Explicit Helfgott type growth in free products and in limit groups

Abstract

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 A3 of A satisfies |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| 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 δ=1 such that any finite subset A of G either satisfies |A3|⩾c|A|1+δ or generates a virtually nilpotent group.