{"id":"348eeec3-f748-45e7-9cb0-d9b850c8d3b6","arxiv_id":"2607.20273","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A sparse product-neighbourhood condition implies fixed price one for discrete and locally compact groups, yielding new cases such as products of lcsc groups and lattices in exotic buildings.","lead":"This paper proves a flexible new condition that forces a group to have \"fixed price one\" — meaning all its free measure-preserving actions need exactly one generator on average. The condition covers products of locally compact groups, higher-rank lattices, and automorphism groups of affine buildings, including recently constructed exotic ones.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Locally compact proof hinges on unverified Abért–Mellick point-process theorems; if the Delone factor or Poisson-maximality fails for mixed discrete/continuous products, Corollary 1.7 falls.","rationale":"I read the manuscript in good faith. The discrete proof (Theorem 1.1) is self-contained, correct in its probabilistic estimates, and the use of the Abért–Weiss maximality of Bernoulli cost is standard. The locally compact proof (Theorem 1.3) is coherent given the point-process framework of [1], and the steps from the sparse product-neighbourhood condition to the cost inequality (4) are logically sound: the anchor-map construction, the Mecke-formula bound for blue–blue edges, and the bad-vertex estimate all check out. The product lemma (Lemma 2.3) is also correct and establishes the needed F_n for Corollary 1.7. The most load-bearing point is the reliance on [1] for the Delone factor, Poisson maximality, and the free-action-to-point-process representation. The paper does not reproduce these results and does not explicitly confirm that the hypotheses of [1] are met for all compactly generated unimodular lcsc groups, especially mixed discrete/continuous products. This is the same weakest assumption identified by the reader. Since this is a question of external support rather than an internal flaw, the reader's ACCEPT verdict with moderate confidence remains appropriate; the concern can be settled by checking the cited theorems in the product case.","tokens_in":9856,"tokens_out":46942,"duration_ms":403258,"concrete_test":"Inspect [1, Cor. 4.12, Prop. 4.13, Prop. 4.18] and verify their hypotheses for G = H × K, where H is a non-discrete compactly generated unimodular lcsc group (e.g., SL_2(R)) and K is an infinite discrete group. Concretely, exhibit an equivariant Delone-set factor of a unit-intensity Poisson process on such a product (or identify the exact hypothesis in [1] that covers this case). If the hypotheses hold, the concern is resolved; if the cited propositions exclude this case, Theorem 1.3 requires an additional argument for mixed discrete/continuous products.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central new theorem for locally compact groups (Thm 1.3) is not self-contained. Its proof invokes substantial results from Abért–Mellick [1]: the point-process cost formula cost(Π)-1 = inf_G(e(G)-int(Π)) (Eq. 3), the fact that Poisson processes realize maximal cost ([1, Thm 1.2]), and the representation of every free action as a point process ([1, Thm 1.1]). Most delicate is Prop. 3.1, which builds a locally finite connected S-bounded factor graph H on a Delone-set factor of a unit-intensity Poisson process; it cites [1, Cor. 4.12, Props. 4.13, 4.18] for the existence of the Delone set. The paper does not verify that these constructions apply to every compactly generated unimodular lcsc group, in particular to products H×K where H is continuous and K is discrete. If the Delone-factor construction requires extra hypotheses (e.g., no compact open subgroups, connectedness, or positive-dimensionality), then the estimate (4) and hence Thm 1.3, Cor. 1.7, and the lattice consequences would not be established for such groups. This is a concern about the proof's foundation, not an internal inconsistency; the discrete Thm 1.1 proof is self-contained and correct.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":10098,"tokens_out":25553,"duration_ms":232893,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Theorem 1.3 / §3"},{"comment":"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.","section":"Prop. 3.1 / §3.2"},{"comment":"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'.","section":"Lemma 2.3"},{"comment":"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.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The discrete theorem is solid and the paper is a valuable contribution. My main request is clarification: the locally compact proof relies on published results of [1] and should state the precise hypotheses being imported, and the discrete case of Theorem 1.3 should be explicitly reduced to Theorem 1.1. I do not see a reason to doubt the correctness of the cited Abért–Mellick machinery, so I do not regard the stress-test concern as a fatal objection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead 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.\n\nThe 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.\n\nMinor 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].\n\nVerdict: 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.\n\nRecommendation: 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.","headline":"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.","tokens_in":10639,"tokens_out":4498,"would_cite":true,"duration_ms":41164,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A20","20F65","22D40","22E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["fixed price one","cost of group actions","product-neighbourhood growth","locally compact groups","lattices","affine buildings","amenable groups","point processes"],"falsifier":"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","tokens_in":9681,"feed_emoji":"📐","tokens_out":9571,"duration_ms":95264,"temperature":0.7,"pith_summary":"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.","feed_headline":"Sparse product-neighbourhood growth forces fixed price one","feed_subtitle":"In discrete and locally compact groups, tiny FSF⁻¹-to-|F|² ratios guarantee all free actions have the same cost: one.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Tiny neighbourhood ratios force fixed price one","One ratio test: every free action costs one","Cost one proven for lattices and amenable groups","Shrinking sets pin down action cost to one"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Tiny neighbourhood ratios force fixed price one","One ratio test: every free action costs one","Cost one proven for lattices and amenable groups","Shrinking sets pin down action cost to one"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000186,"raw_usage":{"total_tokens":1097,"prompt_tokens":615,"completion_tokens":482,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":359,"completion_tokens_details":{"reasoning_tokens":423}},"tokens_in":359,"tokens_out":482,"duration_ms":6073,"temperature":1.0,"reasoning_tokens":423,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T10:21:54.979494+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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","supporting_citations":[],"review_version":1}