  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 $|A+A| \leq K|A|$. Then $A$ can be covered by at most
$(2K)^{O(m^3)}$ translates of a subgroup of $H \leq G$ of cardinality at most
$|A|$. The argument is a variant of that used in the case $G = \mathbf{F}_2^n$
in a recent paper of the authors.
