Marton’s conjecture in abelian groups with bounded torsion

We prove a Freiman–Ruzsa-type theorem with polynomial bounds in arbitrary abelian groups with bounded torsion, thereby proving (in full generality) a conjecture of Marton. Specifically, let G be an abelian group of torsion m (meaning m g = 0 for all g ∈ G ) and suppose that A is a non-empty subset of G with | A + A | ≤ K | A | . Then A can be covered by at most ( 2 K ) O ( m 3 ) translates of a subgroup H ≤ G of cardinality at most | A | . The argument is a variant of that used in the case G = F 2 n in a recent paper of the authors.