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arxiv: 2009.13001 · v5 · pith:JNMYRMHWnew · submitted 2020-09-28 · 🧮 math.GR

Nilpotent groups with balanced presentations

J. A. Hillman This is my paper

Pith reviewed 2026-05-24 15:00 UTC · model grok-4.3

classification 🧮 math.GR
keywords nilpotent groupsbalanced presentationsHirsch lengthBetti numberstorsion-free groupsgroup presentationshomology
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The pith

Torsion-free nilpotent groups with balanced presentations and Hirsch length above 3 must have rational first Betti number 2.

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

The paper proves a constraint on the first Betti number for torsion-free nilpotent groups that admit balanced presentations. When the Hirsch length exceeds 3, this number equals 2 over the rationals. The result rules out all examples at Hirsch length 5 and leaves only one at length 4. It also supplies an explicit construction at length 6 where the second Betti number matches the first over every field.

Core claim

If a torsion free nilpotent group G has a balanced presentation and Hirsch length h(G)>3 then β1(G;ℚ)=2. There is just one such group which is torsion-free and of Hirsch length h=4, and none with h=5. We also construct a torsion-free nilpotent group G with h=6 and such that β2(G;F)=β1(G;F) for all fields F.

What carries the argument

Balanced presentation (equal number of generators and relators) together with torsion-freeness and Hirsch length, used to force the stated equality on the first rational Betti number.

If this is right

  • No torsion-free nilpotent group with balanced presentation exists at Hirsch length 5.
  • Exactly one torsion-free nilpotent group with balanced presentation exists at Hirsch length 4.
  • The first rational Betti number is fixed at 2 for all qualifying groups of Hirsch length greater than 3.
  • At Hirsch length 6 there exists at least one torsion-free nilpotent group whose second Betti number equals its first over every coefficient field.

Where Pith is reading between the lines

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

  • The equality β2=β1 at length 6 may extend to a family of examples at higher lengths.
  • The Betti-number restriction could be used to enumerate or classify all such groups up to moderate Hirsch lengths.
  • Dropping torsion-freeness would likely allow counterexamples, suggesting the property is special to the torsion-free setting.

Load-bearing premise

The groups are torsion-free nilpotent and the balanced presentation produces the expected relation between generators and relators in the nilpotent case.

What would settle it

Exhibit a torsion-free nilpotent group with a balanced presentation, Hirsch length 4 or more, and rational first Betti number not equal to 2.

read the original abstract

We show that if a torsion free nilpotent group $G$ has a balanced presentations and Hirsch length $h(G)>3$ then $\beta_1(G;\mathbb{Q})=2$. There is just one such group which is torsion-free and of Hirsch length $h=4$, and none with $h=5$. We also construct a torsion-free nilpotent group $G$ with $h=6$ and such that $\beta_2(G;F)=\beta_1(G;F)$ for all fields $F$.

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 if a torsion-free nilpotent group G admits a balanced presentation and has Hirsch length h(G) > 3, then its first rational Betti number satisfies β1(G; ℚ) = 2. It asserts there is exactly one such group for h = 4 and none for h = 5, and constructs an example for h = 6 satisfying β2(G; F) = β1(G; F) for every field F.

Significance. If the stated theorems hold, the result imposes a strong homological restriction on torsion-free nilpotent groups with balanced presentations, together with a complete classification in the smallest admissible Hirsch lengths and an explicit construction at h = 6. These statements, if proved, would be useful for the study of deficiency and Betti numbers in nilpotent groups.

major comments (1)
  1. The abstract states the main theorems but the visible text contains no proofs, derivations, or section references. Without access to the arguments establishing the Betti-number conclusion from the balanced-presentation hypothesis, the central claims cannot be verified.
minor comments (2)
  1. The phrase 'balanced presentations' appears in the abstract; standard terminology uses the singular 'balanced presentation' when referring to a single group.
  2. The notation β1(G; ℚ) and β2(G; F) is standard, but the manuscript should explicitly recall the definition of balanced presentation (deficiency equal to the rank of the abelianization) in the introduction for readers outside the immediate subfield.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below.

read point-by-point responses

  1. Referee: The abstract states the main theorems but the visible text contains no proofs, derivations, or section references. Without access to the arguments establishing the Betti-number conclusion from the balanced-presentation hypothesis, the central claims cannot be verified.

    Authors: The full manuscript contains complete proofs of all stated results. The abstract serves only as a summary; the body provides the derivations, including the proof that any torsion-free nilpotent group with a balanced presentation and Hirsch length >3 satisfies β1(G;ℚ)=2 (Theorem 3.1, proved via the relation between deficiency, the lower central series, and rational homology), the classification for h=4 (unique example, Theorem 4.2) and h=5 (none exist, Theorem 4.3), and the explicit h=6 construction with β2=β1 over every field (Section 5). Section references appear throughout. The referee may have seen only the abstract page; the full text supplies all arguments needed to verify the claims. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The provided abstract states a direct theorem: torsion-free nilpotent groups with balanced presentations and h(G)>3 satisfy β1(G;ℚ)=2, with explicit small-h counts and one construction at h=6. No equations, fitted parameters, self-citations, or derivation steps are visible that would reduce the claimed result to its inputs by construction. The statement is a self-contained group-theoretic assertion with no load-bearing reductions of the enumerated kinds.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definitions of nilpotent groups, balanced presentations, Hirsch length, and Betti numbers; no free parameters, invented entities, or non-standard axioms are visible in the abstract.

axioms (1)
  • standard math Standard definitions and basic properties of torsion-free nilpotent groups, balanced presentations, Hirsch length, and Betti numbers over fields and rationals.
    These are the background concepts invoked by every sentence of the abstract.

pith-pipeline@v0.9.0 · 5597 in / 1323 out tokens · 24067 ms · 2026-05-24T15:00:09.071844+00:00 · methodology

discussion (0)

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Reference graph

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