  We show the connection between the relative Dehn function of a finitely
generated metabelian group and the distortion function of a corresponding
subgroup in the wreath product of two free abelian groups of finite rank.
Further, we show that if a finitely generated metabelian group $G$ is an
extension of an abelian group by $\mathbb Z$ the relative Dehn function of $G$
is polynomially bounded. Therefore, if $G$ is finitely presented, the Dehn
function is bounded above by the exponential function up to equivalence.
