Asymmetry of $\ell^{2}$-cohomology via skewed F{\o}lner geometry

Nachi Avraham-Re'em; Zemer Kosloff

arxiv: 2605.12360 · v3 · pith:ARXBKH7Nnew · submitted 2026-05-12 · 🧮 math.GR · math.DS

Asymmetry of ell²-cohomology via skewed F{o}lner geometry

Nachi Avraham-Re'em , Zemer Kosloff This is my paper

Pith reviewed 2026-05-19 17:24 UTC · model grok-4.3

classification 🧮 math.GR math.DS

keywords ℓ²-cohomologyDirichlet structuresnilpotent groupsFølner sequencesleft schemesvirtually abelian groupsasymmetryBernoulli shifts

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The pith

For finitely generated nilpotent groups, left and right ℓ²-Dirichlet subspaces coincide exactly when the group is virtually abelian.

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

The paper proves that for finitely generated nilpotent groups the left and right ℓ²-Dirichlet subspaces on the group are equal if and only if the group is virtually abelian. A reader would care because this links the group's algebraic property of virtual commutativity directly to an analytic distinction in the associated Dirichlet forms, even though the left and right regular representations are unitarily equivalent. The work introduces skewed Følner geometry through left schemes to establish this equivalence and to construct examples of asymmetric Bernoulli shifts on non-virtually abelian nilpotent groups, which are the first such examples for amenable groups.

Core claim

The authors prove that for a finitely generated nilpotent group G the two canonical ℓ²-Dirichlet structures D₂(G,λ) and D₂(G,ρ) arising from the left and right regular actions coincide if and only if G is virtually abelian. They achieve this by developing left schemes that combine summability of left boundaries with displacement under right translation, and by refining them to recurrent left schemes that produce Bernoulli schemes with nonsingular weakly mixing left shifts but singular right shifts.

What carries the argument

The left scheme, a skewed Følner-geometric mechanism combining summability of left boundaries with displacement under right translation.

If this is right

Non-virtually abelian finitely generated nilpotent groups admit Bernoulli schemes whose left shift is nonsingular and weakly mixing while the right shift is singular.
The techniques extend to amenable wreath products over ℤ and solvable Baumslag-Solitar groups.
ℓ²-asymmetry in the virtually cyclic case comes from one-sided commensurated ends.

Where Pith is reading between the lines

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

This indicates that the asymmetry phenomenon may be present in other classes of amenable groups where similar Følner properties hold.
Explicit calculations on the integer Heisenberg group could confirm the predicted asymmetry in its Dirichlet spaces.
The classification suggests virtual abelianness acts as a sharp criterion for symmetry in these ℓ²-structures within the nilpotent category.

Load-bearing premise

The groups under consideration are finitely generated and nilpotent, which ensures the required commutator structure and properties of Følner sequences for left schemes to reveal the asymmetry.

What would settle it

Observe a finitely generated nilpotent group that is not virtually abelian but has equal left and right ℓ²-Dirichlet subspaces, or a virtually abelian one where they differ.

read the original abstract

We study the two $\ell^{2}$-Dirichlet structures on a countable group $G$ arising from the left and right regular actions on $\mathbb{R}^{G}$. Although the two regular representations are unitarily equivalent, their $\ell^{2}$-Dirichlet subspaces of $\mathbb{R}^{G}$ need not coincide. Our main result gives a complete classification of this asymmetry for countable amenable groups: $$\mathcal{D}_{2}\left(G,\lambda\right)=\mathcal{D}_{2}\left(G,\rho\right)\quad\Longleftrightarrow\quad G \text{ is an FC-group}.$$ The proof is based on a skewed F{\o}lner-geometric mechanism, called a left scheme, combining summability of left boundaries with displacement under a right translation. We develop this mechanism generally, and demonstrate it concretely in the Heisenberg group and amenable wreath products over $\mathbb{Z}$. We also show that this mechanism has a dynamical counterpart in the theory of nonsingular Bernoulli shifts: every countable amenable group that is not an FC-group admits Bernoulli schemes whose left shift is nonsingular, conservative and weakly mixing, whereas the right shift by some element is singular.

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

For finitely generated nilpotent groups the left and right ℓ²-Dirichlet spaces coincide exactly when the group is virtually abelian, shown via a new left-scheme construction on skewed Følner sequences.

read the letter

The main takeaway is that for finitely generated nilpotent groups, the ℓ²-Dirichlet spaces coming from the left and right regular actions are equal if and only if the group is virtually abelian. The paper reaches this by introducing left schemes that combine left-boundary summability with right-translation displacement, then refining them to recurrent versions that produce the desired asymmetric actions.

Referee Report

1 major / 2 minor

Summary. The paper proves that for finitely generated nilpotent groups G, the left and right ℓ²-Dirichlet spaces coincide (D₂(G,λ) = D₂(G,ρ)) if and only if G is virtually abelian. It introduces left schemes (combining summability of left boundaries with right-translation displacement) and refines them to recurrent left schemes to produce Bernoulli actions where the left shift is nonsingular and weakly mixing while the right shift is singular. The techniques extend to amenable wreath products over ℤ and solvable Baumslag-Solitar groups; the virtually cyclic case is classified separately via one-sided commensurated ends.

Significance. If the central equivalence holds, the work supplies the first constructions of Bernoulli schemes over amenable groups exhibiting this left-right asymmetry in singularity and mixing properties. The skewed Følner-geometric mechanism links ℓ²-asymmetry directly to virtual commutativity, providing a new geometric criterion in the nilpotent setting that complements existing algebraic characterizations. The robustness to wreath products and Baumslag-Solitar groups suggests broader applicability within amenable group theory.

major comments (1)

[Introduction and §3] The abstract and introduction sketch the construction of recurrent left schemes from left schemes via the commutator filtration of nilpotent groups, but the load-bearing step—showing that right-translation displacement forces singularity of the right Bernoulli shift while preserving nonsingularity on the left—requires explicit verification that the Dirichlet energy remains controlled under the refinement (see the paragraph following the definition of recurrent left schemes).

minor comments (2)

[§1] Notation for the left and right regular actions (λ and ρ) is introduced clearly in the abstract but should be restated with the precise definition of the Dirichlet form D₂ in the first section for readers unfamiliar with ℓ²-cohomology.
[Final section] The classification of the virtually cyclic case via one-sided commensurated ends is stated as an additional result; a short remark comparing its mechanism to the left-scheme approach would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for identifying this point in the construction of recurrent left schemes. We agree that an explicit verification of Dirichlet energy control is needed and will incorporate it in the revision.

read point-by-point responses

Referee: [Introduction and §3] The abstract and introduction sketch the construction of recurrent left schemes from left schemes via the commutator filtration of nilpotent groups, but the load-bearing step—showing that right-translation displacement forces singularity of the right Bernoulli shift while preserving nonsingularity on the left—requires explicit verification that the Dirichlet energy remains controlled under the refinement (see the paragraph following the definition of recurrent left schemes).

Authors: We agree that the load-bearing step requires a more explicit verification. In the revised manuscript we will expand the paragraph immediately following the definition of recurrent left schemes to include a direct computation. Using the commutator filtration, we will show that the right-translation displacement bounds the Dirichlet energy from below on the right while the left energy remains finite, thereby forcing singularity of the right Bernoulli shift and preserving nonsingularity and weak mixing on the left. This addition will make the argument self-contained without changing the overall proof strategy. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives the central equivalence D₂(G,λ) = D₂(G,ρ) ⇔ G virtually abelian for f.g. nilpotent groups from explicit constructions of left schemes (summability of left boundaries combined with right-translation displacement) and their refinement to recurrent left schemes. These rely on standard nilpotency properties (polynomial growth, commutator filtration) and Følner geometry, which are independent of the target statement. The virtually abelian direction follows directly from left/right actions coinciding up to finite-index subgroups. No steps reduce by definition, fitted parameters, or self-citation chains to the result itself; the argument is self-contained against external benchmarks in geometric group theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Relies on standard facts about nilpotent groups and Følner sequences; introduces two new geometric objects whose independent evidence is the proof itself.

axioms (1)

domain assumption Finitely generated nilpotent groups admit Følner sequences whose boundaries and commutator structure control summability and displacement.
Invoked to link virtual commutativity to equality of Dirichlet spaces.

invented entities (2)

left scheme no independent evidence
purpose: Mechanism that combines summability of left boundaries with displacement under right translation.
New object used to prove the main equivalence.
recurrent left scheme no independent evidence
purpose: Refinement that produces asymmetric Bernoulli schemes.
Used to construct the first such schemes over amenable groups.

pith-pipeline@v0.9.0 · 5805 in / 1297 out tokens · 50202 ms · 2026-05-19T17:24:37.824831+00:00 · methodology

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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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

We prove that for finitely generated nilpotent groups this ℓ²-asymmetry is governed by virtual commutativity: D₂(G,λ)=D₂(G,ρ) ⇔ G is virtually abelian. The proof introduces a skewed Følner-geometric mechanism, called a left scheme...
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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

left scheme ... summability of normalized left boundaries, and strong displacement under right translation by a distinguished infinite order element

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