Weak commutativity between two isomorphic polycyclic groups

An operator of weak commutativity between isomorphic groups, H and H-psi was defined by Sidki as chi(H) = < H H-psi vertical bar [h, h(psi)] = 1 for all h is an element of H >, where psi : h bar right arrow h(psi) for all h is an element of H defines an isomorphism. It is known that the operator chi preserves group properties such as finiteness, solubility, and also nilpotency for finitely generated groups. We prove in this work that chi preserves the properties of being polycyclic or polycyclic-by-finite. As a consequence of this result, we conclude that the non-abelian tensor square H circle times H of a group H as defined by Brown and Loday preserves the property of being polycyclic-by-finite. This last result extends work of Blyth and Morse who proved that H circle times H is polycyclic if H is polycyclic.