Explicit Helfgott type growth in free products and in limit groups

We adapt Safin's 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 A(3) of A satisfies vertical bar A(3)vertical bar >= (1/7776)vertical bar A vertical bar(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 vertical bar A(3)vertical bar satisfies the above inequality unless A generates a free abelian group. This gives rise to many infinite groups G where there exist c > 0 and delta = 1 such that any finite subset A of G either satisfies vertical bar A(3)vertical bar >= c vertical bar A vertical bar(1+delta) or generates a virtually nilpotent group. (c) 2013 Elsevier Inc. All rights reserved.