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;<= is convex if, for all f <= g in S, every element h E G satisfying f <= h <= 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.

Automated summary

This paper studies bi-orders on groups and proves that the derived subgroup is convex under any bi-order, and studies the convex hull of the derived subgroup of a free metabelian group of higher rank. As an application, it is proved that the space of bi-orders of a non-abelian free metabelian group of finite rank is homeomorphic to the Cantor set, and it is shown that no bi-order for these groups can be recognized by a regular language.