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 $mg=0$ 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 (2K)O(m³) translates of a subgroup of H ≤ G of cardinality at most $|A|$. The argument is a variant of that used in the case G = F₂ⁿ in a recent paper of the authors.