The isomorphism problem for finitely generated bi-orderable groups
Filippo Calderoni, Adam Clay
TL;DR
The paper investigates the isomorphism problem for finitely generated bi-orderable groups through descriptive set theory, constructing Polish spaces for left- and bi-orderable groups and showing that the bi-orderable case is weakly universal. The core method encodes the conjugacy relation of a non-abelian free group within bi-orderable quotients via central extensions, enabling a weak Borel reduction from the universal countable Borel equivalence relation to isomorphism on BO. This yields strong anti-classification results, suggesting that classifying finitely generated left- or bi-orderable groups is as complex as classifying all finitely generated groups. The work also introduces a concrete central-extension construction H(G,P) that preserves bi-orderability of quotients and supports the necessary cocycle-compatible automorphisms, paving the way for future explorations of universality and cocycle phenomena in this setting.
Abstract
We analyze the classification problem for finitely generated orderable groups from the viewpoint of descriptive set theory. We define the space of finitely generated left-orderable groups, and the subspace of finitely generated bi-orderable groups using spaces of relative cones, and show that both spaces are Polish. We use this setup to show that the isomorphism relation on the space of finitely generated bi-orderable groups is weakly universal.