Orders On Free Metabelian Groups

TL;DR

This paper investigates bi-orders on free metabelian groups, showing the derived subgroup's convexity, characterizing the space of bi-orders, and proving the non-regularity of bi-order recognition.

Contribution

It establishes the convexity of the derived subgroup in free metabelian groups and characterizes the topology of the bi-order space.

Findings

Derived subgroup is convex under any bi-order.

The space of bi-orders is homeomorphic to the Cantor set.

Bi-orders cannot be recognized by regular languages.

Abstract

A bi-order on a group $G$ is a total, bi-multiplication invariant order. A subset $S$ in an ordered 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 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-order of 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.