2011/09/01 by J. O. Button, Button, J. O.
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Limits and Structures in Graph Theory #math.CO #math.GR
paper · pdf · doi:10.48550/arxiv.1109.0244
Minor reordering; ends with some questions
openalex publication_date 2011/09/01 · arxiv created 2011/09/27 · arxiv updated 2011/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.