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arxiv: 2605.20397 · v1 · pith:ES7ES6WBnew · submitted 2026-05-19 · 🧮 math.GR · math.LO· math.OA

An ell² Obstruction for Elementary Embeddings of Hyperbolic Groups

Connor MacMahon This is my paper

Pith reviewed 2026-05-21 07:22 UTC · model grok-4.3

classification 🧮 math.GR math.LOmath.OA
keywords ℓ² Betti numberselementary embeddingshyperbolic groupsfirst-order logictorsion-free groupsexistential embeddingsgroup invariants
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The pith

The first ℓ² Betti number of a group is non-decreasing under various embeddings from first-order logic.

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

This paper shows that the first ℓ² Betti number of a group does not decrease when the group is embedded into a larger one via embeddings that come from first-order logic. It establishes a strict increase in this number for elementary embeddings of non-abelian proper subgroups inside torsion-free hyperbolic groups, relying on a classification of those inclusions. The same non-decreasing behavior is shown for existential embeddings of any finitely generated group. A sympathetic reader might care because this creates a numerical way to rule out certain logical embeddings between groups.

Core claim

The first ℓ² Betti number of a group is non-decreasing under various embeddings arising from first order logic. Strict inequality is proved for elementary embeddings of non-abelian proper subgroups within torsion free hyperbolic groups using a classification of such inclusions. The monotonicity is further demonstrated for existential embeddings of arbitrary finitely generated groups.

What carries the argument

The first ℓ² Betti number as a group invariant that remains non-decreasing or increases under first-order logic embeddings.

If this is right

  • Elementary embeddings preserve or increase the value of the first ℓ² Betti number.
  • There is a strict increase in the first ℓ² Betti number for elementary embeddings of non-abelian proper subgroups of torsion-free hyperbolic groups.
  • The first ℓ² Betti number is non-decreasing under existential embeddings of finitely generated groups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar monotonicity results might be provable for other classes of groups if their subgroup inclusions can be classified.
  • This invariant could help determine if an embedding between groups satisfies first-order logic properties by comparing the Betti numbers.
  • The approach may extend to other homological invariants of groups.

Load-bearing premise

A classification of inclusions of non-abelian proper subgroups in torsion-free hyperbolic groups applies directly to the elementary embeddings considered in the paper and yields the strict inequality for the first ℓ² Betti number.

What would settle it

Discovery of an elementary embedding between a non-abelian proper subgroup and a torsion-free hyperbolic group where the first ℓ² Betti number does not increase strictly would contradict the main result.

Figures

Figures reproduced from arXiv: 2605.20397 by Connor MacMahon.

Figure 1
Figure 1. Figure 1: A centered splitting with surface Σ, bottom groups B1, B2, B3, and with the associated graph of groups Γ depicted. Reproduced from [1] [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A maximal tree T within the graph of groups Γ from [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

The first $\ell^2$ Betti number of a group is non-decreasing under various embeddings arising from first order logic. Strict inequality is proved for elementary embeddings of non-abelian proper subgroups within torsion free hyperbolic groups using Perin's classification of such inclusions. The monotonicity is further demonstrated for existential embeddings of arbitrary finitely generated groups.

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

1 major / 2 minor

Summary. The paper claims that the first ℓ² Betti number β₁⁽²⁾ is non-decreasing under embeddings arising from first-order logic. It proves a strict inequality β₁⁽²⁾(H) < β₁⁽²⁾(G) for elementary embeddings of non-abelian proper subgroups H < G in torsion-free hyperbolic groups, derived from Perin's classification of such inclusions. Monotonicity is also shown for existential embeddings of arbitrary finitely generated groups.

Significance. If the central claims hold, the result supplies an ℓ² obstruction to elementary embeddings in the class of torsion-free hyperbolic groups, linking model-theoretic embeddings to analytic invariants with geometric content. The monotonicity statements for both elementary and existential embeddings broaden the scope beyond hyperbolic groups. Credit is due for grounding the argument in Perin's existing classification and standard properties of ℓ² Betti numbers rather than introducing new ad-hoc constructions.

major comments (1)
  1. [proof of the strict inequality for elementary embeddings (near the invocation of Perin's classification)] The derivation of the strict inequality β₁⁽²⁾(H) < β₁⁽²⁾(G) from Perin's classification (invoked for the main theorem on torsion-free hyperbolic groups) lacks an explicit case-by-case bridge. The classification identifies structural types of inclusions, yet the manuscript does not spell out why each type forces a drop in the first ℓ² Betti number (e.g., via the formula β₁⁽²⁾(F_r) = r-1 or Euler-characteristic relations). This step is load-bearing for the strict-inequality claim.
minor comments (2)
  1. [preliminaries] Notation for ℓ² Betti numbers is introduced without a brief reminder of the normalization or the precise definition used (e.g., whether it is the von Neumann dimension of the reduced homology).
  2. [section on existential embeddings] The statement of monotonicity for existential embeddings would benefit from a short example illustrating a concrete pair of groups where the inequality is strict.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the paper's contributions, and constructive feedback. We address the major comment below and will strengthen the exposition accordingly in the revised version.

read point-by-point responses

  1. Referee: [proof of the strict inequality for elementary embeddings (near the invocation of Perin's classification)] The derivation of the strict inequality β₁⁽²⁾(H) < β₁⁽²⁾(G) from Perin's classification (invoked for the main theorem on torsion-free hyperbolic groups) lacks an explicit case-by-case bridge. The classification identifies structural types of inclusions, yet the manuscript does not spell out why each type forces a drop in the first ℓ² Betti number (e.g., via the formula β₁⁽²⁾(F_r) = r-1 or Euler-characteristic relations). This step is load-bearing for the strict-inequality claim.

    Authors: We agree that an explicit case-by-case bridge would improve clarity and make the argument more self-contained. Perin's classification partitions the possible inclusions into finitely many structural types (free factors, certain amalgamated products, and HNN extensions compatible with hyperbolicity). In the revised manuscript we will add a dedicated paragraph or short subsection immediately following the invocation of the classification. For each type we will verify the strict drop using the formula β₁⁽²⁾(F_r) = r−1 together with the known additivity of ℓ² Betti numbers under free products and the Euler-characteristic relation χ⁽²⁾(G) = 1 − β₁⁽²⁾(G) for torsion-free hyperbolic groups. When the embedding is proper and the ambient group is non-abelian, at least one of these relations forces β₁⁽²⁾(H) < β₁⁽²⁾(G). This addition addresses the load-bearing step without altering any theorems. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external Perin classification and standard ℓ² Betti properties

full rationale

The paper establishes monotonicity of the first ℓ² Betti number under embeddings from first-order logic and proves strict inequality for elementary embeddings of non-abelian proper subgroups in torsion-free hyperbolic groups by invoking Perin's external classification of such inclusions together with known formulas for ℓ² Betti numbers. No step reduces a claimed prediction or result to a quantity defined or fitted inside the paper itself; the central implication is not obtained by self-definition, renaming, or a load-bearing self-citation chain. The argument is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background facts from geometric group theory and model theory together with one external classification result; no new free parameters or invented entities are introduced.

axioms (2)
  • standard math Standard functoriality and monotonicity properties of the first ℓ² Betti number under group homomorphisms and embeddings.
    Invoked to establish the non-decreasing claim for logical embeddings.
  • domain assumption Perin's classification theorem for inclusions of non-abelian proper subgroups inside torsion-free hyperbolic groups.
    Used as the key external input to obtain strict inequality for elementary embeddings.

pith-pipeline@v0.9.0 · 5572 in / 1368 out tokens · 62999 ms · 2026-05-21T07:22:11.641584+00:00 · methodology

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Reference graph

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