Bi-orderability and generalized torsion elements from the perspective of profinite properties

TL;DR

The paper addresses whether bi-orderability and generalized torsion are profinite properties. It develops a fiber-product construction that converts a left-orderable group into a bi-orderable one and analyzes profinite completions to produce counterexamples. By constructing a finitely presented group $G$ with trivial profinite completion and a residually finite, non-bi-orderable subgroup, the authors produce subgroups whose finite quotients coincide with those of non-bi-orderable or torsion-containing groups, thereby showing that bi-orderability and generalized torsion are not determined by finite quotients. This demonstrates fundamental limits of profinite data in detecting orderability-like properties and introduces a method to generate explicit counterexamples via $F_n*G$–type fiber products and their profinite completions.

Abstract

Using fiber products, we construct bi-orderable groups from left-orderable groups. As an application, we show that bi-orderability is not a profinite property, answering a question of Piwek and Wykowski negatively. We also show that the existence of a generalized torsion element is not a profinite property.

Theorems & Definitions (33)

• Theorem 1.1: Corollary \ref{['Thm-BOPP']}
• Theorem 1.2: Theorem \ref{['Thm-BO']}
• Theorem 1.3: Corollary \ref{['Thm-GTPP']}
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Remark 2.7