{"id":"149536d8-9605-403b-8f06-54183d6b4dde","arxiv_id":"2508.01347","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Measured embedding dimension and volume give upper bounds for Betti number gradients and logarithmic torsion homology gradients of residually finite groups.","lead":"This paper proves that the growth of Betti numbers and torsion in homology along finite-index subgroups of a group is bounded by two new dynamical quantities, the measured embedding dimension and the measured embedding volume. These quantities are computed from measure-preserving actions of the group, giving a new bridge between dynamics and homological growth.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the flatness base-change is supported by a cited freeness theorem and the retraction step is valid; the Corollary 16.6 concern does not land because the diagonal action with a trivial factor remains essentially free.","rationale":"The reader's weakest assumption was flatness of the crossed product ring over ZΓ. I checked this step in detail: the canonical isomorphism in Proposition 2.10 is valid, and the flatness conclusion is supported by the cited freeness of L∞(α,Z) as an abelian group, with the finite-field case trivial. The retraction argument in Theorem 7.6 uses the fundamental lemma in a setting where the target complex D*(i) is not a resolution; however, the existence of a left inverse g*(i) and the homotopy g*(i)∘f*(i) ≃ id follow from projectivity of D_n(i), exactness of C*(i), and the standard uniqueness statement for chain maps from a projective resolution to a resolution. I also examined the Corollary 16.6 issue raised by the reader: the product of an essentially free action with a trivial action is still essentially free, so there is no hidden non-free extension of the setup. Since the central Theorem 1.2 does not appear to have a load-bearing gap, I do not endorse the specific conditional concern. I retain the UNCHANGED verdict only because the proof is long, partly analytic, and not machine-checked, so a conservative conditional stance is defensible; no correction to the paper's mathematics is required by my reading.","tokens_in":77320,"tokens_out":48080,"duration_ms":615504,"concrete_test":"Independently verify the freeness theorem cited as [Ste85] for L∞(X,Z) on a standard probability space; if that theorem is correct, Proposition 2.10 and the base-change step in Theorems 7.6 and 8.6 are sound. A useful secondary check is to recompute the degree-one profinite example for F_d and confirm that medim equals d−1, matching the claimed equality in Proposition 12.3(iv).","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the proof of Theorem 1.2 as sound. The most insecure point in the chain is Proposition 2.10, since Theorems 7.6 and 8.6 need L∞(α) * Γ to be flat over ZΓ. But the claimed canonical isomorphism (L∞(α) * Γ) ⊗_{ZΓ} M ≅ L∞(α) ⊗_Z M is correct for the natural right ZΓ-module structure, and flatness reduces to flatness of L∞(α) over Z. For Z a finite field this is immediate, and for Z = Z the paper cites the known freeness of L∞(α) as an abelian group [Ste85]; no internal gap is apparent. The retraction in Theorem 7.6 is also valid: although D*(i) need not be a resolution, the left inverse g*(i) exists by inductively lifting along the exact complex C*(i) using projectivity of D_n(i), and uniqueness in the fundamental lemma gives g*(i) ∘ f*(i) ≃ id. Finally, the reader's separate concern about Corollary 16.6 appears unfounded: if α is essentially free and β is trivial, then γ·(x,y) = (γx, y), so the stabilizer of (x,y) is trivial for µ-a.e. x; the diagonal action is therefore a standard action in the paper's sense. I did not find a load-bearing defect in the central argument.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two new dynamical invariants, the measured embedding dimension medim_n^Z(α) and the measured embedding volume mevol_n(α), defined as infima over projective chain complexes over crossed product rings admitting an α-embedding. It then proves upper bounds for Betti-number gradients and logarithmic torsion-homology gradients along residual chains of finite-index normal subgroups in terms of these invariants for the profinite completion action (Theorem 1.2). The proof builds an elaborate quantitative homological algebra framework: norms and supports, almost-equality, a Gromov–Hausdorff distance for marked projective complexes, strictification and deformation theorems, and a passage from adapted dynamical embeddings to homological retracts over finite-index subgroups. The paper also states an upper bound on L2-Betti numbers (Theorem 1.3), monotonicity under weak containment, invariance under weak bounded orbit equivalence, and calculations for amenable, free, surface, product, and finite-index-subgroup situations, as well as comparisons with cost and integral foliated simplicial volume.","tokens_in":77616,"tokens_out":32532,"duration_ms":417226,"significance":"If the main estimates hold, the paper creates a genuinely new bridge between measured group theory and homology growth: homology gradients are bounded by an infimum over essentially free dynamical systems, and the invariants have good inheritance properties such as weak-containment monotonicity and weak bounded orbit equivalence invariance. The proofs are unusually explicit, with the full approximation chain from dynamical embeddings to finite-index-retract estimates written out in Sections 3–8. I found the main chain supporting Theorem 1.2 convincing: Proposition 2.10 supplies the needed flatness base change, Theorem 7.6 constructs the retractions, and Theorems 8.4 and 8.5 convert them into gradient estimates. There are no fitted parameters or circular reductions. My concerns are localized to the finite-field scope of the L2-Betti statement and to a missing component in one translation-invariance constant used for weak containment.","major_comments":[{"comment":"The proof of Theorem 8.6 is only valid when the coefficient ring Z is the ring of integers. For Z = F_p, the expression N R ⊗_{ZΓ} C_* used in the proof is not defined: N R is a complex von Neumann algebra and there is no unital ring homomorphism F_pΓ → N R. Consequently the claim b_n^{(2)}(Γ) ≤ medim^{F_p}_n(α) is not proved. This is not merely cosmetic, because Propositions 12.3(iii) and 12.4(iii) invoke Theorem 8.6 with Z allowed to be a finite field. Please either restrict Theorem 1.3/8.6 to integer coefficients and give a separate argument for the finite-field lower bounds in Section 12 (for free and surface groups such bounds can be obtained from Theorem 1.2 together with EMD* and weak containment, but this is not written), or provide a genuine finite-field proof.","section":"Theorem 8.6 / Section 8.2"},{"comment":"The translation-invariant constant κδ(f_*) in Theorem 15.29 is defined as a maximum involving Q(f0),...,Q(fn), but an n-chain map (Definition 4.3) has components f0,...,f_{n+1}, and κ_n(f_*) from Definition 4.14 includes ∥f_{n+1}∥. As written, the inequality κ_n(f_*) ≤ κδ(f_*) used to apply Theorem 4.15 is not justified. This is a local but real gap in the proof of weak-containment monotonicity (Theorem 15.30). It is repaired by adding Q(f_{n+1}) to the defining maximum and adjusting the constants in Lemma 15.32 and Theorem 15.30 accordingly.","section":"Theorem 15.29 / Section 15.5"}],"minor_comments":[{"comment":"The quantity ν_n(D_*) is listed twice with identical definition; one of the two entries appears intended to be a different quantity, perhaps one involving N2 or a distinct norm. Please correct the typo and ensure all subsequent references use the intended quantity.","section":"Definition 4.7"},{"comment":"The invariant mevol_n is introduced only for integer coefficients, but statements such as Theorem 1.4 and Proposition 1.5(i) assert 'mevol_n(α)=0' without repeating the coefficient convention. Please make explicit in each statement that mevol is an integer-coefficient invariant, so readers do not infer a finite-field version.","section":"Theorem 1.4 and Section 11"},{"comment":"The function log^+ is used for zero norms in Proposition 6.4(iv), where K can be 0, but log^+ is never defined at 0. Please state the convention log^+(0)=0 or restrict K to positive values.","section":"Section 6.2, Proposition 6.4"},{"comment":"Corollary 16.6 is correct as written: for α×β with β trivial, the stabilizer of (x,y) equals the stabilizer of x, so essential freeness of α passes to the diagonal action. A one-sentence justification of this point would preempt reader confusion.","section":"Corollary 16.6"}],"recommendation":"major_revision","confidential_remarks":"The core Theorem 1.2 appears sound to me: the flatness base change in Proposition 2.10 and the retraction arguments in Theorems 7.6 and 8.4/8.5 check out, and I did not find a circularity or fitted-parameter problem. The main obstacle is the overbroad finite-field statement of Theorem 8.6, which is used in the Section 12 lower bounds; the missing f_{n+1} component in the Theorem 15.29 constant is easy to patch. I recommend major revision rather than rejection: after restricting or fixing Theorem 8.6 and supplying the finite-field lower-bound argument sketched above, the paper should be publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, it delivers what it promises: genuinely new invariants—measured embedding dimension medim and measured embedding volume mevol—and a clean theorem bounding Betti gradients and logarithmic torsion gradients by these invariants for profinite actions. Second, the central proof is sound as far as I can tell, and the specific concern the reader raised about Corollary 16.6 does not land. If α is essentially free and β is trivial, the diagonal action is still essentially free because the stabilizer of (x,y) is the stabilizer of x, trivial for µ-a.e. x. So that corollary is fine.\n\nThe genuinely new content: the invariant definitions, the dynamical upper bound (Theorem 1.2), monotonicity under weak containment, invariance under weak bounded orbit equivalence, and the exact computations for free and surface groups. The reproofs of known amenable vanishing results in the new language are clean, and the estimates for amalgamated products are a nice bonus. The proofs in Sections 2 through 8 are extensive and, on inspection, the load-bearing steps hold. The flatness of L∞(α)*Γ over ZΓ rests on a cited freeness theorem for L∞(α) as an abelian group; that is acceptable, though it would be nice if the paper spelled out the argument or gave a secondary reference.\n\nSoft spots are mostly about presentation and scope. The paper is very long and the quantitative homological algebra in Sections 3–6 is heavy; I could not verify every estimate line by line, and I doubt many readers will. The equivalence relation ring part is explicitly incomplete—the authors note it is unclear whether general orbit equivalence invariance can be obtained—so that section reads more like a research announcement. There is also some reliance on the authors' own previous work [LLM+] for the cheap rebuilding property, but that is clearly flagged and not circular.\n\nWho this is for: anyone working on homology growth, L2-Betti numbers, or measured group theory. It deserves a serious referee. A careful referee will need time on the technical sections, but I do not see a load-bearing defect. Recommend acceptance with minor revisions, mostly requests to streamline and to add a short remark explaining the essential freeness in Corollary 16.6.","headline":"The new measured embedding invariants and the dynamical upper bounds for homology growth are real and the central argument holds up; the apparent flaw in Corollary 16.6 is not a flaw.","tokens_in":78142,"tokens_out":2043,"would_cite":true,"duration_ms":29713,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A20","20J05","16S35","20E26","20E18"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that for residually finite groups of type FP_{n+1}, the asymptotic Betti numbers and logarithmic torsion sizes along a chain of finite-index normal subgroups are bounded above by the measured embedding dimension and…","keywords":["measured embedding dimension","measured embedding volume","logarithmic torsion homology growth","Betti number gradient","L2-Betti numbers","weak containment","weak bounded orbit equivalence","profinite completion"],"falsifier":"For a free group F_d with a residual chain, the paper computes medim^Z_1 of the profinite completion to be d−1; if one could construct a 1-dimensional α-embedding with dimension strictly less than d−1, Theorem 1.2 would force the first Betti gradient below d−1, contradicting the known rank gradient. Alternatively, one could test flatness directly by checking whether L∞(bΓ,Z) ⊗_ZΓ C_* is exact for the standard free resolution of a concrete group such as SL_3(Z) in degree 1.","tokens_in":77112,"feed_emoji":"📐","tokens_out":5404,"duration_ms":67837,"temperature":0.7,"pith_summary":"The paper proves a 'cheap embedding principle' for homology growth: if a group's action on its profinite completion can be embedded into a small algebraic complex, then the group's Betti number gradients and logarithmic torsion homology gradients along finite-index subgroups are bounded by the size of that complex. This gives dynamical upper bounds that work uniformly for integer and finite-field coefficients, and it separates torsion growth estimates from Betti number estimates. The authors show the principle is sharp in concrete cases such as free groups and surface groups, where the measured embedding dimension matches known rank gradients. They also develop a quantitative homological algebra over crossed product rings to make the comparison between dynamical systems and finite-index subgroups precise.","feed_headline":"Homology growth bounded by the profinite action's size","feed_subtitle":"The cheap embedding principle: Betti and torsion gradients lie under dynamical embedding dimension and volume.","key_machinery":"The central object is the crossed product ring L∞(α,Z) ∗ Γ of a standard measure-preserving action, together with marked projective chain complexes and a quantitative homological algebra of 'almost' complexes and maps. The key identity-like tool is the logarithmic norm lognorm, a refined version of dimension times log of operator norm that controls torsion in cokernels and is compatible with approximate equalities and Gromov–Hausdorff distances between complexes. Deformation and strictification theorems show that any complex close to an adapted one can be turned into an honest chain complex without changing its size much, and the density of cylinder sets in the profinite completion allows passage from dynamical complexes to ones associated with deep enough finite-index subgroups.","core_discovery":"The central discovery is that the asymptotic homological size of a residually finite group along a residual chain is controlled by the size of its profinite completion action, measured through two new invariants: the measured embedding dimension and the measured embedding volume. These invariants are defined as infima over marked projective chain complexes over the crossed product ring L∞(α,Z) ∗ Γ that admit an α-embedding from a free resolution of the trivial module. The main theorem, Theorem 1.2, asserts that the upper Betti gradient is at most the measured embedding dimension whenever Z is the integers or a finite field, and the upper logarithmic torsion gradient is at most the measured embedding volume when Z is the integers. The proof works by approximating any embedding by one adapted to cylinder sets, discretising it to complexes over individual finite-index subgroups, and then using homological retracts together with torsion estimates. The same machinery also yields an upper bound for L2-Betti numbers by the measured embedding dimension.","pith_inferences":["The decoupling of torsion growth from Betti growth suggests that vanishing of mevol can be verified independently of L2-Betti numbers, which may allow torsion growth vanishing proofs in cases where L2-Betti numbers are non-zero, such as products with amenable factors.","Since medim and mevol are monotone under weak containment, the framework points toward a dynamical strategy for the conjecture replacing Q-rank by R-rank in semisimple Lie groups: construct cheap embeddings for the relevant lattice actions rather than computing gradients directly.","The cylinder-set approximation underlying the main theorem implies that medim and mevol of a profinite completion can be approximated numerically by looking at deep enough finite-index subgroups, offering a computable route to upper bounds on homology growth.","If the measured embedding invariants behave like cost and integral foliated simplicial volume for ergodic decompositions, one would expect medim and mevol of an arbitrary action to equal the essential supremum over its ergodic components, which the paper only establishes in an approximate, one-sided form."],"forward_implications":["For residually finite groups of type FP_{n+1}, upper Betti number gradients over Z and over finite fields, as well as logarithmic torsion homology gradients, are all bounded by dynamical invariants of the profinite completion action.","Amenable groups have vanishing measured embedding dimension and volume for every standard action, recovering and refining known vanishings of Betti gradients over every field and of logarithmic torsion growth.","For free groups of rank d and surface groups of genus g, the measured embedding dimension in degree 1 of the profinite completion equals d−1 and 2g−2 respectively, exactly matching L2-Betti numbers and rank gradients.","L2-Betti numbers of a group of type FP_{n+1} are always bounded by the measured embedding dimension of any standard action, giving a dynamical upper bound independent of the action's choice.","Measured embedding dimension and volume are multiplicative under weak bounded orbit equivalence, yielding proportionality results for hyperbolic 3-manifolds where these invariants scale with volume."],"supporting_citations":[{"why":"Supplies the freeness of L∞(α) as an abelian group, which the paper uses to prove that the crossed product module is flat over the group ring.","marker":"[Ste85]"},{"why":"Provides the density of cylinder sets in the profinite completion, which the approximation and deformation arguments rely on.","marker":"[Löh20b]"},{"why":"Gives Gabber's estimate for torsion in cokernels in terms of log norms of matrix entries, a key tool for the logarithmic torsion bounds.","marker":"[Sou99]"},{"why":"Supplies the explicit Gabber-type torsion estimate and the earlier principle that retractions control homology and torsion growth.","marker":"[Sau16]"},{"why":"Provides the description of L2-Betti numbers via the orbit equivalence relation ring, used in the proof of the L2-Betti bound.","marker":"[Sau05]"},{"why":"Introduces the cheap rebuilding property that motivates the measured embedding invariants and provides the vanishing result for SL_d(Z) that the paper seeks to generalize.","marker":"[ABFG25]"}],"fun_headline_variants":["Cheap embeddings cap homology growth","Homology growth bounded by profinite action size","Measured embedding dimension sets homology bounds","Cheap embedding principle: dynamics control homology","New invariants limit Betti and torsion growth"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the function space L∞ of the action being flat over the group ring, so that tensoring a free resolution with it still gives a resolution; if that flatness fails, the retraction arguments behind the inequalities collapse.","fun_headline_variants_meta":{"raw":{"variants":["Cheap embeddings cap homology growth","Homology growth bounded by profinite action size","Measured embedding dimension sets homology bounds","Cheap embedding principle: dynamics control homology","New invariants limit Betti and torsion growth"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000121,"raw_usage":{"total_tokens":986,"prompt_tokens":734,"completion_tokens":252,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":350,"completion_tokens_details":{"reasoning_tokens":187}},"tokens_in":350,"tokens_out":252,"duration_ms":3483,"temperature":1.0,"reasoning_tokens":187,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T05:41:37.535290+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a free group F_d with a residual chain, the paper computes medim^Z_1 of the profinite completion to be d−1; if one could construct a 1-dimensional α-embedding with dimension strictly less than d−1, Theorem 1.2 would force the first Betti gradient below d−1, contradicting the known rank gradient. Alternatively, one could test flatness directly by checking whether L∞(bΓ,Z) ⊗_ZΓ C_* is exact for the standard free resolution of a concrete group such as SL_3(Z) in degree 1.","supporting_citations":[],"review_version":1}