On centrally orderable groups (Q2569427)

The authors study the class \(\mathcal C\) of all orderable groups with all orders having central convex jumps and the class \({\mathcal C}_2\) of all orderable groups all of whose two generator subgroups belong to \(\mathcal C\). Closures of \(\mathcal C\) and \({\mathcal C}_2\) under elementary equivalence, direct sums and full Cartesian products, homomorphic images, and residuals are investigated. It is proved that: a) every metabelian group \(G\) which has a central order can be embedded in a metabelian group \(\overline{G}\) belonging to \({\mathcal C}\) so that each central order on \(G\) can be extended to \(\overline{G}\); b) every finitely generated metabelian group has a central order if and only if it is residually torsion-free nilpotent.