The Dehn function of Baumslag's metabelian group. (Q431071)

Baumslag's group \(\Gamma\) is presented by \(\langle a,s,t\mid [a,a^t]=1,\;[s,t]=1,\;a^s=aa^t\rangle\). It is a finitely presented metabelian group with the derived subgroup of infinite rank. The subgroup \(\langle a,t\rangle\) of \(\Gamma\) is isomorphic to the wreath product \(\mathbb Z\text{\,wr\,}\mathbb Z\) of two infinite cyclic groups. Introducing the relation \(\langle a^m=1\rangle\), where \(m\geq 2\), gives a family \(\Gamma_m=\langle\Gamma\mid a^m=1\rangle\), in which the subgroup \(\langle a,t\rangle\) is isomorphic to \(\mathbb Z_m\text{\,wr\,}\mathbb Z\) of the finite cyclic group of order \(m\) and the infinite cyclic group. It is shown in the paper that the Dehn function \(D_\Gamma(n)\) is equivalent to \(2^n\), but for all \(m\) the Dehn function \(D_{\Gamma_m}(n)\) is less or equivalent to \(n^4\).