Solvable groups with polynomial Dehn functions

Given a finitely presented group H, finitely generated subgroup B of H, and a monomorphism ψ: B → H, we obtain an upper bound of the Dehn function of the corresponding HNN-extension G = 〈H, t | t−1Bt = ψ(B)〉 in terms of the Dehn function of H and the distortion of B in G. Using such a bound, we construct first examples of non-polycyclic solvable groups with polynomial Dehn functions. The constructed groups are metabelian and contain the solvable Baumslag-Solitar groups. In particular, this answers a question posed by Birget, Ol’shanskii, Rips, and Sapir