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 $g\in G$) and suppose that $A$ is a non-empty subset of $G$ with $\mathrm{\mid }A+A\mathrm{\mid }\le K\mathrm{\mid }A\mathrm{\mid }$. Then $A$ can be covered by at most $(2K{)}^{O(m^3)}$ translates of a subgroup of $H\le G$ of cardinality at most $\mathrm{\mid }A\mathrm{\mid }$. The argument is a variant of that used in the case $G={\mathbf{F}}_2^n$ in a recent paper of the authors.