{"id":"79e9145d-a3b1-4b46-b5f5-72117762a166","arxiv_id":"2607.07940","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Irreducible higher-rank lattices have virtually isotopically projectively regularizable birational actions on complex projective surfaces, via cohomological vanishing and a cube-complex fixed-point theorem.","lead":"Higher-rank lattices acting by birational maps on complex projective surfaces are virtually isotopically projectively regularizable. The proof gives a new fixed-point theorem for cube complexes with solvable hyperplane stabilizers and avoids relying on property FW.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The manuscript’s logical skeleton is classical arithmetic-group cohomology plus CAT(0) cube geometry plus known Cremona dynamics. The reduction to Jonq via infinite-dimensional hyperbolic geometry is the only non-elementary step, and it is handled by Monod’s superrigidity plus the Blanc–Cantat gap, both of which are correctly invoked. The new vanishing result is weaker than full property FW yet sufficient for the solvable-hyperplane case needed on the Jonquières complex; the induction on Q-rank and the use of Reeder’s isomorphism are standard and appear free of gaps. Because the reader already identified the same (reasonable) weakest link and assigned ACCEPT with high confidence, no adjustment is warranted.","tokens_in":15214,"tokens_out":489,"duration_ms":6441,"concrete_test":"Independently re-derive the vanishing H_i(A,St(G,F))=0 for i>δ(G) in Prop. 2.2.1 by induction on Q-rank, verifying the Mackey decomposition (2.12)–(2.14) and the rank identity δ(L)=δ(G) for Levi factors; if the induction step fails for a proper parabolic of Q-rank 1, the cube fixed-point theorem (and hence Thm A) collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem A) rests on a transparent chain: cohomological vanishing for Steinberg modules of solvable subgroups (Prop. 2.2.1 + Borel–Serre duality) yields the cube fixed-point theorem (Thm D), which is applied to the Jonquières complex after the elementary-action reduction of Lemma 4.2.1. The reader’s flagged weakest link—the gap-property contradiction for non-elementary actions via Monod–Py and density—is standard and correctly applied; no exotic continuous action of a higher-rank factor can arise because such factors have property (T). The remaining steps (hyperplane stabilizers inject into point stabilizers in PGL(2,C) and are therefore solvable; fixed vertices give projective regularizability; absence of loxodromics/Halphen twists upgrades to virtual isotopic regularizability) are local and checkable. No internal inconsistency or hidden unboundedness assumption appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves that for an irreducible lattice Γ in a product of non-compact center-free simple linear algebraic groups of total real rank ≥2, every homomorphism ρ:Γ\to Bir(S) into the birational group of a complex projective surface is virtually isotopically projectively regularizable (Theorem A). The argument proceeds by establishing a cohomological vanishing result (Theorem C) for H^{1}(Γ,F[Γ/K]) when K is solvable, via Borel–Serre duality and an inductive vanishing theorem for Steinberg modules of solvable subgroups of arithmetic groups (Proposition 2.2.1). This yields a fixed-point theorem for actions on CAT(0) cube complexes whose hyperplane stabilizers are solvable (Theorem D). Superrigidity for non-elementary actions on infinite-dimensional real hyperbolic spaces (Theorem E, from Monod and Caprace–Lytchak) reduces the Cremona action to the de Jonquières group; the fixed-point theorem is then applied to the Jonquières complex of Lonjou–Przytycki–Urech. The proof deliberately avoids direct appeal to property FW, covering lattices for which FW is not yet known.","tokens_in":15390,"tokens_out":936,"duration_ms":8848,"significance":"The result completes the regularizability half of the birational Zimmer program in dimension two for all higher-rank lattices, including those with rank-one factors (e.g., irreducible lattices in PSL(2,R)^{n}\times PGL(2,C)^{m} and in PU(1,n)^{m}) where property FW remains open. Combined with Margulis superrigidity and the classification of connected algebraic group actions on surfaces, it yields a complete list of possible infinite birational actions up to conjugacy. The cohomological vanishing and cube fixed-point theorems are of independent interest and give a clean alternative route that bypasses FW. The chain is explicit (exact sequences, Shapiro lemma, induction on Q-rank) and rests on standard black-box inputs (Borel–Serre, Monod, Blanc–Cantat gap property).","major_comments":[],"minor_comments":[{"comment":"In the proof of Theorem E (pp. 10–11), the extension of Tf_{0} from the linear span of C to the closed subspace E\rho is asserted to be continuous and isometric; a one-line appeal to density of the span and continuity of B would make the argument fully self-contained.","section":null},{"comment":"Lemma 4.2.1 invokes the gap property of Blanc–Cantat [7, Cor. 2.7(2)] after density of the projection of Γ; a brief parenthetical that the translation-length function is continuous on Isom(HE) would clarify why lengths can be made arbitrarily small.","section":null},{"comment":"Section 6 treats the Hirzebruch-surface case by Whitehead’s lemma; the reference [43, Cor. 7.8.10] is for Lie-algebra cohomology, so a short remark that the cocycle is continuous (hence smooth) would remove any ambiguity.","section":null},{"comment":"Typographical: “dimen-sion” (p. 2), “uniformirreducible” (p. 3), and occasional missing spaces after commas in displayed sequences (e.g., (2.1)).","section":null},{"comment":"The statement of Theorem B excludes PSL(3,R) and PGL(3,C); a one-sentence reminder that these appear only via Aut^{0}(P^{2}) after regularization would help the reader.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is ready for acceptance. The reader’s flagged weakest link (gap-property contradiction via Monod–Py) is standard and correctly applied; higher-rank factors cannot act non-elementarily on HE by property (T). No load-bearing gaps remain. Fit for a top journal in dynamics/geometry is excellent."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper closes the two-dimensional birational Zimmer program for every irreducible higher-rank lattice, including those whose property FW is still open. The new pieces are Theorem C (H^{1}(Γ, F[Γ/K]) = 0 for solvable K, via Steinberg modules and Borel–Serre duality) and the resulting cube fixed-point theorem D that only needs solvable hyperplane stabilizers. Those feed into the Jonquières complex after a Monod-superrigidity reduction, giving virtual isotopic projective regularizability (Theorem A).\n\nWhat works: the vanishing argument is clean. Induction on Q-rank, Reeder’s induction isomorphism, Shapiro, and the rank estimate δ(G) = r - q give the vanishing for free once you have the exact sequence of the Tits building. The cocycle argument that turns vanishing into a bounded orbit (hence a fixed vertex by Gerasimov) is short and standard. The reduction to Jonq via elementary actions on HE is the usual Blanc–Cantat + Monod–Py package; the gap-property contradiction is correctly applied and higher-rank factors are blocked by property (T). Hyperplane stabilizers inject into point stabilizers in PGL(2,C) and are therefore solvable, so D applies directly. Citations are appropriate and the self-citations are background black boxes.\n\nSoft spots are minor. Continuity of the linear extension of isometries from the Caprace–Lytchak subset to the whole H\rho is sketched rather than fully expanded, but the calculation is elementary. The final upgrade from projective to virtual isotopic regularizability leans on the absence of loxodromics and Halphen twists inside Jonq, which is already in the literature. Nothing load-bearing is missing.\n\nThis is for people working on algebraic Zimmer, Cremona groups, or property FW. The math is classical and checkable line-by-line. I would send it to referees without hesitation; it advances the field cleanly.","headline":"Solid arithmetic-group vanishing that finishes surface Cremona rigidity for all higher-rank lattices without needing FW.","tokens_in":15983,"tokens_out":518,"would_cite":true,"duration_ms":5672,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37F99","14E07","22E40","20F65","53C24"],"pacs":[],"model":"grok-4.5","headline":"Higher-rank lattices acting by birational maps on surfaces always become regularizable after a finite-index change and conjugation.","keywords":["higher-rank lattices","birational transformations","CAT(0) cube complexes","regularizability","Zimmer program","Steinberg modules","Jonquières complex","property FW"],"falsifier":"An explicit infinite image homomorphism from an irreducible lattice in PSL(2,R)^{n} × PGL(2,C)^{m} (n+m ≥ 2) into Bir(S) whose image cannot be conjugated into Aut^{0} of any smooth projective surface, or a continuous non-elementary action of a higher-rank simple factor on the infinite-dimensional hyperbolic space that produces positive translation lengths without violating the gap property.","tokens_in":16089,"feed_emoji":"📐","tokens_out":1019,"duration_ms":10602,"temperature":0.7,"pith_summary":"This paper shows that any action of an irreducible higher-rank lattice by birational transformations of a complex projective surface can be conjugated, after passing to a finite-index subgroup, into the connected automorphism group of a smooth projective surface. The argument first reduces the problem, via superrigidity on infinite-dimensional hyperbolic spaces, to the de Jonquières subgroup, then proves a cohomological vanishing theorem that forces any action of the lattice on a CAT(0) cube complex with solvable hyperplane stabilizers to fix a vertex. That fixed-point theorem is applied to the Jonquières complex, yielding the regularizability statement for every higher-rank lattice, including those for which property FW is still open. The result completes the dimension-two case of the birational Zimmer program for all higher-rank lattices.","feed_headline":"Higher-rank lattices always regularize on surfaces","feed_subtitle":"Any birational action becomes algebraic after finite-index change and conjugation","key_machinery":"A cohomological vanishing theorem (Theorem C) asserting H^{1}(Γ′, F[Γ′/K]) = 0 for a finite-index torsion-free subgroup Γ′ and every solvable subgroup K, obtained from Borel–Serre duality and Steinberg modules; this vanishing forces fixed vertices for cube-complex actions with solvable hyperplane stabilizers (Theorem D), which is then applied to the Jonquières complex.","core_discovery":"For an irreducible lattice Γ in a product of non-compact center-free simple linear algebraic groups of total real rank at least 2, every homomorphism from Γ into the birational group of a complex projective surface is virtually isotopically projectively regularizable: after a finite-index subgroup and a birational conjugation the image lies in the connected component of the automorphism group of a smooth projective surface.","pith_inferences":["The same cohomological vanishing may extend regularizability results to other CAT(0) cube complexes that encode birational or automorphism groups in higher dimension, once analogous complexes with solvable hyperplane stabilizers are available.","The bypass of property FW suggests that other rigidity statements previously proved only under FW could be recovered for all higher-rank lattices by similar Steinberg-module vanishing arguments.","If an exotic continuous action of a higher-rank factor on infinite-dimensional hyperbolic space were discovered, the reduction step would fail and new non-regularizable examples might appear."],"forward_implications":["Every birational action of a higher-rank lattice on a surface is virtually conjugate to an algebraic action of a connected algebraic group, so the possible actions are classified by known lists of algebraic group actions on surfaces.","No infinite birational action exists when none of the simple factors is PSL(2,R), PGL(2,C), PU(1,2), PSL(3,R) or PGL(3,C).","The same vanishing implies fixed points for any CAT(0) cube-complex action of such a lattice whose hyperplane stabilizers are solvable, without invoking property FW.","Lattices for which property FW remains open still satisfy the regularizability conclusion in dimension two."],"fun_headline_variants":["Higher-rank lattices regularize birational surface actions","Rank ≥2 lattices force algebraic birational maps on surfaces","Higher-rank lattice actions on surfaces become virtually regular","Birational maps by higher-rank lattices virtually algebraize","Cube rigidity yields regularizable surface actions for lattices"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The reduction to the de Jonquières group rests on the claim that a non-elementary action on the infinite-dimensional hyperbolic space attached to the Cremona group would produce arbitrarily small translation lengths, contradicting a known gap property; that step uses density of lattice projections and a classification of continuous actions of rank-one groups.","fun_headline_variants_meta":{"raw":{"variants":["Higher-rank lattices regularize birational surface actions","Rank ≥2 lattices force algebraic birational maps on surfaces","Higher-rank lattice actions on surfaces become virtually regular","Birational maps by higher-rank lattices virtually algebraize","Cube rigidity yields regularizable surface actions for lattices"]},"model":"grok-4.5","effort":"low","cost_usd":0.004266,"raw_usage":{"total_tokens":1203,"prompt_tokens":642,"num_sources_used":0,"completion_tokens":79,"cost_in_usd_ticks":42660000,"prompt_tokens_details":{"text_tokens":642,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":482,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":642,"tokens_out":79,"duration_ms":5279,"temperature":1.0,"reasoning_tokens":482,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T15:03:26.510511+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"An explicit infinite image homomorphism from an irreducible lattice in PSL(2,R)^{n} × PGL(2,C)^{m} (n+m ≥ 2) into Bir(S) whose image cannot be conjugated into Aut^{0} of any smooth projective surface, or a continuous non-elementary action of a higher-rank simple factor on the infinite-dimensional hyperbolic space that produces positive translation lengths without violating the gap property.","supporting_citations":[],"review_version":1}