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

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.

💡 Research Summary

The paper investigates whether two subtle group properties—bi‑orderability and the existence of a generalized torsion element—can be detected from the profinite completion of a finitely generated residually finite group. A property that can be read off from the set of all finite quotients (equivalently, from the profinite completion) is called a “profinite property.” While many classical properties such as amenability, Kazhdan’s property (T) or Property FA are known not to be profinite, the status of bi‑orderability had remained open, and it was explicitly asked by Piwek and Wykowski.

The authors answer this question negatively by constructing explicit counter‑examples using fiber products. The key technical device is a construction that turns any non‑trivial finitely generated left‑orderable group (G) into a bi‑orderable group (P) via a fiber product. Let (F_n) be a free group of rank (n) (where (n) is the number of generators of (G)). Define two epimorphisms \