{"id":"77350fcf-8e3a-4073-b6ad-b4d3f66ae0c4","arxiv_id":"2608.09796","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sigma-compact etale groupoids receive a canonical coarse metric; the new fiberwise amenability notions recover group and metric amenability, imply invariant measures, coincide for minimal groupoids, and yield a Følner-paradoxical dichotomy.","lead":"The paper introduces fiberwise amenability for etale groupoids, a coarse-geometric notion that recovers amenability of the acting group for transformation groupoids and metric amenability for coarse spaces. The payoff is a Følner-versus-paradoxical dichotomy for minimal etale groupoids, which the authors intend to use in a sequel on almost elementariness.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Local Slice Lemma proof is incomplete: the map f is not well-defined on the stated domain.","rationale":"The reader correctly identified Lemma 5.11 as the load-bearing premise, but did not spot the specific defect in its proof. The lemma is indeed central: Theorem 5.14 (minimal implies ubiquitous) uses Lemma 5.12, which invokes Lemma 5.11 to clone Følner sets across nearby fibers, and Proposition 5.18 uses the same mechanism for coarse groupoids. Without a valid Local Slice Lemma, the conclusion that fiberwise amenability implies ubiquitous fiberwise amenability for minimal groupoids does not follow from the written proof. However, the defect is local and repairable: intersecting the neighborhood U with the intersection of the sources of the chosen bisections gives a well-defined f, and the subsequent compactness/continuity argument preserves both the covering property (5.2) and the ε-distortion estimate. Thus the mathematical claim is likely true, but the paper should be accepted only after this patch is supplied. Other components of the central argument, including Theorem A's existence/uniqueness of coarse continuous length functions and Propositions 3.8/3.9 underlying the dichotomy, appear sound; no further load-bearing objection was identified.","tokens_in":25763,"tokens_out":48899,"duration_ms":405121,"concrete_test":"Re-derive Lemma 5.11 with U' = U ∩ ⋂_{x∈\\bar B_ρ(u,S)} s(U_x) in place of U, and verify that for every v∈U' the equalities f(\\bar B_ρ(u,S)×{v}) = \\bar B_ρ(v,S) and the ε-distortion bound still hold. If this verification succeeds, Theorem 5.14 and Proposition 5.18 are supported; if it fails, exhibit an explicit étale groupoid and a point v∈U' for which the transported set is not the full ball, which would invalidate the cloning mechanism used in the minimal equivalence theorem.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 5.11 (Local Slice Lemma) is load-bearing: it drives Lemma 5.12 and hence Theorem 5.14 and Proposition 5.18. As printed, the proof is not well-defined. For each x in the finite ball \\bar B_ρ(u,S), the authors choose an open bisection U_x containing x and set f_x = (s|_{U_x})^{-1}. They then define L = ℓ^{-1}([0,S]) \\setminus ∪_x U_x and U = G^{(0)} \\setminus s(L), and claim U = {v : \\bar B_ρ(v,S) ⊆ ∪_x U_x} (equation (5.1)), so that f(x,v)=f_x(v) is defined for all (x,v) in \\bar B_ρ(u,S)×U. But f_x is defined only on s(U_x). From (5.1) we only learn that for v∈U the set \\bar B_ρ(v,S) is covered by the U_x, and in particular v itself lies in some U_x; it does not follow that v∈s(U_x) for every x∈\\bar B_ρ(u,S). Thus f(x,v) is undefined whenever v∉s(U_x). The gap is repairable: replace U by U' = U ∩ ⋂_{x∈\\bar B_ρ(u,S)} s(U_x), which is still an open neighborhood of u; the covering (5.1) remains valid on U', and the rest of the argument (injectivity, (5.2), metric estimates) goes through unchanged. Since the repair is straightforward, I do not regard the claim as false, but the printed proof of a central technical lemma is incomplete and should be patched before acceptance.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces fiberwise amenability and ubiquitous fiberwise amenability for étale groupoids, viewed through a canonical coarse metric structure. The authors prove that every σ-compact étale groupoid admits a proper continuous length function unique up to coarse equivalence (Theorem A), that fiberwise amenability with compact unit space yields an invariant probability measure (Theorem B), that for minimal groupoids fiberwise amenability coincides with its ubiquitous variant (Theorem C), and that these notions yield a Følner-versus-paradoxical dichotomy (Theorem D). The framework connects metric amenability of coarse spaces with fiberwise amenability of coarse groupoids and distinguishes the new notion from topological amenability.","tokens_in":26053,"tokens_out":11384,"duration_ms":91919,"significance":"If the technical gaps are patched, this is a valuable contribution to the coarse geometry of étale groupoids. The canonical coarse metric construction is natural and extends the classical passage from countable groups to proper invariant metrics. The invariant-measure consequence and the Følner-paradoxical dichotomy are likely to be useful in the study of almost finiteness, pure infiniteness, and the authors' planned notion of almost elementariness. The paper is generally well organized, with explicit statements of dependencies between lemmas, and it correctly situates the new notions relative to known examples such as transformation groupoids and coarse groupoids.","major_comments":[{"comment":"The proof of the Local Slice Lemma is incomplete because the map f is not defined on the stated domain. For v in U, equation (5.1) only ensures that v lies in the union of the source sets s(U_x) over x in the finite ball, not that v belongs to s(U_x) for every such x. Hence f(x,v) = f_x(v) is undefined when v is outside s(U_x). The gap is repairable: define U' = U ∩ ⋂_{x∈\\bar B_ρ(u,S)} s(U_x). Since each s(U_x) is an open neighborhood of u, U' is an open neighborhood of u, and the covering condition (5.1) remains valid on U'. The rest of the proof, including injectivity and the metric estimates, then goes through. Because Lemma 5.11 is used in Lemma 5.12, Theorem 5.14, and Proposition 5.18, this patch is necessary before the main applications can be accepted.","section":"Lemma 5.11"},{"comment":"The proof of Theorem D contains an incorrect reduction: it claims that by enlarging K we may assume K = ℓ^{-1}(r) for some r ≥ 0. An arbitrary compact set need not be contained in a level set of ℓ, and the metric propositions being invoked (Propositions 3.8 and 3.9) concern sublevel balls ℓ^{-1}([0,r]), not level sets. The reduction should presumably read K = ℓ^{-1}([0,r]) (or a sublevel set containing the original K). Without this correction, the conversion of Propositions 3.8 and 3.9 into the groupoid setting does not yield the stated conclusion about |KF| or about sets of the form Kx.","section":"Theorem 5.22"}],"minor_comments":[{"comment":"In the proof of Proposition 3.9, the inequality immediately after applying Lemma 3.7 says d(x_i,x_j) > r, but disjointness of the closed balls \\bar B(x_i,r) requires d(x_i,x_j) > 2r. The use of N_X(2r) in the choice of k indicates that the intended application of Lemma 3.7 is with s = 2r, so the displayed inequality should be corrected to d(x_i,x_j) > 2r.","section":"Proposition 3.9"},{"comment":"In the continuity argument after equation (4.3), the displayed equality (f^{-1}([1,N]))^j ∩ (˚δ^{(j)})^{-1}({y}) = {η_i^{(j)}(y) : i=1,...,m_j} is not literally true as stated, since some η_i^{(j)}(y) may lie outside f^{-1}([1,N])^j. The minimum formula for ℓ(y) remains valid because any tuple with ˚f^{(j)} < N automatically lies in f^{-1}([1,N])^j and tuples with larger ˚f^{(j)} do not affect the minimum, but the equality should be clarified.","section":"Theorem 4.11"},{"comment":"In Remark 5.17, the statement that ℓ^{-1}([0,r)) is contained in the compact set E_r is slightly imprecise: the containment is of the intersection of ℓ^{-1}([0,r)) with the dense subset Y × Y, and one should say that the closure is contained in E_r to justify properness. The intended argument is clear, but the wording invites confusion.","section":"Section 5.2"}],"recommendation":"major_revision","confidential_remarks":"The two major comments are both repairable: the Local Slice Lemma needs a straightforward domain restriction, and Theorem 5.22 needs the level set replaced by a sublevel set. However, because Lemma 5.11 is a central technical tool invoked throughout Section 5 and Theorem 5.22 is one of the headline results, I recommend a major revision rather than acceptance in the present form. The paper is otherwise well written and the mathematical program is attractive; I would be willing to review a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuinely new framework, not a repackaging. The canonical coarse length function (Theorem A) is a real extension of Struble and of Oyono-Oyono-Yu, and the Følner–paradoxical dichotomy (Theorem D) is a useful structural tool. The paper deserves a serious referee and, after small patches, acceptance.\n\nWhat is new: Definition 5.4 (fiberwise and ubiquitous fiberwise amenability) is not in prior literature; Theorem B (invariant measure from fiberwise amenability on compact unit space) and Theorem C (equivalence for minimal groupoids) are solid. The coarse geometry machinery in Sections 3–4 is mostly standard but carefully assembled, and the construction in Theorem 4.11 via the n-ary multiplication maps and Lemma 2.7 is convincing. I think the main claims hold.\n\nSoft spots:\n- Lemma 5.11 (Local Slice Lemma) as printed has a real domain gap. The proof defines f(x,v)=f_x(v) for (x,v) in \\bar B(u,S)×U, but (5.1) only gives \\bar B(v,S)⊆∪_x U_x, so v need not lie in the source of the particular U_x used for coordinate x. The fix is immediate: replace U by U ∩ ⋂_x s(U_x), still an open neighborhood of u, and the rest of the argument goes through. Because the lemma is load-bearing for Lemma 5.12 and Theorem 5.14, the proof should be patched before publication.\n- Proposition 3.9 has a radius typo: to get disjoint r-balls from Lemma 3.7, the centers need distance > 2r, not > r. The surrounding choice of N_X(2r) shows the intended statement.\n- Minor: the proof of Proposition 3.8 writes F as M ∪ (⊔ F_{x_i}) without explaining overlaps. Since the F_{x_i} are pairwise disjoint and |F| ≥ Σ|F_{x_i}|, the estimate still works, but a clarifying sentence would help.\n- Self-citation is not an issue here; the cited sequel [MW20] is motivational rather than load-bearing.\n\nVerdict: accept after minor revision. The framework is worth referee time and will likely be cited for Theorem A and Theorem D.","headline":"Genuinely new coarse-geometric framework for etale groupoids; the main results hold up, but the Local Slice Lemma has a real, repairable domain gap that should be patched before acceptance.","tokens_in":26623,"tokens_out":6050,"would_cite":true,"duration_ms":50786,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22A22","46L35","51F30","37A55","37B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every σ-compact étale groupoid carries a canonical invariant fiberwise extended metric, and minimal ones split into a Følner-like regime and a paradoxical regime according to fiberwise amenability.","keywords":["coarse geometry","fiberwise amenability","étale groupoids","ubiquitous fiberwise amenability","Følner sets","metric amenability","invariant measures","almost finiteness"],"falsifier":"Build a minimal σ-compact étale groupoid with compact unit space whose canonical fiber metric is amenable, yet for some $R,\\varepsilon$ the nearest $(R,\\varepsilon)$-Følner sets to some units lie at radii tending to infinity; Theorem 5.14 predicts this cannot happen. A direct place to probe is the cloning step: test Lemma 5.11 on a groupoid where the length function has no largest value below $R+\\varepsilon$ in a source fiber, to see whether balls of radius $S$ just below $R+\\varepsilon$ can still be cloned exactly onto all nearby fibers.","tokens_in":25520,"feed_emoji":"📏","tokens_out":8759,"duration_ms":74909,"temperature":0.7,"pith_summary":"The paper is trying to establish that amenability of an étale groupoid is a fiberwise coarse-geometric phenomenon, not just a global topological one. It proves that every σ-compact étale groupoid admits a canonical invariant fiberwise extended metric, unique up to coarse equivalence, and it defines fiberwise amenability as metric amenability of the source fibers under that metric. On compact unit spaces, fiberwise amenability forces the existence of a groupoid-invariant probability measure; for minimal groupoids it coincides with the stronger ubiquitous version, producing a genuine Følner-versus-paradoxical dichotomy. A reader should care because this gives a common geometric generalization of amenability of discrete groups and metric amenability of coarse spaces, and because the dichotomy is designed as a tool for distinguishing finite-like from infinite-like groupoids in the sequel the authors announce.","feed_headline":"Minimal étale groupoids are Følner or paradoxical","feed_subtitle":"A new coarse-geometric notion unifies group amenability and metric amenability and yields invariant measures.","key_machinery":"The load-bearing object is the canonical invariant fiberwise extended metric $\\rho_{\\mathcal G}$ induced by a coarse continuous length function $\\ell$ on the groupoid $\\mathcal G$: $\\rho(x,y)=\\ell(xy^{-1})$ when $s(x)=s(y)$, and $\\rho(x,y)=\\infty$ otherwise. Theorem A constructs $\\ell$ from an arbitrary proper continuous function on $\\mathcal G\\setminus\\mathcal G^{(0)}$ and shows any two coarse length functions are coarsely equivalent, so the metric is intrinsic. Uniform local finiteness makes every metric ball in a source fiber finite, and the Local Slice Lemma clones a ball in one source fiber homeomorphically onto nearby fibers with arbitrarily small metric distortion; this cloning mechanism carries Følner sets between fibers and is what upgrades fiberwise amenability to ubiquitous fiberwise amenability for minimal groupoids.","core_discovery":"The central claim is that amenability of an étale groupoid is encoded in the large-scale geometry of its source fibers. Theorem A constructs a proper continuous length function on every σ-compact étale groupoid and proves any two such length functions are coarsely equivalent, so the induced invariant fiberwise extended metric is canonical. Theorem C shows that for minimal groupoids, fiberwise amenability, meaning the existence of $(K,\\varepsilon)$-Følner sets for every compact $K$ and $\\varepsilon>0$, is equivalent to ubiquitous fiberwise amenability, where such Følner sets appear uniformly in a compact enlargement of every unit. Theorem D then turns this into a dichotomy: in the ubiquitous fiberwise amenable case every finite set can be enlarged into a Følner set inside a fixed compact enlargement, while in the non-fiberwise amenable case any compact set has arbitrarily many disjoint translated copies packed into a bounded enlargement.","pith_inferences":["The paper leaves implicit that fiberwise amenability is a coarse invariant of the groupoid: because the canonical metric is unique up to coarse equivalence, any property defined through it does not depend on the auxiliary continuous function used to build the length function.","A testable extension is to push the same dichotomy beyond minimal groupoids by isolating the role of recurrence, since the proof of Theorem 5.14 uses minimality only to transport a Følner set from one unit to all units.","The entourage formulation sketched in Remark 4.16 suggests the definitions could extend verbatim to non-σ-compact or non-Hausdorff étale groupoids, where continuous length functions may fail to exist but the coarse structure is still present.","One consequence the paper points toward but does not prove is that, for transformation groupoids, the dichotomy reinstates group amenability as the dividing line between finite-like and infinite-like behaviour of the associated C*-algebras."],"forward_implications":["For a transformation groupoid $X\\rtimes\\Gamma$, fiberwise amenability is equivalent to amenability of the acting group $\\Gamma$, not to topological amenability of the action.","For the coarse groupoid of a uniformly locally finite extended metric space, fiberwise amenability recovers metric amenability, and ubiquitous fiberwise amenability recovers its ubiquitous version.","A fiberwise amenable σ-compact étale groupoid with compact unit space has at least one invariant probability measure on the unit space.","Almost finiteness for ample groupoids implies ubiquitous fiberwise amenability, and the dichotomy in Theorem 5.22 is intended as a tool for the announced sequel on almost-elementariness.","There are minimal principal almost finite ample groupoids that are ubiquitously fiberwise amenable yet not topologically amenable, showing the new notion does not imply topological amenability."],"supporting_citations":[{"why":"Supplies the classical group-theoretic construction of a proper left-invariant metric, which is the model for Theorem A.","marker":"[Str74, Roe03]"},{"why":"Introduces metric amenability, the coarse-space notion that fiberwise amenability extends to groupoids.","marker":"[BW92]"},{"why":"Studies amenability of coarse spaces and supplies the equivalent Følner criteria used in Proposition 3.6.","marker":"[ALL W18b]"},{"why":"Constructs the coarse groupoid of a metric space and connects topological amenability to property A, providing the backdrop for Proposition 5.18.","marker":"[STY02]"},{"why":"Provides the foundational treatment of measured groupoid amenability that the new fiberwise notion complements.","marker":"[Ren80]"},{"why":"Gives topological amenability for locally compact groupoids and group actions, the notion fiberwise amenability is contrasted with.","marker":"[ADR00]"},{"why":"Defines almost finiteness for ample étale groupoids, which the paper shows implies ubiquitous fiberwise amenability.","marker":"[Mat12]"},{"why":"Produces the minimal principal almost finite ample groupoids that are not topologically amenable, used in Example 5.23.","marker":"[Ele18]"},{"why":"Constructs free actions with invariant probability measures by nonamenable groups, showing the converse of Theorem B fails.","marker":"[HM06]"}],"fun_headline_variants":["Følner or paradoxical: minimal étale groupoids split","Fiberwise amenability: coarse-geometric key to groupoid amenability","Minimal étale groupoids: Følner or paradoxical, never both","For transformation groupoids, fiberwise amenability equals group amenability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the local slice lemma, that a finite metric ball in one source fiber can be copied homeomorphically to every nearby fiber with arbitrarily small metric distortion, because without that cloning step fiberwise amenability would not imply ubiquitous fiberwise amenability even for minimal groupoids.","fun_headline_variants_meta":{"raw":{"variants":["Følner or paradoxical: minimal étale groupoids split","Fiberwise amenability: coarse-geometric key to groupoid amenability","Minimal étale groupoids: Følner or paradoxical, never both","For transformation groupoids, fiberwise amenability equals group amenability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001005,"raw_usage":{"total_tokens":4231,"prompt_tokens":910,"completion_tokens":3321,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":3241}},"tokens_in":526,"tokens_out":3321,"duration_ms":22213,"temperature":1.0,"reasoning_tokens":3241,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:37:03.463080+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build a minimal σ-compact étale groupoid with compact unit space whose canonical fiber metric is amenable, yet for some $R,\\varepsilon$ the nearest $(R,\\varepsilon)$-Følner sets to some units lie at radii tending to infinity; Theorem 5.14 predicts this cannot happen. A direct place to probe is the cloning step: test Lemma 5.11 on a groupoid where the length function has no largest value below $R+\\varepsilon$ in a source fiber, to see whether balls of radius $S$ just below $R+\\varepsilon$ can still be cloned exactly onto all nearby fibers.","supporting_citations":[],"review_version":1}