Explicit Helfgott type growth in free products and in limit groups

We adapt Safinʼs result on powers of sets in free groups to obtain Helfgott type growth in free products: if A is any finite subset of a free product of two arbitrary groups then either A   is conjugate into one of the factors, or the triple product A3A3 of A   satisfies |A3|⩾(1/7776)|A|2|A3|⩾(1/7776)|A|2, or A generates an infinite cyclic or infinite dihedral group. We also point out that if A   is any finite subset of a limit group then |A3||A3| satisfies the above inequality unless A generates a free abelian group. This gives rise to many infinite groups G   where there exist c>0c>0 and δ=1δ=1 such that any finite subset A of G   either satisfies |A3|⩾c|A|1+δ|A3|⩾c|A|1+δ or generates a virtually nilpotent group.