Explicit Helfgott type growth in free products and in limit groups

TL;DR

This paper extends Helfgott type growth results to free products and limit groups, showing that finite subsets either generate small or virtually nilpotent groups or exhibit significant growth in their triple product.

Contribution

It adapts Safin's results to free products and limit groups, establishing new growth inequalities and characterizing the structure of subsets in these groups.

Findings

Finite subsets in free products either generate small or virtually nilpotent groups or exhibit large triple product growth.

In limit groups, similar growth inequalities hold unless the subset generates a free abelian group.

Many infinite groups have subsets with either polynomial growth or virtually nilpotent structure.

Abstract

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 size of the triple product AAA of A is at least 1/7776 times the square of |A|, 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 |AAA| 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 d=1 such that any finite subset A of G has the property that either |AAA| is at least c times (|A| to the power of 1+d) or it generates a virtually nilpotent group.