If a finite subset A of a free group has non-commuting elements, then |A^3|>|A|^2/(log|A|)^O(1)

You need to know: Group, free group, notation $|A|$ for the number of elements in a finite set A, $O(1)$ and $o(1)$ notations.

Background: For subsets A and B of group G denote $A \cdot B=\{g\in G \,|\,g=ab, \, a\in A, \, b \in B\}$ . Denote $A^3 = A \cdot A \cdot A$ .

The Theorem: On 18th June 2007, Alexander Razborov submitted to the Annals of Mathematics a paper in which he proved that, if A is a finite subset of a free group $F_m$ with at least two noncommuting elements, then $|A^3| \geq \frac{|A|^2}{(\log|A|)^{O(1)}}$ .

Short context: For sets of integers A,B, let $A+B=\{a+b \,|\, a\in A, \, b \in B\}$ . If A is an arithmetic progression, then $|A+A|<2|A|$ . A deep 1973 theorem of Freiman describes all possible examples of sets A such that $|A+A|\leq k|A|$ for fixed k. For many applications, it is important to derive the structure of A from a weaker estimate of the form $|A+A|\leq |A|^{1+o(1)}$ , but this is open. For subsets A of arbitrary group G, it is impossible to deduce the structure of A is $|A^2|$ is small, but sometimes possible if $|A^3|$ is small. The Theorem implies that in free groups $|A^3|$ is small only if $ab=ba$ for all $a,b \in A$ .