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Non-residually finite $\tilde{C}_2$-lattices

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Pith's one-line read Five finite triangle complexes with exotic ~C2 universal covers produce the first known non-residually finite uniform lattices on irreducible Euclidean buildings.

desk verdict 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. read the letter →

arxiv 2509.05054 v1 pith:CENV7LSO submitted 2025-09-05 math.GR

Thomas Titz Mite , Stefan Witzel This is my paper · ORCID
classification math.GR MSC 20E2622E4051E2420F65
keywords non-residuallyfiniteEuclideanbuildings~C2exoticBMWgroupsresidualnormalsubgrouppropertyKazhdan(T)
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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].

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [§4.1, proof of Theorem 4.1; cf. §4.4] 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.
  2. [§4.1, §4.4, and Introduction, p. 2] 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.
  3. [§4.3, §4.4, and Appendix A] 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.
minor comments (5)
  1. [Theorem 4.1] 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).
  2. [§4.5] Typo: 'Mouffang boundary' should be 'Moufang boundary'.
  3. [§B.1] The definition of γ reads 'let γ be such that γ^2 − α + 12'; an '= 0' is missing from this equation.
  4. [Appendix A] The sentence 'the automorphism group of the complexes Y^3_k is always 8' would be clearer as 'has order 8'.
  5. [§4.6] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivation is self-contained and independent of the paper's own conclusions.

full rationale

The main claim of non-residual finiteness is obtained by embedding known non-residually finite BMW lattices (from Radu and Janzen–Wise/Caprace) into the newly constructed irreducible C2~-buildings. The embedding is asserted via Lemma 2.2 and the claim that the subdivided product-of-trees subcomplex is locally convex; whether that geometric assertion is true is a correctness question, not a circular one, since it is not derived from the target conclusion and no equation or definition identifies the premise with the conclusion. The building property is checked against the standard Tits local-to-global theorem using explicitly listed vertex links. The finite-residual computations and automorphism-group computations are independent verifications and are explicitly flagged as computer-assisted. The only self-citation in the paper, [Wit17], appears in a historical list of known non-arithmetic lattices and is not load-bearing for any proof. The appendix's faithfulness proof for Γ_R relies on an external 2-adic representation argument and is again independent. No step fits the enumerated circularity patterns: no parameter is fitted and then renamed a prediction, no uniqueness theorem by the present authors is invoked, and no known result is repackaged as a new derivation. The acknowledged computer checks are not circular; they are reproducibility limitations.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central theorem rests on standard building and CAT(0) machinery plus two external non-residual finiteness results for product-of-trees lattices; the new input is the explicit complexes, presentations, and transfer argument. The largest unverified dependency is the asserted computer verification of finite residuals, automorphism groups, and non-isomorphism, for which no code is provided. No invented entities are postulated; the five buildings are fully specified finite combinatorial data.

free parameters (3)
  • epsilon (SDP parameter, Ozawa method) = 1.29
    Hand-chosen in the semidefinite program for property (T) of ¯Γ2_1; numerical error is controlled rigorously via Lemma 4.18, and this parameter does not enter the non-residual finiteness theorem.
  • omega (numerical error margin) = 0.0001
    Chosen to exceed C*nu = 3.036e-5 in Lemma 4.18 to certify the sum-of-squares decomposition; the final constant uses 1.2899 = 1.29 - 0.0001. Confined to the property (T) computation.
  • greedy search parameters (score estimate, restart step count) = not specified
    Algorithmic choices in the Radu-graph search (Section 5) that shaped which complexes were found; they do not affect verification of the final complexes.
assumptions (6)
  • standard math Tits local-to-global theorem: a thick chamber complex with generalized polygon links has a building universal cover (Theorem 2.3, [Tit81], [CL01]).
    Used to conclude Xq_i = ˜Yq_i is a ~C2 building from the computed links (Section 4, Theorem 4.1).
  • standard math Cartan-Hadamard theorem and pi1-injectivity of locally convex subcomplexes (Theorem 2.1, Lemma 2.2, [BH99]).
    Used to embed pi1(S_R) and pi1(S_JW) into Γq_i, transferring non-residual finiteness (Section 4).
  • domain assumption Radu's results: Γ_R is an irreducible BMW group, is not residually finite, and one of two specified commutators lies in its finite residual (Proposition 3.4, [Rad20, Prop. 5.4]).
    External, load-bearing for the q=2 case; Lemma 3.5 derives (xz)^4 in the finite residual from it.
  • domain assumption Janzen-Wise and Caprace: Γ_JW is irreducible, not residually finite, and contains explicit finite-residual elements (Proposition 3.8, [JW09], [Cap19, Remark 4.20]).
    External, load-bearing for the q=3 cases.
  • ad hoc to paper Three computer verifications: finite residual index computation (equivalently, a stated presentation of the trivial group), ball-reconstruction identification of Aut(Xq_i), and non-isomorphism of the quotients of Xq_i by ˇΓq_i.
    The authors explicitly state (Section 1) these are not verified by hand; no code is shipped, so the reader must accept the computations or re-implement them.
  • domain assumption Kramer-Weiss quasi-isometric rigidity of thick irreducible Euclidean buildings (Theorem 4.13, [KW14]) and Oppenheim's property (T) for lattices on irreducible affine buildings ([Opp]).
    External theorems used for pairwise non-quasi-isometry (Corollary 4.14) and for property (T) of the q=3 lattices (Section 4.6).

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Pith. "Pith review of Non-residually finite $\tilde{C}_2$-lattices." pith.science (2026). https://pith.science/paper/CENV7LSO

@misc{pith2026250905054,
  author       = {Pith},
  title        = {Pith review of: Non-residually finite $\tildeC_2$-lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CENV7LSO}},
  note         = {Machine review of arXiv:2509.05054}
}
read the original abstract

We provide the first examples of lattices on irreducible buildings that are not residually finite. Assuming that the normal subgroup property holds for them (which is expected) five of the lattices are simple.

Figures

Figures reproduced from arXiv: 2509.05054 by the authors.

Figure 1
Figure 1. The square complex SR. Let ΓR denote the BMW-group arising from Example 3.3, i.e. the extension of π1(SR) by D2 = ⟨σA, σX⟩. Consider the universal cover S˜R as homotopy classes of paths in SR starting at v00. Define the following automorphisms of S˜R all of which lie in ΓR. a : [γ] 7→ [a −1 ∗ σA(γ)], b : [γ] 7→ [b −1 ∗ σA(γ)], c : [γ] 7→ [c −1 ∗ σA(γ)], x : [γ] 7→ [x −1 ∗ σX(γ)], y : [γ] 7→ [y −1 ∗ σX(γ)], z : [γ] 7… view at source ↗
Figure 2
Figure 2. The square complex SJW. Define the following automorphisms of S˜ JW in ΓJW. a : [γ] 7→ [a −1 ∗ σA(γ)], b : [γ] 7→ [b −1 ∗ σA(γ)], x : [γ] 7→ [x −1 ∗ σX(γ)], y : [γ] 7→ [y −1 ∗ σX(γ)]. Proposition 3.8 (Janzen–Wise, Caprace). 1. The inclusion {a, b, x, y} → ΓJW induces an iso￾morphism ΓJW ∼= ⟨a, b, x, y | axay, ax−1 by−1 , ay−1 b −1x −1 , bxb−1 y −1 ⟩. 2. The BMW-action (ΓJW, SJW) is irreducible and the group ΓJW is n… view at source ↗
Figure 3
Figure 3. The GAB Y 2 1 , the sub complex generated by the triangles t1, . . . , t18 is isomorphic to a subdivision of SR. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The links in Y 2 1 of the vertices u1, u2, u3, u4, u5, v, and w. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: The edge path (g1 ∗ g −1 2 ∗ g3 ∗ g −1 4 ) is an (A1 × A1)-boundary. Now let G be the set of edges in Y from v to w (oriented in this way) and call edge paths in Y that lift to an (A1 × A1)-boundary also (A1 × A1)-boundaries. Consider the following words in F(G) R1 := …
Figure 6
Figure 6. Figure 6: The (A1 × A1)-boundaries around u1, u3, u4 and u5 are cycles of length four in the indicated graphs. Corollary 4.4. 1. The lattice Γ 2 1 is presented by generators h2, . . . , h15 subject to the relations h4h −1 8 h2, h−1 4 h5h −1 2 , h4h −1 3 h6, h−1 4 h6h −1 3 , h7h …
Figure 7
Figure 7. Figure 7: The combinatorial convex hull of two vertices [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: A perfect Radu graph admitting two exact covers. [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]

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