Virtual rational Betti numbers of abelian-by-polycyclic groups

Let 1→A→G→Q→11→A→G→Q→1 be an exact sequence of groups, where A is abelian, Q   is polycyclic and ⨂Qk(A⊗ZQ) is finitely generated as QQQQ-module via the diagonal Q  -action for k≤2mk≤2m. Moreover we assume that if G   is not metabelian, then it is of type FP3FP3. Our main result is thatsupU∈A⁡dimQ⁡Hj(U,Q)<∞ for 0≤j≤m, where AA is the set of all subgroups of finite index in G.