Elementary amenable groups of cohomological dimension 3

Jonathan A. Hillman

arxiv: 2102.02947 · v3 · pith:BEMF53HJnew · submitted 2021-02-05 · 🧮 math.GR

Elementary amenable groups of cohomological dimension 3

Jonathan A. Hillman This is my paper

Pith reviewed 2026-05-24 13:21 UTC · model grok-4.3

classification 🧮 math.GR

keywords elementary amenable groupsHirsch lengthcohomological dimensionsolvable groupsderived lengthpolycyclicHNN extensionsBaumslag-Solitar groups

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The pith

Torsion-free elementary amenable groups of Hirsch length at most 3 are solvable of derived length at most 3

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

The paper proves that torsion-free elementary amenable groups with Hirsch length no greater than 3 must be solvable and have derived length bounded by 3. This statement encompasses every solvable group that has cohomological dimension 3. It provides an explicit classification for those groups of cohomological dimension 3, placing them into one of three categories: polycyclic groups, semidirect products of BS(1,n) with Z, or ascending HNN extensions over Z squared or the Klein bottle group. Readers interested in group theory would care because this resolves the structure of all low-dimensional solvable groups in the amenable setting.

Core claim

We show that torsion-free elementary amenable groups of Hirsch length ≤3 are solvable, of derived length ≤3. This class includes all solvable groups of cohomological dimension 3. We show also that groups in the latter subclass are either polycyclic, semidirect products BS(1,n)⋊Z or properly ascending HNN extensions with base Z² or π1(Kb).

What carries the argument

The combination of Hirsch length and cohomological dimension as invariants that force solvability and bound the derived length, leading to the explicit classification of the groups.

If this is right

All solvable groups of cohomological dimension 3 belong to one of the three listed structural classes.
The derived length of such groups is at most 3.
Elementary amenable groups that are torsion-free cannot have Hirsch length 3 without being solvable.
The classification gives concrete realizations for every solvable group of cohomological dimension 3.

Where Pith is reading between the lines

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

The result may help in studying the possible fundamental groups of 3-manifolds that are solvable.
Similar bounds might apply in related categories of groups with other finiteness conditions.
It suggests that the Hirsch length controls the solvability in the elementary amenable case for small values.

Load-bearing premise

The groups under consideration are torsion-free and elementary amenable, with Hirsch length and cohomological dimension defined via standard constructions.

What would settle it

Discovery of a torsion-free elementary amenable group of Hirsch length 3 whose commutator subgroup requires more than two steps to reach the trivial group, or a solvable group of cohomological dimension 3 outside the polycyclic, BS(1,n) semidirect product, and HNN extension classes.

read the original abstract

We show that torsion-free elementary amenable groups of Hirsch length $\leq3$ are solvable, of derived length $\leq3$. This class includes all solvable groups of cohomological dimension 3. We show also that groups in the latter subclass are either polycyclic, semidirect products $BS(1,n)\rtimes\mathbb{Z}$ or properly ascending HNN extensions with base $\mathbb{Z}^2$ or $\pi_1(Kb)$.

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.

Desk Editor's Note private letter to a colleague

Hillman shows torsion-free elementary amenable groups of Hirsch length at most 3 are solvable of derived length at most 3 and classifies the solvable cd=3 ones into three explicit families.

read the letter

The main result organizes torsion-free elementary amenable groups of Hirsch length ≤3 as solvable of derived length ≤3, and then lists all solvable groups of cohomological dimension 3 as polycyclic, BS(1,n)⋊ℤ, or properly ascending HNN extensions with base ℤ² or the Klein bottle fundamental group. This is a direct extension of earlier work on Hirsch length and cd bounds for amenable groups. The paper does well at pulling the known constructions into one place and showing the bound forces solvability without new counterexamples appearing. The argument relies on standard facts about elementary amenable groups and their invariants, so the contribution is the synthesis and the explicit list rather than new machinery. No circularity or unsupported steps show up in the claim structure. The soft spot is that the classification cases depend on prior results about subgroups and extensions, so any gap in those earlier facts would affect the completeness here; the abstract gives no sign of such a gap. This paper is for specialists in infinite solvable and amenable groups who need a concrete list for low Hirsch length or cd=3 examples. It is worth sending to a serious referee because the classification is sharp and the setting is narrow enough that verification would be straightforward.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that torsion-free elementary amenable groups of Hirsch length ≤3 are solvable of derived length ≤3. This includes all solvable groups of cohomological dimension 3, which are classified as either polycyclic, semidirect products BS(1,n)⋊ℤ, or properly ascending HNN extensions with base ℤ² or π₁(Kb).

Significance. If the results hold, the classification provides a concrete structural description of solvable groups of cohomological dimension 3 and links Hirsch length to solvability in the elementary amenable case. This builds directly on standard definitions and prior results about these invariants without introducing new ad-hoc parameters or entities, which strengthens the contribution to the literature on amenable groups.

minor comments (2)

[Abstract] The abbreviation 'Kb' for the Klein bottle should be introduced or referenced explicitly when first used in the classification statement.
[Introduction] The manuscript should include a brief reminder or citation to the standard definitions of Hirsch length and cohomological dimension for torsion-free groups to aid readers unfamiliar with the precise conventions used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive summary of the results, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives its main theorem from standard definitions of Hirsch length and cohomological dimension together with prior results on elementary amenable groups; the abstract and structure show no reduction of the claimed classification to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The argument is self-contained against external group-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard definitions and background theorems in group theory concerning elementary amenability, Hirsch length, solvability, and cohomological dimension; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)

domain assumption Standard definitions and properties of elementary amenable groups, Hirsch length, derived length, and cohomological dimension as used in the literature on infinite groups.
The statements presuppose these established concepts without re-deriving them.

pith-pipeline@v0.9.0 · 5583 in / 1346 out tokens · 26868 ms · 2026-05-24T13:21:30.942175+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that torsion-free elementary amenable groups of Hirsch length ≤3 are solvable, of derived length ≤3. This class includes all solvable groups of cohomological dimension 3.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

If G is a torsion-free virtually solvable group then c.d.G = h ⇔ G is of type FP ⇔ G is constructible [7].

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

and Bieri, R

Baumslag, G. and Bieri, R. Constructable solvable groups, Math. Z. 151 (1976), 249--257

work page 1976
[2]

and Strebel, R

Bieri, R. and Strebel, R. Almost finitely presentable soluble groups, Comment. Math. Helv. 53 (1978), 258--278

work page 1978
[3]

Carin, V. S. On soluble groups of type A_4 , Mat. Sbornik 94 (1960), 895--914

work page 1960
[4]

Davis, J. F. and Hillman, J. A. Aspherical 4-manifolds with elementary amenable fundamental group, in preparation

work page
[5]

Classification of soluble groups of cohomological dimension two, Math

Gildenhuys, D. Classification of soluble groups of cohomological dimension two, Math. Z. 166 (1979), 21--25

work page 1979
[6]

Hillman, J. A. and Linnell, P. A. Elementary amenable groups of finite Hirsch length are locally-finite by virtually solvable, J. Aust. Math. Soc. 52 (1992), 237--241

work page 1992
[7]

Kropholler, P. H. Cohomological dimension of soluble groups, J. Pure Appl. Alg. 43 (1986), 281--287

work page 1986
[8]

Robinson, D. J. S. A Course in the Theory of Groups, GTM 80, Springer-Verlag, Berlin -- Heidelberg -- New York (1982)

work page 1982
[9]

Wall, C. T. C. Finiteness conditions on CW complexes. II, Proc. Roy. Soc. Ser. A 295 (1966), 129--139

work page 1966
[10]

Homological Dimension of Discrete Groups, Queen Mary College Mathematics Notes (1976)

Bieri, R. Homological Dimension of Discrete Groups, Queen Mary College Mathematics Notes (1976)

work page 1976
[11]

and Strebel, R

Gildenhuys, D. and Strebel, R. On the cohomological dimension of soluble groups, Canad. Math. Bull. 24 (1981), 385--392

work page 1981
[12]

Kropholler, P. H. Soluble groups of type (FP)_ have finite torsion-free rank, Bull. London Math. Soc. 25 (1993), 558--566

work page 1993
[13]

Mal'cev, A. I. On some classes of infinite solvable groups, Mat. Sbornik 70 (1951), 567--588. If G is FP_2 and G/G' is infinite then G is an HNN extension H*_ with finitely generated base H BS78 , and the extension is ascending since G is solvable. Clearly h(H)=h(G)-1=2 , and c.d.G c.d.H+1 . In fact h(H) must be 2, for otherwise H Z and c.d.G=2 . In our n...

work page 1951

[1] [1]

and Bieri, R

Baumslag, G. and Bieri, R. Constructable solvable groups, Math. Z. 151 (1976), 249--257

work page 1976

[2] [2]

and Strebel, R

Bieri, R. and Strebel, R. Almost finitely presentable soluble groups, Comment. Math. Helv. 53 (1978), 258--278

work page 1978

[3] [3]

Carin, V. S. On soluble groups of type A_4 , Mat. Sbornik 94 (1960), 895--914

work page 1960

[4] [4]

Davis, J. F. and Hillman, J. A. Aspherical 4-manifolds with elementary amenable fundamental group, in preparation

work page

[5] [5]

Classification of soluble groups of cohomological dimension two, Math

Gildenhuys, D. Classification of soluble groups of cohomological dimension two, Math. Z. 166 (1979), 21--25

work page 1979

[6] [6]

Hillman, J. A. and Linnell, P. A. Elementary amenable groups of finite Hirsch length are locally-finite by virtually solvable, J. Aust. Math. Soc. 52 (1992), 237--241

work page 1992

[7] [7]

Kropholler, P. H. Cohomological dimension of soluble groups, J. Pure Appl. Alg. 43 (1986), 281--287

work page 1986

[8] [8]

Robinson, D. J. S. A Course in the Theory of Groups, GTM 80, Springer-Verlag, Berlin -- Heidelberg -- New York (1982)

work page 1982

[9] [9]

Wall, C. T. C. Finiteness conditions on CW complexes. II, Proc. Roy. Soc. Ser. A 295 (1966), 129--139

work page 1966

[10] [10]

Homological Dimension of Discrete Groups, Queen Mary College Mathematics Notes (1976)

Bieri, R. Homological Dimension of Discrete Groups, Queen Mary College Mathematics Notes (1976)

work page 1976

[11] [11]

and Strebel, R

Gildenhuys, D. and Strebel, R. On the cohomological dimension of soluble groups, Canad. Math. Bull. 24 (1981), 385--392

work page 1981

[12] [12]

Kropholler, P. H. Soluble groups of type (FP)_ have finite torsion-free rank, Bull. London Math. Soc. 25 (1993), 558--566

work page 1993

[13] [13]

Mal'cev, A. I. On some classes of infinite solvable groups, Mat. Sbornik 70 (1951), 567--588. If G is FP_2 and G/G' is infinite then G is an HNN extension H*_ with finitely generated base H BS78 , and the extension is ascending since G is solvable. Clearly h(H)=h(G)-1=2 , and c.d.G c.d.H+1 . In fact h(H) must be 2, for otherwise H Z and c.d.G=2 . In our n...

work page 1951