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

Subgroup Distortion and the Relative Dehn Functions of Metabelian Groups