Submission date: 26 October 2022

Abstract:

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, ≤) is convex if for all f ≤ g in S, every element h ∈ G satisfying f ≤ h ≤ 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.

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Full text arXiv 2210.14630: pdf, ps.