Orders on free metabelian groups

A bi-order on a group 𝐺 is a total, bi-multiplication invariant order. A subset 𝑆 in an ordered group ( G , ⩽ ) (G,\leqslant) is convex if, for all f ⩽ g f\leqslant g in 𝑆, every element h ∈ G h\in G satisfying f ⩽ h ⩽ g f\leqslant h\leqslant g belongs to 𝑆. In this paper, we show that the derived subgroup of the free metabelian group of rank 2 is convex with respect to any bi-order. Moreover, we study the convex hull of the derived subgroup of a free metabelian group of higher rank. As an application, we prove that the space of bi-orders of a non-abelian free metabelian group of finite rank is homeomorphic to the Cantor set. In addition, we show that no bi-order for these groups can be recognised by a regular language.