A Product-Neighbourhood Criterion for Fixed Price One

Raz Slutsky

A Product-Neighbourhood Criterion for Fixed Price One

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Pith's one-line read This paper establishes a sparse product-neighbourhood criterion that forces fixed price one for discrete and locally compact groups, and uses it to prove fixed price one for products, higher-rank lattices, and affine-building automorphism g

desk verdict A clean, genuinely useful criterion for fixed price one with a self-contained discrete proof; the locally compact applications need a referee to verify the heavy external machinery in full generality. read the letter →

arxiv 2607.20273 v1 pith:IWBO7F55 submitted 2026-07-22 math.GR math.DS

Raz Slutsky This is my paper · ORCID

classification math.GRmath.DS MSC 37A2020F6522D4022E40

keywords fixedpriceonecostofgroupactionsproduct-neighbourhoodgrowthlocallycompactgroupslatticesaffinebuildingsamenablepointprocesses

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish a broad quantitative criterion for when all essentially free probability-measure-preserving actions of a group have the same minimal generating cost, namely one. The criterion asks only for finite (or compact) sets F_n containing the identity such that the size of the product-neighbourhood set F_n S F_n^{-1} is negligible compared with |F_n|^2. It proves the criterion in the discrete case and in the locally compact unimodular case, where it also forces fixed price one for every lattice in the group. From this it derives fixed price one for direct products of two noncompact compactly generated unimodular locally compact groups, for higher-rank semisimple groups over local fields, for cocompact automorphism groups of higher-rank affine buildings, and for amenable groups. A sympathetic reader would care because it unifies and extends many previously separate fixed-price-one results under one checkable hypothesis.

What carries the argument

The central object is the product-neighbourhood ratio α(F) = |F S F^{-1}| / |F|^2, and its Haar-measure analogue in the locally compact setting. The mechanism is a random-sparse-graph construction: colour points of the group (or of a Poisson process) as selected with small probability p, mark as 'bad' any unselected point whose F-translate avoids all selected points, then connect selected points whose F-patches nearly touch and connect bad points to nearby selected anchors. The resulting graph is connected and equivariant, and its expected degree at the identity is bounded in terms of p^2 α(F) plus exponentially small error terms. Tuning p against α(F) makes the expected cost tend to one, an

What would settle it

A concrete counterexample would be a finitely generated group Γ with a finite generating set S and finite sets F_n containing e such that |F_n S F_n^{-1}| / |F_n|^2 → 0, together with an essentially free probability-measure-preserving action of Γ whose cost is strictly greater than 1; the theorem rules out such a pair. In the locally compact version, the same would have to be found for a unimodular lcsc group and a lattice of it. One could search among groups already known to have fixed price greater than one, testing whether any sequence of sets in them satisfies the sparse product-neighbourh

Watch

Extended reading notes

Core claim

Theorem: if a finitely generated group Γ has a finite symmetric generating set S and a sequence of finite sets F_n containing e with |F_n S F_n^{-1}| / |F_n|^2 tending to 0, then Γ has fixed price one: every essentially free probability-measure-preserving action of Γ generates its orbit relation with cost exactly one. The locally compact analogue says the same for a noncompact compactly generated unimodular locally compact second countable group G when λ(F_n S F_n^{-1}) / λ(F_n)^2 tends to 0 for compact Borel sets F_n, and then every lattice in G also has fixed price one. The proof builds an equivariant connected random graph whose expected cost is forced close to one; because Bernoulli shif

Load-bearing premise

The discrete theorem leans on a known theorem that Bernoulli actions have maximal cost, and the locally compact theorem imports an entire point-process theory of cost — in particular that Poisson processes have maximal cost and that every free measure-preserving action can be represented as a point process — so if that imported theory has a case it does not cover, the product and lattice corollaries do not follow.

Editorial extensions

If this is right

Every product G1 × G2 of two noncompact, compactly generated, unimodular locally compact second countable groups has fixed price one, and so does every lattice in such a product — this answers an open question about products.
Connected semisimple groups of rank at least two over local fields, including positive characteristic, have fixed price one, as do their lattices.
Closed unimodular subgroups acting cocompactly on locally finite thick regular affine buildings of Euclidean rank at least two have fixed price one; this includes lattices in recently constructed exotic buildings.
Every infinite finitely generated amenable group satisfies the criterion, giving a direct proof of its fixed price one that does not rely on a structural classification theorem for amenable equivalence relations.
Any group satisfying the criterion has vanishing first ℓ²-Betti number and zero rank gradient along any residual chain of finite-index subgroups.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The criterion is quantitative: one could in principle search computationally for sets F_n in a given finitely generated group and certify fixed price one, making the invariant a priori checkable in cases where growth is not fully described.
The reliance on maximality of Bernoulli and Poisson costs suggests the method is sharp where it applies; if a group satisfies the sparse ratio condition it must have fixed price one, so groups that might fail fixed price one are exactly those where every F_n has a non-negligible product-neighbourhood ratio.
The locally compact proof uses unimodularity essentially (through mass transport and Haar-measure symmetry); testing whether a non-unimodular analogue holds would show whether the criterion is a genuinely general phenomenon or tied to unimodular groups.
The metric corollary gives a concrete growth threshold — ball families satisfying |B(2R)| / |B(R)|^2 → 0 force fixed price one — which connects the invariant to volume-growth dichotomies and is a natural place to look for counterexamples or extensions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Read this one: Slutsky has found a sparse product-neighbourhood condition that implies fixed price one, proves it from scratch in the discrete case, then pushes it to locally compact groups with applications to products and to lattices in affine buildings, including exotic C2 examples and an answer to an Abért–Mellick question. The discrete proof is the gem: Bernoulli shift, patch graph, Abért–Weiss, and a clean estimate that makes the cost tend to one. The criterion also recovers amenable groups and Gaboriau's weak normality criterion, with honest credit to Khezeli and to Frączyk–Mellick–Wilkens for the earlier cases.

The soft spot is the locally compact version. Theorem 1.3 is stated for all noncompact compactly generated unimodular lcsc groups, and Corollary 1.7 uses it for products including mixed discrete/continuous factors. The proof in Section 3, however, assumes from the start that G is nondiscrete and leans on substantial machinery from Abért–Mellick: the point-process cost formula, Poisson maximality, and especially Proposition 3.1, which produces a Delone-set factor and a compactly supported connected graph from a Poisson process. The paper cites [1] for these but never verifies that every group in the theorem's scope satisfies the hypotheses of those cited results. The stress-test worry about products of a continuous and a discrete group is not addressed in the text. This is a presentation gap rather than an internal contradiction — the discrete part of Theorem 1.3 is already covered by Theorem 1.1, and mixed products are nondiscrete, so Section 3 would apply if the point-process theory is as general as [1] claims. But a referee needs to check that Prop 3.1 actually works in that mixed setting. If yes, the paper is complete; if no, Corollary 1.7 needs a separate proof.

Minor issue: Section 3 says "nondiscrete" while Theorem 1.3 does not; the mismatch should be cleaned up. The building applications use growth estimates that look plausible, and the lattice consequence follows from the induction formula in [9].

Verdict: this deserves a serious referee. The discrete half is solid and will be cited regardless; the locally compact half is important but has an external-hypothesis gap that needs chasing down.

Recommendation: send it to a referee who knows the Abért–Mellick paper, and ask them to verify Prop 3.1 in the full generality of Theorem 1.3.

Referee Report

0 major / 4 minor

Summary. This paper proves a flexible sufficient criterion for fixed price one. In the discrete setting (Theorem 1.1), a finitely generated group Γ with finite symmetric generating set S has fixed price one if there are finite sets F_n containing the identity with |F_n S F_n^{-1}|/|F_n|^2 → 0. The proof selects points from a Bernoulli shift, forms patches F, builds a connected equivariant graph on selected and 'bad' points, bounds its cost using elementary probability, and applies the Abért–Weiss theorem that Bernoulli actions have maximal cost. Applications include weak normality, amenability, finite-subgroup q-normality, and direct products of infinite finitely generated groups. The locally compact theorem (Theorem 1.3) is proved using the Abért–Mellick point-process framework: from a unit-intensity Poisson process one obtains a Delone factor graph (Proposition 3.1), then a sparse anchored process via the Mecke formula and mass transport; the resulting cost bound gives fixed price one for G and for lattices in G. Consequences cover higher-rank groups over local fields, automorphism groups of affine buildings, and products of two noncompact compactly generated unimodular lcsc groups (Corollary 1.7).

Significance. The discrete criterion is elegant, self-contained, and genuinely flexible, with complete estimates and no fitted parameters. It recovers Gaboriau's weak normality criterion, amenability, and direct-product cases, and provides a uniform proof of several existing results. The locally compact version and Corollary 1.7 answer a question of Abért–Mellick and extend recent work of Frączyk–Mellick–Wilkens and Mellick. If the cited point-process machinery of [1] is accepted, the applications to higher-rank groups and affine buildings are substantial. The main strength is the conceptual unification and the simplicity of the discrete proof.

minor comments (4)

[Theorem 1.3 / §3] The theorem is stated for every noncompact compactly generated unimodular lcsc group, but §3 begins with 'Throughout this section, G is a noncompact, nondiscrete...' and the proof of Theorem 1.3 does not explicitly handle the discrete case. Since a compactly generated discrete group is finitely generated, this case is already covered by Theorem 1.1 (and Corollary 2.4), so the gap is local, but a sentence should be added to make the scope precise.
[Prop. 3.1 / §3.2] Proposition 3.1 is load-bearing for Theorem 1.3 and is asserted to be 'contained in [1]'. The proof sketches the key steps, but a reader cannot verify from this paper alone that the cited results (Cor. 4.12, Props. 4.13, 4.18 of [1]) apply to every compactly generated unimodular lcsc group, including products with a discrete factor. Please add a precise statement of which hypotheses are used and a remark confirming that the stated generality of Theorem 1.3 is covered. This is a transparency issue rather than an internal inconsistency: the reliance on a published source is legitimate.
[Lemma 2.3] The submultiplicativity claim for b_i is not fully demonstrated. The convolution argument as written gives submultiplicativity up to a constant depending on K_i unless additional care is taken; since equation (2) already absorbs a constant b_i(3), the proof is salvageable, but the statement or proof should be adjusted to 'quasi-submultiplicative'.
[Throughout] Minor typos: 'le e∈S' in Theorem 1.3 should be 'let e∈S'; equation numbers (1) and (2) are used both in §2.2 and §3.3, which is confusing.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the criterion is a new sufficient condition proved using standard external theorems; applications are independent.

full rationale

The paper's central claim (Theorems 1.1 and 1.3) is a conditional criterion: if a group admits sets F_n with sparse product-neighbourhood, then it has fixed price one. The discrete proof constructs equivariant graphs from Bernoulli shifts and bounds their expected degree; the only external input is Abért–Weiss's theorem that Bernoulli actions have maximal cost [4], an independent published result by other authors. The locally compact proof invokes the point-process cost framework of Abért–Mellick [1]: formula (3) for cost in terms of factor graphs, maximal cost of Poisson processes, representation of free actions by point processes, and the Delone factor construction in Proposition 3.1. These are external, non-circular results; they are not results of the present paper, nor are they derived from the conclusion. The paper does not fit any parameter to the target conclusion or rename a known result as a new one. Corollaries about products and lattices are deduced by verifying the hypothesis (Lemma 2.3), not by assuming the conclusion. Even though the locally compact proof is not self-contained and depends on substantial external theorems, dependency on other work is not circularity in the sense of this review. No self-citation of the author is load-bearing, and no hypothesis is defined in terms of fixed price one. Hence no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

The central theorems rest on standard external results in measured group theory and point-process theory (Abért–Weiss, Abért–Mellick, Gaboriau, Mecke, mass transport). No numbers are fitted to data and no new structural entities are postulated; the bad points, anchors, and patch graphs are internal proof devices.

assumptions (10)

domain assumption Bernoulli actions have maximal cost among essentially free pmp actions (Abért–Weiss)
Used at the end of Theorem 1.1 (Section 2.1) to pass from cost of Bernoulli shifts to bound for all free actions.
domain assumption Gaboriau's compression formula (cost(R)-1 = P(A)(cost(R|A)-1))
Used in Section 2.1 to relate the cost of the Bernoulli relation to its restriction to the complete section A.
domain assumption Cost of a pmp action of an lcsc group equals inf over factor graphs of (edge intensity - intensity) [1, Def 4.1]
Equation (3) in Section 3.2 is the basis of the locally compact proof.
domain assumption Poisson processes have maximal cost and every free pmp lcsc action is isomorphic to a point process [1, Thm 1.1, 1.2]
Used in Theorem 1.3 to reduce cost computation of arbitrary actions to cost of Poisson-process-related factor graphs.
domain assumption A unit-intensity Poisson process admits an equivariant Delone set and connected factor graph with edges in S [1, Cor 4.12, Prop 4.13, 4.18]
Proposition 3.1 is proved using these results from [1].
standard math Mecke formula for Poisson processes [18, Thm 4.4]
Used in Section 3.2 to bound blue-blue edge intensity by p^2 λ(FSF^{-1}).
domain assumption Mass-transport principle for unimodular locally compact groups
Used in Section 3.2 to bound edge intensity of the subgraph of bad vertices.
domain assumption Cartan integration / Macdonald volume estimates for semisimple groups over local fields [6,21]
Used in Corollary 1.5 to verify the sparse product-neighbourhood condition via λ(F_R) ≍ e^R R^{rank-1}.
domain assumption Spherical growth estimate for affine buildings: ∑_{L(w)≤T} q_w ≍ e^T T^{r-1} [5,14]
Used in Corollary 1.6 to verify the sparse condition for automorphism groups of affine buildings.
standard math Ruzsa triangle inequality for finite subsets of groups
Used in the proofs of Corollaries 1.9 and 1.2 to build Følner-type sets satisfying the sparse condition.

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Pith. "Pith review of A Product-Neighbourhood Criterion for Fixed Price One." pith.science (2026). https://pith.science/paper/IWBO7F55

@misc{pith2026260720273,
  author       = {Pith},
  title        = {Pith review of: A Product-Neighbourhood Criterion for Fixed Price One},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IWBO7F55}},
  note         = {Machine review of arXiv:2607.20273}
}

read the original 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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Works this paper leans on

26 extracted references · 6 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Abért and S

M. Abért and S. Mellick,Point processes, cost, and the growth of rank in locally compact groups, Israel J. Math. 251(2022), 48–155, doi:10.1007/s11856-022-2445-9

doi:10.1007/s11856-022-2445-9 2022

[2] [9]

Frączyk, S

M. Frączyk, S. Mellick, and A. Wilkens,Poisson–Voronoi tessellations and fixed price in higher rank, Ann. of Math., to appear; arXiv:2307.01194

arXiv

[3] [2]

Abért and N

M. Abért and N. Nikolov,Rank gradient, cost of groups and the rank versus Heegaard genus problem, J. Eur. Math. Soc. (JEMS)14(2012), no. 5, 1657–1677, doi:10.4171/JEMS/344

doi:10.4171/jems/344 2012

[4] [3]

Abért and L

M. Abért and L. M. Tóth,Uniform rank gradient, cost and local-global convergence, Trans. Amer. Math. Soc. 373(2020), no. 4, 2311–2329, doi:10.1090/tran/8008

work page doi:10.1090/tran/8008 2020

[5] [4]

Abért and B

M. Abért and B. Weiss,Bernoulli actions are weakly contained in any free action, Ergodic Theory Dynam. Systems33(2013), no. 2, 323–333, doi:10.1017/S0143385711000988

doi:10.1017/s0143385711000988 2013

[6] [5]

Abramenko, J

P. Abramenko, J. Parkinson and H. Van Maldeghem,Distance regularity in buildings and structure constants in Hecke algebras, J. Algebra481(2017), 158–187, doi:10.1016/j.jalgebra.2017.03.001

work page doi:10.1016/j.jalgebra.2017.03.001 2017

[7] [6]

Benoist and J.-F

Y. Benoist and J.-F. Quint,Random Walks on Reductive Groups, Ergebnisse der Mathematik und ihrer Grenzge- biete, vol. 62, Springer, Cham, 2016, doi:10.1007/978-3-319-47721-3

doi:10.1007/978-3-319-47721-3 2016

[8] [7]

Bevilacqua and L

E. Bevilacqua and L. Bowen,Metric criteria for fixed price of countable groups, arXiv:2510.05459, 2025

arXiv 2025