"Varopoulos paradigm": Mackey property vs. metrizability in topological groups

Abstract: The class of all locally quasi-convex (lqc) abelian groups contains all locally convex vector spaces (lcs) considered as topological groups. Therefore it is natural to extend classical properties of locally convex spaces to this larger class of abelian topological groups. In the present paper we consider the following well known property of lcs: "A metrizable locally convex space carries its Mackey topology ". This claim cannot be extended to lqc-groups in the natural way, as we have recently proved with other coauthors (\cite{AD}, \cite{DMPT}). We say that an abelian group $G$ satisfies the \emph{Varopoulos paradigm} (VP) if any metrizable locally quasi-convex topology on $G$ is the Mackey topology. Surprisingly VP - which is a topological property of the group - is characterizes an algebraic feature, namely being of finite exponent.