirmatively. No finite-generation hypothesis on the subgroups is required. A separating nonabelian cocycle defines a faithful action on a free product with one additional infinite cyclic factor. We construct such cocycles for free-group automorphisms using one marker word in each enlarged free factor.