On centrally orderable groups

AbstractIn [V.V. Bludov, A.M.W. Glass, A.H. Rhemtulla, Ordered groups in which all convex jumps are central, J. Korean Math. Soc. 40 (2003) 225–239] we considered the class C of all orderable groups with all orders having central convex jumps and the class C2 of all orderable groups all of whose two generator subgroups belong to C. Both these classes contain all locally nilpotent torsion-free groups. We proved that every soluble-by-finite group belonging to C2 must be locally nilpotent, but there is a two-generator metabelian group belonging to C∖C2 (whence it is not locally nilpotent).In this paper we consider the closure of C and C2 under elementary equivalence, direct sums and full Cartesian products, homomorphic images, and residual properties. We further show that every metabelian group G which has a central order can be embedded in a metabelian group G˜ belonging to C so that each central order on G can be extended to G˜. This is achieved by ascending HNN-extensions. We further show that a finitely generated metabelian group has a central order if and only if it is residually torsion-free nilpotent