Orders On Free Metabelian Groups

A bi-order on a group $G$ is a total, bi-multiplication invariant order. Such an order is regular if the positive cone associated to the order can be recognised by a regular language. A subset $S$ in an orderable group $(G,\leqslant)$ is convex if for all $f\leqslant g$ in $S$, every element $h\in G$ satisfying $f\leqslant h \leqslant g$ belongs to $S$. In this paper, we study the convex hull of the derived subgroup of a free metabelian group with respect to a bi-order. As an application, we prove that non-abelian free metabelian groups of finite rank do not admit a regular bi-order while they are computably bi-orderable.Comment: 19 Pages, 1 figure. Comments are welcome