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arxiv: 2510.10673 · v2 · submitted 2025-10-12 · 🧮 math.GR · math.LO

The isomorphism problem for finitely generated bi-orderable groups

Filippo Calderoni , Adam Clay This is my paper

Pith reviewed 2026-05-18 08:10 UTC · model grok-4.3

classification 🧮 math.GR math.LO
keywords bi-orderable groupsleft-orderable groupsisomorphism relationdescriptive set theoryBorel spacerelative conesweakly universalfinitely generated groups
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The pith

The isomorphism relation on finitely generated bi-orderable groups is weakly universal.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the classification problem for finitely generated orderable groups from the viewpoint of descriptive set theory. It sets up a standard Borel space for finitely generated left-orderable groups and a corresponding subspace for bi-orderable groups by using spaces of relative cones. The authors then prove that the isomorphism relation on the finitely generated bi-orderable groups is weakly universal. This matters because it gives a precise description of how hard it is to tell whether two such groups are the same up to isomorphism.

Core claim

We analyze the standard Borel space of finitely generated left-orderable groups, and the subspace of finitely generated bi-orderable groups using spaces of relative cones. We use this setup to show that the isomorphism relation on finitely generated bi-orderable groups is weakly universal.

What carries the argument

The standard Borel space of finitely generated bi-orderable groups defined via spaces of relative cones, which parametrizes the groups along with their bi-orderings to allow analysis of their isomorphisms.

Load-bearing premise

The construction of the standard Borel space of finitely generated bi-orderable groups via spaces of relative cones correctly captures the isomorphism relation without introducing extraneous identifications or missing essential orderings.

What would settle it

Finding two finitely generated bi-orderable groups that are isomorphic but not related through the relative cone space construction would show the space does not fully capture the isomorphism relation.

read the original abstract

We analyze the classification problem for finitely generated orderable groups from the viewpoint of descriptive set theory. We analyze the standard Borel space of finitely generated left-orderable groups, and the subspace of finitely generated bi-orderable groups using spaces of relative cones. We use this setup to show that the isomorphism relation on finitely generated bi-orderable groups is weakly universal.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript analyzes the classification problem for finitely generated orderable groups from the viewpoint of descriptive set theory. It constructs the standard Borel space of finitely generated left-orderable groups and the subspace consisting of finitely generated bi-orderable groups via spaces of relative cones. Using this setup, the paper shows that the isomorphism relation on finitely generated bi-orderable groups is weakly universal.

Significance. If the central result holds, it would establish that the isomorphism problem for finitely generated bi-orderable groups is weakly universal among Borel equivalence relations, indicating maximal complexity within the descriptive set theoretic framework. This provides a new perspective on the decidability and classification difficulties for ordered groups and introduces the relative-cone encoding as a technical tool that could apply to related problems in left-orderable groups.

major comments (2)
  1. The construction of the standard Borel space of finitely generated bi-orderable groups via spaces of relative cones: it must be shown explicitly that this encoding distinguishes distinct bi-orderings on the same underlying group and separates non-isomorphic ordered groups, so that the induced equivalence relation coincides with genuine ordered-group isomorphism rather than a quotient. This is load-bearing for the weak-universality claim, as any collapse would restrict the result to a coarser relation.
  2. The transfer of weak universality from the ambient space of left-orderable groups to the bi-orderable subspace: the argument requires a clear reduction or embedding lemma showing that the universality property survives restriction to the subspace without additional assumptions on the cone data.
minor comments (1)
  1. The abstract states the main result but provides no indication of the key technical steps or verification of the cone construction; adding one sentence on the role of relative cones would improve accessibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these important points regarding the technical foundations of our construction. We address each major comment below and will incorporate clarifications in a revised version.

read point-by-point responses

  1. Referee: The construction of the standard Borel space of finitely generated bi-orderable groups via spaces of relative cones: it must be shown explicitly that this encoding distinguishes distinct bi-orderings on the same underlying group and separates non-isomorphic ordered groups, so that the induced equivalence relation coincides with genuine ordered-group isomorphism rather than a quotient. This is load-bearing for the weak-universality claim, as any collapse would restrict the result to a coarser relation.

    Authors: We agree that an explicit verification is necessary to ensure the equivalence relation is precisely the isomorphism relation on ordered groups. The relative-cone encoding in Section 2 is constructed so that each bi-ordering on a fixed group corresponds to a distinct cone (via the positive cone and its conjugates), and isomorphisms of ordered groups induce Borel maps between cones. To make this fully rigorous and address the concern directly, we will add a new lemma (Lemma 2.7) in the revised manuscript proving injectivity on bi-orderings and separation of non-isomorphic ordered groups. This confirms the induced equivalence relation matches genuine ordered-group isomorphism. revision: yes

  2. Referee: The transfer of weak universality from the ambient space of left-orderable groups to the bi-orderable subspace: the argument requires a clear reduction or embedding lemma showing that the universality property survives restriction to the subspace without additional assumptions on the cone data.

    Authors: The weak universality for bi-orderable groups is obtained by a Borel embedding of the bi-orderable space into the left-orderable space that preserves the isomorphism relation, allowing the universality to transfer directly. We will strengthen the argument by adding an explicit embedding lemma (Lemma 4.3) in the revised version. This lemma will show that the restriction of the equivalence relation to the bi-orderable subspace inherits weak universality from the ambient space, with no further assumptions required on the cone data beyond those already used in the construction. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation is self-contained via explicit Borel-space construction

full rationale

The paper constructs the standard Borel space of finitely generated bi-orderable groups explicitly via spaces of relative cones on generating sets, then derives the weak universality of the isomorphism relation from that setup. No step reduces by definition to its own output, no fitted parameters are relabeled as predictions, and no load-bearing premise rests solely on a self-citation chain whose content is unverified outside the paper. The argument remains independent of the target universality statement and is presented as a theorem derived from the descriptive-set-theoretic encoding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result relies on standard axioms of descriptive set theory and Borel reducibility; no free parameters or invented entities are mentioned.

axioms (1)
  • standard math Standard axioms of ZFC set theory and the existence of standard Borel spaces for countable structures.
    Invoked implicitly when defining the space of finitely generated groups and the isomorphism relation.

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discussion (0)

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Reference graph

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