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- Title
Orders on free metabelian groups.
- Authors
Wang, Wenhao
- Abstract
A bi-order on a group 퐺 is a total, bi-multiplication invariant order. A subset 푆 in an ordered group (G , ⩽) is convex if, for all f ⩽ g in 푆, every element h ∈ G satisfying f ⩽ h ⩽ g belongs to 푆. 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.
- Subjects
FREE groups; FINITE groups; HOMEOMORPHISMS
- Publication
Journal of Group Theory, 2024, Vol 27, Issue 3, p485
- ISSN
1433-5883
- Publication type
Article
- DOI
10.1515/jgth-2022-0203