{"id":"9c0e4e36-c161-461b-a5f5-0c4b7286ae58","arxiv_id":"2608.05037","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For higher-rank locally symmetric spaces and affine buildings, generator number and logarithmic first-homology torsion grow sublinearly in volume when the injectivity radius is large or along Benjamini-Schramm convergent sequences.","lead":"The paper proves that torsion-free lattices in higher-rank symmetric spaces with large injectivity radius have small algebraic complexity: their generator number and first-homology torsion are bounded by volume times a decaying factor. The proof introduces a new random-cover construction, called scaffolded Poisson processes, and also proves the expected vanishing of normalized torsion along Benjamini-Schramm convergent sequences.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Affine-building Theorem 1.5 rests on unverified generality of the Parkinson regularity and volume-growth estimates in Lemma 7.1; the main manifold theorems are unaffected.","rationale":"I checked the heart of the paper, the proof of Theorem 1.1, in detail. The deterministic scaffold, the Poisson sampling, the expected edge count in Proposition 3.3, and the topological consequences in Lemma 2.5 are internally consistent; equations (1)-(3) are sufficient, and the constant choices are not circular. The reader's weakest-assumption list already mentioned the affine-building volume-growth lemma as a source of risk, so the agreement is partial rather than full. I isolate Lemma 7.1 because it is the single place where a prominently advertised result, Theorem 1.5 with its claimed application to exotic affine buildings, depends on a compressed citation rather than a derivation. If Parkinson's theorems do not apply in the claimed generality, the affine-building theorem loses its support, while the locally symmetric manifold theorems remain valid. A conditional acceptance is therefore the most precise verdict: the main results are accepted, but Theorem 1.5 should be marked as pending verification of the regularity and volume-growth hypotheses in Lemma 7.1.","tokens_in":17426,"tokens_out":35779,"duration_ms":444343,"concrete_test":"Check the full hypotheses of Parkinson [30, Theorems 2.4 and 5.15]: do they cover every thick locally finite irreducible affine building with a cocompact automorphism group, including non-Moufang exotic rank-2 buildings? If yes, add the short verification to Lemma 7.1; if no, either supply a proof of the two-sided annulus bound (9) for all such buildings or restrict Theorem 1.5 to Bruhat-Tits buildings and regular buildings. A useful sanity computation is the ball-volume growth of a non-Moufang rank-2 building with a cocompact automorphism group, compared with the claimed exponent (r-1)/2.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The self-contained scaffolded-Poisson argument for Theorem 1.1 is internally sound: the radius choices in Lemma 2.2, the edge decomposition in Proposition 3.3, and the application of Lemma 2.5 all check out. The weakest load-bearing point is the affine-building extension, specifically Lemma 7.1. It asserts two-sided uniform ball-volume bounds with exponent (r-1)/2 and a uniform injectivity-radius lower bound for every locally finite thick affine building admitting a compact quotient. The proof reduces to equation (9), but for rank r_j >= 2 factors it simply cites [30, Theorem 2.4] for regularity and [30, Theorem 5.15] for the annulus estimate, without verifying that those theorems apply to all exotic rank-2 buildings, including non-Moufang cocompact cases, rather than only to strongly transitive or regular ones. Since the Poisson estimates in Section 7 require (9) as both an upper and a lower bound uniformly in the centre, a single factor with different spherical growth would change the exponent t^{(1-r)/2}(log t)^2 and break the Benjamini-Schramm diagonal argument. This does not endanger Theorem 1.1; it makes Theorem 1.5 conditional on the cited regularity and volume-growth facts.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a 'scaffolded Poisson process' construction to build random good covers of higher-rank locally symmetric spaces and affine buildings. The main theorem (Theorem 1.1) asserts that for every torsion-free lattice with global injectivity radius R large, both the generator number d(Γ) and log|H_1(M;Z)_tors| are bounded by C_X vol(M) R^{(1-r)/2} (log R)^2, where r≥2 is the real rank. The proof samples sparse Poisson balls on the manifold and fills missed regions with small scaffold balls, then uses the Mecke formula, ball-volume estimates, the nerve lemma, and Gabber's inequality to control the expected number of edges in the cover. From this, the paper derives Benjamini–Schramm vanishing of normalized first-homology torsion (Theorem 1.2), a sublinear rank theorem for general orbifolds (Theorem 1.3), and analogous statements for affine buildings (Theorem 1.5). The manifold-theoretic core is a genuine proof: the constants depend only on X and the key inequalities are derived rather than assumed.","tokens_in":17607,"tokens_out":8029,"duration_ms":96942,"significance":"If correct, Theorem 1.1 gives the first general quantitative sublinear bound for generator rank and first-homology torsion in higher-rank locally symmetric spaces under a large injectivity-radius assumption, and Theorem 1.2 answers the Abért–Gelander–Nikolov question on vanishing of normalized first-homology torsion along Benjamini–Schramm convergent sequences. The proof is elegant and largely self-contained: it avoids model-specific geometry and reduces everything to ball-volume growth and a Poisson-process bookkeeping. The scaffolded-cover idea is a genuine improvement over deterministic covers, since it makes the expected nerve sparse without losing coverness. The main caveat is the affine-building extension, whose proof relies on external regularity and volume-growth estimates that are not verified in full generality. The manifold theorems are not affected by this caveat.","major_comments":[{"comment":"Lemma 7.1(i) asserts two-sided uniform ball-volume bounds (9) for every locally finite thick affine building admitting a compact quotient. For rank r_j ≥ 2 factors the proof simply cites [30, Theorem 2.4] for regularity and [30, Theorem 5.15] for the annulus estimate, but it does not verify that these theorems apply to all exotic cocompact affine buildings, including non-Moufang and non-strongly-transitive rank-2 cases. Since the Poisson estimates in Section 7 require (9) uniformly in the centre as both an upper and a lower bound, a single factor with different spherical growth would change the exponent t^{(1-r)/2}(log t)^2 and break the Benjamini–Schramm diagonal argument. Please either provide a direct proof of (9) under the stated hypotheses or state explicitly the additional assumptions on the building that make the cited results applicable.","section":"Section 7, Lemma 7.1"},{"comment":"The compact BS-convergent case depends on deep external estimates: the Dobrowolski injectivity-radius bound from [14, Proposition 2.3] and the exponential thin-part decay [14, Theorem D]. The proof uses these only through the trace-field degree d_n and constants said to depend on X, but the precise hypotheses under which these estimates hold are not stated. In particular, the text says 'By Margulis arithmeticity' without recalling the exact form of the theorem needed for irreducible lattices in semisimple groups of rank at least two. Please state the required hypotheses and verify that every compact torsion-free BS-convergent sequence of irreducible lattices in this setting satisfies them.","section":"Section 4.4, Lemma 4.6"},{"comment":"The noncompact case of Theorem 1.2 uses a diagonal argument to choose t_n so that log t_n · vol((M_n)^{<3ρ(t_n)+b_X})/vol(M_n) → 0. The argument is described only in one sentence, but it is load-bearing because t_n must simultaneously exceed the threshold of Proposition 4.3 and make the thin-part ratio small. Please spell out the construction with an explicit choice of index intervals, for example t_n = j on ranges where the ratio is at most 1/(j log(j+2)), and justify that the resulting t_n can be taken to tend to infinity while preserving all earlier bounds.","section":"Section 4.2, proof of Theorem 1.2"}],"minor_comments":[{"comment":"The statement says 'every torsion-free lattice Γ<G' but the proof in Section 3 assumes M is compact; for noncompact finite-volume quotients the global injectivity radius is zero, so the hypothesis InjRad(M) ≥ R_X is never satisfied. It would be clearer to write 'compact torsion-free lattice' explicitly.","section":"Theorem 1.1"},{"comment":"References [12] and [13] appear to be the same paper by Frączyk, 'Growth of mod-2 homology in higher-rank locally symmetric spaces', listed twice with slightly different formatting. One duplicate should be removed.","section":"References [12] and [13]"},{"comment":"The notation v_-(s), v_+(s) for building ball volumes is useful, but the proof of Lemma 7.1(i) could state explicitly that for the ℓ2 product metric the volume entropy h_B = (∑ h_j^2)^{1/2} follows from a Laplace-method calculation, since the displayed 'after decomposing a product ball' formula is not immediate for the ℓ2 ball.","section":"Section 7, notation"},{"comment":"In the random graph G_t, the choice of the particular Poisson point for edges of type (iii) is arbitrary; the text says the estimates do not depend on the choices. It would be helpful to note that the graph connectivity argument uses only that at least one such edge is present for each retained covered vertex, so any deterministic choice works.","section":"Section 6, edge types"}],"recommendation":"major_revision","confidential_remarks":"The manifold-theoretic results are sound and would be a strong contribution on their own. The main risk is the affine-building theorem, which is currently conditional on unverified generality of the cited Parkinson estimates. I recommend asking the author to either prove the required ball-volume bounds directly or explicitly qualify Theorem 1.5. The self-citations do not carry the proof of the main theorem, and I see no circularity in the construction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: this paper delivers the first general vanishing theorem for normalized H1-torsion in cocompact higher-rank lattices, and the construction—scaffolded Poisson covers—is a real step beyond the Poisson–Voronoi method. I think the manifold results (Theorems 1.1–1.3) are solid. The affine-building extension is shakier.\n\nWhat is new: the bound max{d(Γ), log|H1_tors|} ≤ C_X vol(M) R^{(1-r)/2}(log R)^2 for torsion-free lattices with injectivity radius R is exactly the kind of quantitative vanishing that was missing. The proof does not tune parameters to the target. The Poisson intensity is 1/v(t), the radii are fixed from the volume growth, and the nerve edge count is bounded by expectation, with constants universal in X. I checked Lemma 2.2 and Propositions 3.3, 4.3, 4.5; the inequalities hold. The use of Gabber's estimate to get the torsion log bound from a sparse nerve is clean. The BS-convergent versions and the orbifold rank theorem (Theorem 1.3) are also substantial; the graph lift argument in Section 6 is nice. The import of Gelander's core, Frączyk–Hurtado–Raimbault, and Dobrowolski is appropriate, and the paper says clearly when it relies on them.\n\nThe soft spot is Section 7. Theorem 1.5 claims the vanishing for all locally finite thick affine buildings with compact quotients, including exotic ones. Lemma 7.1 is load-bearing: it needs two-sided uniform ball volume growth with exponent (r-1)/2 and a uniform injectivity radius lower bound. For rank ≥2 factors, the proof cites Parkinson [30] for regularity and the annulus estimate, but does not check that those theorems apply to every thick cocompact building, including non-Moufang ones. If one factor has slightly different spherical growth, the exponent changes and the diagonal argument collapses. This makes Theorem 1.5 conditional rather than proved. The manifold theorems do not depend on this.\n\nAlso minor: the phrase 'answers a question of Abért–Gelander–Nikolov' is appropriate for Theorem 1.2, but the torsion statement is for degree one only; the paper says that clearly. The references are fine; self-citations are not load-bearing.\n\nWho should read this: anyone working on homology growth, fixed price, or generator rank of higher-rank lattices. It deserves a serious referee. I would send it out, but tell the referee to pay special attention to Lemma 7.1 and to ask the author to either prove the volume estimates for arbitrary thick buildings or restrict the statement to the class where Parkinson applies.","headline":"Strong new vanishing theorems for homology torsion and generator growth in higher-rank locally symmetric spaces; the manifold results are solid, but the affine-building extension relies on an unverified citation for exotic buildings.","tokens_in":18200,"tokens_out":2717,"would_cite":true,"duration_ms":28072,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E40","20F65","51E24"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs random open covers of higher-rank locally symmetric spaces and proves they bound both the minimal generator number and the logarithm of first-homology torsion by $C_X\\,\\mathrm{vol}(M)\\,R^{(1-r)/2}(\\log R)^2$, with…","keywords":["homology torsion","higher-rank lattices","Benjamini–Schramm convergence","random open covers","Poisson point process","generator rank","first homology","affine buildings"],"falsifier":"Take a sequence of torsion-free cocompact lattices $\\Gamma_i<G$ with injectivity radii $R_i\\to\\infty$ and compute the ratio $\\max\\{d(\\Gamma_i),\\log|H_1(M_i;\\mathbb Z)_{\\mathrm{tors}}|\\}/\\bigl(\\mathrm{vol}(M_i)\\,R_i^{(1-r)/2}(\\log R_i)^2\\bigr)$; if the limsup is infinite, Theorem 1.1 is false. For the normalized vanishing assertions, it suffices to exhibit any Benjamini–Schramm convergent sequence along which $\\log|H_1(M_i;\\mathbb Z)_{\\mathrm{tors}}|/\\mathrm{vol}(M_i)$ does not tend to zero.","tokens_in":17166,"feed_emoji":"🎲","tokens_out":13385,"duration_ms":138735,"temperature":0.7,"pith_summary":"Higher-rank locally symmetric spaces are thought to be algebraically small: their homology torsion and generating complexity should be negligible relative to volume. This paper proves a quantitative form of that expectation. It builds a random cover—a coarse scaffold of small balls plus large balls thrown down by a Poisson process—whose nerve is a good model of the manifold and whose expected number of edges is sublinear in volume. If the injectivity radius is at least a constant depending only on the symmetric space, both the minimal number of generators of the lattice and the logarithm of the order of the torsion subgroup of first integral homology are bounded by $C_X\\,\\mathrm{vol}(M)\\,R^{(1-r)/2}(\\log R)^2$, with $r$ the real rank. It follows that along Benjamini–Schramm convergent sequences the normalized torsion and normalized generator rank vanish, and the same holds for compact quotients of affine buildings; this confirms the predicted degree-one integral torsion vanishing in higher rank.","feed_headline":"Random covers force torsion in H1 to vanish in higher rank","feed_subtitle":"A scaffolded Poisson process keeps the nerve sublinear in volume, bounding generators and H1 torsion.","key_machinery":"The central object is the scaffolded Poisson cover: a maximal $1$-separated set of points (the scaffold) with radius-$2$ balls guarantees coverage, and an independent Poisson point process of intensity $1/v(t)$ sprinkles large radius-$\\rho(t)$ balls, where $\\rho(t)=t+h^{-1}\\log\\log t+c$ and $v(s)$ is the ball volume in the symmetric space. A large ball is retained wherever it meets the scaffold, and the small scaffold ball is retained only at missed points; the threshold $\\rho(t)$ is tuned by the asymptotics $v(s)\\simeq e^{hs}s^{(r-1)/2}$. The resulting cover is good—every nonempty finite intersection is contractible—and its nerve models the manifold. The expected number of nerve edges is computed by three terms: Poisson–Poisson pairs via the standard expectation formula for unordered pairs of a Poisson process, scaffold–scaffold edges via a polynomial tail bound on missed centres, and mixed edges via independence between the inner ball and the surrounding annulus. The topological conversion is that a good cover with $E$ edges gives $d(\\pi_1M)\\leq E$ and $\\log|H_1(M;\\mathbb Z)_{\\mathrm{tors}}|\\leq E\\log\\sqrt3$, so the sublinear edge count is the whole mechanism. This is the only place where higher-rank geometry enters: the exponent $(r-1)/2$ in the volume growth is what makes the edge count decay.","core_discovery":"The central claim is that a scaffolded Poisson process yields a good cover of any torsion-free quotient $M=\\Gamma\\backslash X$ whose nerve has, in expectation, $O(\\mathrm{vol}(M))\\,R^{(1-r)/2}(\\log R)^2$ edges when $R=\\mathrm{InjRad}(M)$ is large. Because the nerve of a good cover is homotopy equivalent to $M$, a one-skeleton edge count $E$ bounds both invariants: $d(\\Gamma)\\leq E$ and $\\log|H_1(M;\\mathbb Z)_{\\mathrm{tors}}|\\leq E\\log\\sqrt{3}$. The paper's quantitative heart, Proposition 3.3, computes the expected edge count: Poisson–Poisson pairs contribute of order $\\mathrm{vol}(M)\\,t^{-(r-1)/2}(\\log t)^2$, missed scaffold centres contribute polynomially small terms, and mixed pairs contribute another logarithmic factor; the only geometric input is the ball-volume asymptotics $v(s)\\asymp e^{hs}s^{(r-1)/2}$ of the symmetric space. Choosing $t\\approx R/4$ gives Theorem 1.1. For Benjamini–Schramm convergent sequences the same construction runs on a thick region, with the thin part controlled by imported good-cover and injectivity-radius estimates; the building analogue uses uniform ball-volume growth and a uniform lower bound on translation lengths in place of manifold geometry.","pith_inferences":["Editorial inference: the mechanism should transfer to other families of quotients with ball-volume growth of the form $e^{hs}s^{(r-1)/2}$ and a bounded-degree good cover of the thin part, so the results are likely not special to symmetric spaces or affine buildings.","Editorial inference: the logarithmic factor $(\\log R)^2$ probably is an artifact of the chosen radius $\\rho(t)=t+h^{-1}\\log\\log t+c$; a sharper tail bound for missed scaffold points could plausibly remove it, matching the conjectural shape $R^{(1-r)/2}$.","Editorial inference: the same nerve-edge estimate should have higher-degree analogues, with degree $q$ controlled by $(q+1)$-tuple intersection counts rather than pair counts; the paper only carries out degree one, but nothing in the construction seems degree-specific."],"forward_implications":["In a normal residual tower of a fixed cocompact higher-rank lattice, logarithmic systole growth upgrades Theorem 1.1 to an explicit bound $\\max\\{d(\\pi_1 M),\\log|H_1(M;\\mathbb Z)_{\\mathrm{tors}}|\\} \\leq C\\,\\mathrm{vol}(M)(\\log\\mathrm{vol}(M))^{(1-r)/2}(\\log\\log\\mathrm{vol}(M))^2$.","For any torsion-free Benjamini–Schramm convergent sequence of irreducible higher-rank manifolds, the volume-normalized logarithm of first-homology torsion, the generator rank, and the first-homology dimension over every field all vanish simultaneously.","When $G$ is simple, the Benjamini–Schramm hypothesis is automatic along any sequence of lattices whose covolumes tend to infinity, so the vanishing theorem applies to all such torsion-free sequences without further assumptions.","The same scaffolded construction proves the analogous vanishing for compact quotients of thick affine buildings of Euclidean rank at least two, including exotic buildings and quotients arising from semisimple groups over local fields."],"supporting_citations":[{"why":"Supplies the Poisson–Voronoi method and sublinear generator bound for torsion-free sequences that this paper's scaffolded cover extends.","marker":"[15]"},{"why":"Supplies the bounded-degree good covers and exponential thin-part decay used to control the compact Benjamini–Schramm convergent case.","marker":"[14]"},{"why":"Supplies the compact homotopy core for noncompact quotients, used to reduce torsion estimates to a core.","marker":"[16]"},{"why":"Supplies the volume-versus-rank core scaffold with uniform bounded degree and positive injectivity radius used for general orbifolds.","marker":"[17]"},{"why":"Supplies the regularity and uniform ball-volume growth in rank-two affine-building factors used for the building volume estimates.","marker":"[30]"},{"why":"Supplies the uniform translation-length lower bound giving a positive injectivity radius for torsion-free building quotients.","marker":"[10]"},{"why":"Supplies the standard Poisson expectation formula used to compute expected numbers of Poisson–Poisson pairs.","marker":"[24]"},{"why":"Supplies the nerve lemma converting a good cover into a homotopy-equivalent simplicial complex.","marker":"[21]"},{"why":"Supplies the conjecture predicting degree-one integral torsion vanishing that the paper confirms in higher rank.","marker":"[9]"},{"why":"Supplies the question about normalized first-homology torsion and the sublinear generator conjecture that the paper answers.","marker":"[3]"}],"fun_headline_variants":["Scaffolded Poisson covers enforce torsion vanishing in higher rank","Torsion in H1 dies under sparse random covers in higher rank","Random covers bound nerve edges, forcing H1 torsion to vanish","Higher-rank symmetric spaces: torsion-free via scaffolded Poisson","Sparse covers give injectivity radius control of H1 torsion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on imported estimates—bounded-degree good covers, exponential thin-part decay, positive injectivity-radius lower bounds, and uniform ball-volume growth—holding with constants depending only on $X$ (or on the building $B$); if any of those fails for some admissible family, the stated theorems do not follow from this proof.","fun_headline_variants_meta":{"raw":{"variants":["Scaffolded Poisson covers enforce torsion vanishing in higher rank","Torsion in H1 dies under sparse random covers in higher rank","Random covers bound nerve edges, forcing H1 torsion to vanish","Higher-rank symmetric spaces: torsion-free via scaffolded Poisson","Sparse covers give injectivity radius control of H1 torsion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000225,"raw_usage":{"total_tokens":1529,"prompt_tokens":1071,"completion_tokens":458,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":687,"completion_tokens_details":{"reasoning_tokens":372}},"tokens_in":687,"tokens_out":458,"duration_ms":5625,"temperature":1.0,"reasoning_tokens":372,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:51:58.338235+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a sequence of torsion-free cocompact lattices $\\Gamma_i<G$ with injectivity radii $R_i\\to\\infty$ and compute the ratio $\\max\\{d(\\Gamma_i),\\log|H_1(M_i;\\mathbb Z)_{\\mathrm{tors}}|\\}/\\bigl(\\mathrm{vol}(M_i)\\,R_i^{(1-r)/2}(\\log R_i)^2\\bigr)$; if the limsup is infinite, Theorem 1.1 is false. For the normalized vanishing assertions, it suffices to exhibit any Benjamini–Schramm convergent sequence along which $\\log|H_1(M_i;\\mathbb Z)_{\\mathrm{tors}}|/\\mathrm{vol}(M_i)$ does not tend to zero.","supporting_citations":[{"cited_title":"Topological complexity of arithmetic locally symmetric spaces.Invent","cited_arxiv_id":null,"evidence_quote":"Supplies the bounded-degree good covers and exponential thin-part decay used to control the compact Benjamini–Schramm convergent case."},{"cited_title":"Homotopy type and volume of locally symmetric manifolds.Duke Math","cited_arxiv_id":null,"evidence_quote":"Supplies the compact homotopy core for noncompact quotients, used to reduce torsion estimates to a core."},{"cited_title":"Volume versus rank of lattices.J","cited_arxiv_id":null,"evidence_quote":"Supplies the volume-versus-rank core scaffold with uniform bounded degree and positive injectivity radius used for general orbifolds."},{"cited_title":"Buildings and hecke algebras.J","cited_arxiv_id":null,"evidence_quote":"Supplies the regularity and uniform ball-volume growth in rank-two affine-building factors used for the building volume estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the uniform translation-length lower bound giving a positive injectivity radius for torsion-free building quotients."},{"cited_title":"Cambridge University Press, Cambridge, 2018","cited_arxiv_id":null,"evidence_quote":"Supplies the standard Poisson expectation formula used to compute expected numbers of Poisson–Poisson pairs."},{"cited_title":"The asymptotic growth of torsion homology for arithmetic groups","cited_arxiv_id":null,"evidence_quote":"Supplies the conjecture predicting degree-one integral torsion vanishing that the paper confirms in higher rank."}],"review_version":1}