{"id":"d2289ba9-975e-45d1-b874-ddfee1c97017","arxiv_id":"2509.05054","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"First examples of non-residually finite lattices on irreducible buildings, constructed explicitly as fundamental groups of five ~C2 triangle complexes.","lead":"This paper builds five new geometric spaces called ~C2 buildings, together with their symmetry groups, and proves these groups are the first known lattices on irreducible buildings that are not residually finite, meaning they cannot be approximated by finite groups. The construction starts from known non-residually finite groups acting on simpler spaces (products of trees) and embeds them into the new buildings; several verification steps are done by computer.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Non-residual finiteness hinges on an unproven local-convexity assertion for the embedded product-of-trees subcomplex, which is more load-bearing than the acknowledged computer checks.","rationale":"The reader's weakest_assumption identifies three computer-assisted verification steps (finite residual check, automorphism group reconstruction, and pairwise non-isomorphism of quotients). Those are real and well described. However, the most load-bearing concern for the central claim of non-residual finiteness is the unproven local-convexity statement used to embed the non-residually finite BMW lattice into Γ^q_i. If that embedding is not a local isometry, then even the basic conclusion that Γ^q_i is not residually finite fails, irrespective of the computer checks. The authors do not supply a proof or a reference for 'non-positively curved (thus locally convex)'; the implication is not a standard theorem for arbitrary non-positively curved chamber subcomplexes. The concern is concrete and can be settled by finite graph computations in the links (e.g. Figure 4). Since the paper gives enough data to perform this check, the appropriate editorial outcome remains CONDITIONAL: the main theorem should not be accepted as fully established until the local convexity/embedding assertion is verified by hand or by a small explicit computation. Thus the reader's verdict is unchanged, but for a different and more fundamental reason than the one highlighted in the reader's analysis.","tokens_in":38824,"tokens_out":16328,"duration_ms":179577,"concrete_test":"For each vertex of the subcomplex \\dot{S}_R (and of the S_JW subdivisions), compute its link L0 inside the subcomplex and its link L inside the ambient Y^q_i, then check that L0 is an isometrically embedded subgraph of L: for every pair of vertices in L0, the graph distance in L0 must equal the graph distance in L. For Y^2_1 this can be done directly from Figure 4; in particular the K_{3,3} link of u1 (or u2) inside the subcomplex must be geodesically convex in the displayed link graph. If any pair has a shorter path in L than in L0, the inclusion is not a local isometry, Lemma 2.2 does not apply, and the injection of Γ_R fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 4.1 (proof of Theorem 4.1) the authors state: 'Since \\dot{S}_R is non-positively curved (thus locally convex in Y^2_1) its universal cover embeds into X^2_1 and its fundamental group embeds into Γ^2_1 by Lemma 2.2.' The implication is not valid in general: a non-positively curved subcomplex of a non-positively curved polygonal complex need not be locally convex. Lemma 2.2 explicitly requires Y0 to be locally convex, and non-positive curvature of Y0 alone does not prevent shortcuts in ambient vertex links. The same type of assertion is used for each Y^3_k embedding a subdivision of S_JW. If the relevant link subgraphs are not convex (isometrically embedded), the universal cover of the subcomplex does not embed as a convex subspace, π1(S_R) (or π1(S_JW)) need not inject into Γ^q_i, and the central non-residual-finiteness claim would not follow from the BMW lattices. This is a hand-checkable geometric step, not one of the three explicitly listed computer exceptions, and it underpins the core claim rather than only the stronger structural conclusions.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs five finite triangle complexes Y^2_1, Y^3_1, ..., Y^3_4 and claims that their universal covers X^q_i are exotic Euclidean buildings of type \\tilde{C}_2 and that the fundamental groups \\Gamma^q_i are the first known non-residually finite uniform lattices on irreducible two-dimensional Euclidean buildings. The proof embeds subdivisions of known non-residually finite Burger–Mozes–Wise square complexes—Radu's complex S_R for q=2 and Janzen–Wise's complex S_JW for q=3—into the new complexes, then uses CAT(0) local convexity (Lemma 2.2) to transfer non-residual finiteness. The paper also computes finite-residual indices, identifies full automorphism groups, proves pairwise non-isomorphism and non-quasi-isometry of the five examples, and establishes property (T) for the q=2 lattice. Several central verification steps are explicitly acknowledged as computer-dependent.","tokens_in":39144,"tokens_out":6877,"duration_ms":74136,"significance":"If the construction and verifications are correct, this is a major advance: it would supply the first non-residually finite uniform lattices on irreducible Euclidean buildings, a question explicitly open in the literature, and the finite residuals \\check{\\Gamma}^q_i would have no proper finite-index subgroups and would be simple under the expected normal subgroup property. The paper also contributes an explicit combinatorial search method for non-positively curved chamber complexes, a normal-form algorithm for the resulting lattices, and very explicit data for the complexes and presentations. A notable strength is the paper's transparency: the introduction lists exactly which parts are not hand-verifiable. However, that same transparency exposes three load-bearing computational steps for which no code, data, or certificates are supplied, and the proof of the central embedding contains an unproved local-convexity assertion.","major_comments":[{"comment":"The proof states: 'Since \\dot{S}_R is non-positively curved (thus locally convex in Y^2_1) its universal cover embeds into X^2_1 and its fundamental group embeds into Γ^2_1 by Lemma 2.2.' This implication is not valid in general: a non-positively curved subcomplex of a non-positively curved polygonal complex need not be locally convex, and Lemma 2.2 explicitly requires local convexity of the subcomplex. The same unproved local-convexity step is used for the S_JW subdivisions in the q=3 cases (§4.4). The authors need to verify—or provide an explicit link-checking argument—that the subdivision of S_R is locally convex in Y^2_1 and that each S_JW subdivision is locally convex in its Y^3_k. Without this, Lemma 2.2 cannot be invoked, and the embedding of the non-residually finite BMW groups into the Γ^q_i is not established.","section":"§4.1, proof of Theorem 4.1; cf. §4.4"},{"comment":"The paper's finite-residual claims are load-bearing for Theorem A: for q=2, 'Γ^2_1 is the finite residual' is reduced to checking that adding the relation (g_1 g_6^{-1})^4 to the presentation of Proposition 4.3 gives the cyclic group of order 2, and the authors state this has not been carried out by hand; for q=3, the indices [Γ^3_k : \\check{Γ}^3_k] = 4 or 8 depend on a computer verification that the normal closure of explicit loops is of finite index. No code, machine-readable certificate, or independent proof is provided. Since these computations determine the claim that the \\check{Γ}^q_i have no proper finite-index subgroups, they need to be made reproducible (code and data) or replaced by hand-checkable derivations.","section":"§4.1, §4.4, and Introduction, p. 2"},{"comment":"The identification of Aut(X^q_i) rests on computer reconstruction of balls in the Cayley complex and on a rigidity assertion: for X^2_1 this is the claim that the automorphism group of a radius-4 ball pointwise fixes the radius-2 ball (§4.3), and for X^3_k it relies on a computer check satisfying Lemma A.1. Pairwise non-isomorphism of the q=3 buildings is similarly reduced to a computer comparison of the finite quotients \\check{Γ}^3_k \\backslash X^3_k. These are exactly exceptions 2 and 3 listed in the introduction, and they support the 'exotic' assertion, the pairwise non-isomorphism claim, and the non-quasi-isometry claim via Corollary 4.14. Without code or a written derivation, these central conclusions cannot be independently checked.","section":"§4.3, §4.4, and Appendix A"}],"minor_comments":[{"comment":"The statement says 'Its fundamental group Γ^2_1 = π_1(X^2_1)', but Γ^2_1 is the fundamental group of the quotient complex: it should read π_1(Y^2_1).","section":"Theorem 4.1"},{"comment":"Typo: 'Mouffang boundary' should be 'Moufang boundary'.","section":"§4.5"},{"comment":"The definition of γ reads 'let γ be such that γ^2 − α + 12'; an '= 0' is missing from this equation.","section":"§B.1"},{"comment":"The sentence 'the automorphism group of the complexes Y^3_k is always 8' would be clearer as 'has order 8'.","section":"Appendix A"},{"comment":"The semidefinite programming verification of property (T) for ¯Γ^2_1 is interesting but is not reproducible from the text: no optimizer output, Gram matrix, or code is included. Since property (T) is also attributed to the cited preprint [Opp], this is not a blocking issue for the main theorem, but the data should be made available if the quantitative claim is to be checked.","section":"§4.6"}],"recommendation":"major_revision","confidential_remarks":"The central claim of the paper is attractive and potentially very important, but I am not comfortable with the current level of verification. The local-convexity gap in §4.1 directly affects the transfer of non-residual finiteness and must be fixed by an explicit argument or link inspection. In addition, the three computer-dependent steps are not reproducible because no code, data, or certificates are supplied; this is particularly serious for the index computations and the non-isomorphism claims that are part of Theorem A. I would advise the editor to require supplementary material and an explicit treatment of the local-convexity assertion before considering the paper for publication. The authors' own statement that the theorem is 'verified without a computer with the following exceptions' should be taken as a clear indication of what needs to be supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper delivers what the title promises: the first examples of non-residually finite lattices on irreducible Euclidean buildings, of type ~C2. The main idea is genuinely good. The authors take the known non-RF BMW lattices on products of trees and embed them into newly constructed ~C2 buildings, so that non-residual finiteness is inherited directly. Five finite triangle complexes are given explicitly, with presentations, a normal-form algorithm, and a property (T) computation. If the construction holds, this is a major step toward the BCL conjecture and provides concrete test objects for the normal subgroup property.\n\nNow the soft spots, in proportion.\n\nFirst and most important: the proof of Theorem 4.1 contains the sentence “Since ǐS_R is non-positively curved (thus locally convex in Y^2_1)” and then applies Lemma 2.2 to conclude that the fundamental group of the subcomplex injects. That implication is not valid in general; non-positive curvature of a subcomplex does not imply local convexity in the ambient complex. The universal cover of ǐS_R embeds as a convex subspace only if the link condition is checked. This is load-bearing for the core non-residual-finiteness claim, and it is not one of the three acknowledged computer-check exceptions. I suspect the subcomplex is actually locally convex here—the links are small enough to verify by hand—but the current text does not provide that verification. The same issue appears in the q=3 cases. This needs to be fixed before the main theorem is fully established.\n\nSecond, the reader is right to flag the three computer-assisted steps: the finite-residual index computations, the full automorphism group determination, and the pairwise non-isomorphism for q=3. The authors are explicit that these were not done by hand, and no code or certificates are provided. The complexes are fully specified, so the claims are reproducible in principle, but for a result of this importance the authors should ship scripts or certificates. This affects the stronger claims (index formulas, exoticness, non-isomorphism, and the conditional simplicity), not the basic non-RF statement conditional on the local-convexity fix.\n\nThird, a minor dependency: property (T) for the q=3 lattices relies on an arXiv preprint. That is acceptable but worth noting.\n\nBottom line: the central strategy is sound, the explicit data is extensive, and the paper deserves serious refereeing. But the local-convexity assertion is a genuine gap in the written proof, and the computer verifications need to be publicly reproducible. With those addressed, it is a strong paper.","headline":"First non-residually finite lattices on irreducible buildings, with a clean inheritance argument from BMW lattices—but the current proof has a load-bearing local-convexity gap and some unshipped computer verifications.","tokens_in":39661,"tokens_out":4080,"would_cite":true,"duration_ms":44889,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20E26","22E40","51E24","20F65"],"pacs":[],"model":"deepseek-v4-flash","headline":"Five finite triangle complexes with exotic ~C2 universal covers produce the first known non-residually finite uniform lattices on irreducible Euclidean buildings.","keywords":["non-residually finite","Euclidean buildings","~C2 buildings","exotic buildings","BMW groups","finite residual","normal subgroup property","Kazhdan property (T)"],"falsifier":"Independently run those three checks: decide whether the relevant finite presentation presents the trivial group (in the q=2 case, whether adding (g1 g6^-1)^4 to the presentation of the extended group yields the cyclic group of order 2); recompute automorphism groups of radius-4 balls in each Cayley complex; and compare the quotients of the q=3 buildings by their finite residuals. A different outcome in any one of these would falsify part of Theorem A.","tokens_in":38673,"feed_emoji":"📐","tokens_out":12981,"duration_ms":119055,"temperature":0.7,"pith_summary":"The paper constructs five finite triangle complexes whose universal covers are irreducible two-dimensional Euclidean buildings of type ~C2, i.e. buildings whose chambers are triangles with angles 90°, 45°, 45°. It proves that the fundamental groups of these complexes are the first known uniform lattices on irreducible Euclidean buildings that are not residually finite: each contains a nontrivial element that vanishes in every finite quotient. The construction builds each complex around a known non-residually finite lattice on a product of trees, so the failure of residual finiteness is inherited. The paper also shows the five buildings are pairwise non-isomorphic and quasi-isometrically distinct, and that the finite residual of each lattice has no proper finite-index subgroups; if the expected normal subgroup property holds, those residuals are abstractly simple.","feed_headline":"First non-residually finite lattices on irreducible buildings","feed_subtitle":"Five triangle complexes yield exotic buildings with non-residually finite fundamental groups.","key_machinery":"The load-bearing mechanism is the geometric presentation of Lemma 4.2: for a ~C2 GAB (a chamber complex whose vertex links are generalized polygons) with an involution swapping its two special vertex types, the extension acting regularly on special vertices is presented by the long edges between the two special types, with relations coming from the involution and from filling every (A1 × A1)-boundary, a four-cycle around a non-special vertex. Filling those four-cycles produces a simply connected complex, so the lattice's Cayley graph embeds as the long-edge subgraph of the building. Normal forms—unique reduced paths along the convex hull of two special vertices—make the word problem computab","core_discovery":"Theorem A asserts that there are five finite triangle complexes Y_1^2, Y_1^3, Y_2^3, Y_3^3, Y_4^3 whose universal covers X_i^q are exotic buildings of type ~C2 and whose fundamental groups Γ_i^q are not residually finite; the buildings are pairwise non-isomorphic and the groups pairwise not quasi-isometric. The non-residual finiteness is inherited from BMW groups—groups acting freely and transitively on the vertices of a product of two regular trees—that embed into the new buildings as subcomplexes. For q=2, Γ_1^2 equals its own finite residual; for q=3, the finite residual has index 4 in Γ_1^3 and Γ_2^3 and index 8 in Γ_3^3 and Γ_4^3. Each finite residual has no proper finite-index subgroup","pith_inferences":["The same Radu-graph search could plausibly be run from other non-residually finite BMW groups to produce infinite families of non-residually finite ~C2-lattices, not just the five examples listed.","If the normal subgroup property holds, the finite residuals would become finitely presented infinite simple groups acting cocompactly on irreducible Euclidean buildings, a combination not previously realized.","The three computer checks the paper leaves to the machine—triviality of one finite presentation, rigidity of reconstructed balls, and pairwise comparison of the q=3 quotients—are the natural places to seek independent confirmation or a formal proof."],"forward_implications":["The existence question is settled: uniform lattices on irreducible Euclidean buildings need not be residually finite.","The five lattices represent five distinct quasi-isometry classes, so non-residual finiteness coexists with quasi-isometric rigidity of buildings.","Each finite residual ˇΓ_i^q has no proper finite-index subgroups; under the expected normal subgroup property they are abstractly simple.","Each building X_i^q has discrete full automorphism group, so these are genuinely exotic buildings rather than disguised arithmetic ones.","The q=2 lattice has explicit quantitative property (T)—Kazhdan radius at most 2 and Kazhdan constant at least 0.4147—and the q=3 lattices have property (T) by the cited result [Opp]."],"supporting_citations":[{"why":"Supplies the BMW group on a product of 3-regular trees whose finite residual contains (xz)^4, the source of non-residual finiteness for the q=2 lattice.","marker":"[Rad20]"},{"why":"Supplies the BMW group on a product of 4-regular trees with explicit elements in the finite residual, the source for the q=3 lattices.","marker":"[JW09]"},{"why":"Provides the explicit finite-residual elements and BMW-group background used in the construction.","marker":"[Cap19]"},{"why":"Origin of irreducible lattices on products of trees, the framework that BMW groups realize.","marker":"[BM00]"},{"why":"Establishes the first non-residually finite lattices on products of trees, the phenomenon the paper imports into irreducible buildings.","marker":"[Wis96]"},{"why":"Local approach to buildings: a chamber complex whose vertex links are generalized polygons has a building as universal cover, used to identify the X_i^q as ~C2-buildings.","marker":"[Tit81]"},{"why":"Supplies the CAT(0) background and the lemma that a locally convex subcomplex has injective fundamental group, embedding the BMW lattice into Γ_i^q.","marker":"[BH99]"},{"why":"Coarse equivalence rigidity for thick irreducible Euclidean buildings, used to conclude the five lattices are pairwise not quasi-isometric.","marker":"[KW14]"},{"why":"Sum-of-squares criterion for Kazhdan's property (T), used to prove quantitative property (T) for the q=2 lattice.","marker":"[Oza16]"},{"why":"Cited for property (T) for lattices on affine buildings, which covers the q=3 lattices.","marker":"[Opp]"}],"fun_headline_variants":["First non-residually finite lattices on exotic C2 buildings","Exotic C2 buildings host first non-residually finite lattices","New lattices on irreducible buildings evade residual finiteness","Five triangle complexes give non-residually finite C2 lattices"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The three machine computations the authors explicitly did not check by hand are load-bearing: if any of them is wrong, the claimed finite-residual indices, automorphism groups, or non-isomorphism of buildings fail.","fun_headline_variants_meta":{"raw":{"variants":["First non-residually finite lattices on exotic C2 buildings","Exotic C2 buildings host first non-residually finite lattices","New lattices on irreducible buildings evade residual finiteness","Five triangle complexes give non-residually finite C2 lattices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000386,"raw_usage":{"total_tokens":1794,"prompt_tokens":581,"completion_tokens":1213,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":325,"completion_tokens_details":{"reasoning_tokens":1152}},"tokens_in":325,"tokens_out":1213,"duration_ms":8249,"temperature":1.0,"reasoning_tokens":1152,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T05:41:37.553871+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Independently run those three checks: decide whether the relevant finite presentation presents the trivial group (in the q=2 case, whether adding (g1 g6^-1)^4 to the presentation of the extended group yields the cyclic group of order 2); recompute automorphism groups of radius-4 balls in each Cayley complex; and compare the quotients of the q=3 buildings by their finite residuals. A different outcome in any one of these would falsify part of Theorem A.","supporting_citations":[],"review_version":1}