In 2002 Birget, Ol’shanskii, Rips and Sapir characterized the subgroups of groups with at most a polynomial Dehn function as those groups with NP soluble word problem. In the article under review the authors prove that for every positive integer \(n\) the soluble Baumslag-Solitar group \(\text{BS}(1,n)=\langle t,x\mid txt^{-1}=x^n\rangle\) can be embedded into a finitely presented metabelian group with quadratic Dehn function.
Note that this result cannot be improved since \(B(1,n)\) cannot be embedded into a word hyperbolic group and all other groups have a quadratic lower bound on their Dehn function (see Ol’shanskii, 1991). In the article the authors show also that a Baumslag finitely presented metabelian group \(\langle a,s,t\mid a^p,\;[s,t],\;[a^t,a],\;a^s=a^ta\rangle\) has a quadratic Dehn function (\(p\) is a prime).