Powers of sets in free groups

TL;DR

This paper establishes lower bounds on the growth of powers of finite sets in free groups, demonstrating that the size of A^n grows at least as a constant times |A| to the power of (n+1)/2, with optimal estimates.

Contribution

It proves sharp lower bounds for the size of A^n in free groups, extending understanding of set growth in non-abelian group structures.

Findings

Lower bounds are optimal for each n>0.

Growth rate of A^n is at least proportional to |A|^{(n+1)/2}.

Constants c_n do not depend on the specific set A.

Abstract

We prove that |A^n| > c_n |A|^{[\frac{n+1}{2}]} for any finite subset A of a free group if A contains at least two noncommuting elements, where c_n>0 are constants not depending on A. Simple examples show that the order of these estimates are the best possible for each n>0.