On subsets with small product in torsion-free groups (Q1307444)

The main result is the following: If \(C\) is a generating subset of a nonabelian torsion-free group \(G\) with \(1\in C\) then for all \(B\subseteq G\) such that \(|B|\geq 4\), \(|BC|\geq|B|+|C|+1\). In particular, a finite subset \(X\) with cardinality \(k\geq 4\) satisfies the inequality \(|X^2|\leq 2|X|\) if and only if there are two elements \(x,r\) of \(G\), such that the conditions (i) \(xr=rx\), (ii) \(Xx=\{1,r,\dots,r^k\}\setminus\{c\}\) where \(c\in\{1,r\}\), hold.