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Orders on free metabelian groups

  • Wenhao Wang ORCID logo EMAIL logo
From the journal Journal of Group Theory

Abstract

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 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.

Award Identifier / Grant number: 075-15-2022-265

Funding statement: The work was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2022-265).

Acknowledgements

The author acknowledges Igor Lysenok for many inspirational discussions on this subject and Cristóbal Rivas for pointing out that Corollary 8.4 holds for Conradian orders. The author is also grateful to the referee/s for his/her many useful comments which led to a refinement of the paper and some new results.

  1. Communicated by: Timothy C. Burness

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Received: 2022-12-02
Revised: 2023-10-18
Published Online: 2023-11-30
Published in Print: 2024-05-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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