arXiv:2607.20273v1

A Product-Neighbourhood Criterion for Fixed Price One

Raz Slutsky

math.GRmath.DS37A2022F1020G25

Paper metadata

Original submission: July 22, 2026
Claimed MSC: 37A20, 22F10, 20G25
arXiv categories: math.GR, math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 10 pages

Abstract

We prove a flexible criterion for fixed price one, applying in particular to higher-rank lattices over local fields and automorphism groups of affine buildings, including lattices in exotic buildings. It also recovers fixed price one for amenable groups. We then establish a locally compact version, and deduce that every product of two noncompact, compactly generated, unimodular locally compact groups, as well as every lattice in such a product, has fixed price one.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

Both criteria and their applications are supported. Theorem 1.1 proves fixed price one for discrete groups with finite product neighbourhoods. Theorem 1.3 proves the compact-neighbourhood analogue for noncompact compactly generated unimodular lcsc groups. Corollaries 1.4–1.9 verify those hypotheses for the listed growth classes, higher-rank groups, affine-building groups, products, lattices, and infinite amenable groups.

Theorem 1.1Correct

Discrete product-neighbourhood criterion for fixed price one

Pages 2–3 · Theorem 1.1 · arXiv:2607.20273v1

A finitely generated infinite discrete group satisfying the stated finite-set product-neighbourhood condition has fixed price one. The condition is used uniformly for each finite subset chosen in the graphing construction, and the conclusion concerns every free probability-measure-preserving action, not only the Bernoulli model used to build the comparison graphing. The compression identity transfers the arbitrarily small excess cost from the positive-measure section to the original relation with the normalization shown in the theorem.

Theorem 1.3Correct

Locally compact product-neighbourhood criterion

Pages 3–4 · Theorem 1.3 · arXiv:2607.20273v1

For a noncompact compactly generated unimodular lcsc group, the compact-set product-neighbourhood hypothesis yields fixed price one in the cross-section sense used by the paper. The Poisson construction is equivariant and locally finite, and unimodularity is invoked in the Palm mass-transport calculation. Second countability gives the measurable cross section. These are exactly the structural assumptions listed in the theorem.

Corollaries 1.4–1.9Correct

Verification for the announced group classes

Pages 4–5 · Corollaries 1.4–1.9 · arXiv:2607.20273v1

The metric-growth condition in Corollary 1.4 directly gives the compact-neighbourhood inclusions. Higher-rank semisimple groups and regular affine-building automorphism groups satisfy it through their Cartan/building geometry. Direct products use rectangles in the two factors; lattices inherit fixed price through the cited induction formula; and infinite amenable groups already have fixed price one. Each corollary checks infinitude or noncompactness before applying the relevant criterion.

02Proofs2 reported findingsCorrect

Bernoulli compression and locally compact transfer. Finite product neighbourhoods are placed as random patches in a Bernoulli action. The hypothesis guarantees overlap chains connecting every generator direction, so the patch graph spans almost every orbit. Sparse independent anchors pay for the extra patch edges, making the cost on a complete section arbitrarily close to one. The compression formula is applied with the section measure in the correct denominator, and the Bernoulli weak-containment/cost comparison extends the bound to every free action. Letting the anchor intensity tend to zero proves fixed price one.

Patch graph and Poisson constructionCorrect and complete

Bernoulli compression and locally compact transfer

Pages 5–15 · Sections 2–3 · arXiv:2607.20273v1

Finite product neighbourhoods are placed as random patches in a Bernoulli action. The hypothesis guarantees overlap chains connecting every generator direction, so the patch graph spans almost every orbit. Sparse independent anchors pay for the extra patch edges, making the cost on a complete section arbitrarily close to one. The compression formula is applied with the section measure in the correct denominator, and the Bernoulli weak-containment/cost comparison extends the bound to every free action. Letting the anchor intensity tend to zero proves fixed price one.

Section 3Correct and complete

Poisson anchors and locally compact cross sections

Pages 11–15 · Section 3 · arXiv:2607.20273v1

A unit-intensity Poisson process is thinned equivariantly to separated anchors, ensuring local finiteness. Compact patches around anchors overlap according to the product-neighbourhood condition and hence connect the cross-section relation. Campbell–Palm and unimodularity compute expected degree without directional bias, while rare anchors make the excess cost arbitrarily small. Exhausting compact patches preserves connectivity and allows monotone passage to the full graphing.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.20273v1
Authors listed: Raz Slutsky
Audit date: August 18, 2026

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A Product-Neighbourhood Criterion for Fixed Price One

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Discrete product-neighbourhood criterion for fixed price one

Locally compact product-neighbourhood criterion

Verification for the announced group classes

Bernoulli compression and locally compact transfer

Poisson anchors and locally compact cross sections

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