On subsets with small product in torsion-free groups

Let G be a nonabelian torsion-free group. Let C be a finite generating subs et of G such that 1 is an element of C. We prove that, for all subsets B of G with \B\ greater than or equal to 4, we have \BC\ greater than or equal to \B\ + \C\ + 1. In particular, a finite subset X with cardinality k greater than or equal t o 4 satisfies the inequality \X-2\ less than or equal to 2\X\ if and only i f there are elements x,r is an element of G, such that the following two co nditions hold: (i) xr = rx. (ii) Xx = {1,r,...,r(k)} \ {c} where c is an element of {1, r}.