Abstract
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.