Weak Modularity and $\widetilde{A}_n$ Buildings

Zachary Munro

arxiv: 1906.10259 · v1 · pith:OQPTFHWCnew · submitted 2019-06-24 · 🧮 math.GR · math.CO

Weak Modularity and widetilde{A}_n Buildings

Zachary Munro This is my paper

Pith reviewed 2026-05-25 16:38 UTC · model grok-4.3

classification 🧮 math.GR math.CO

keywords weakly modular graphstilde A_n Coxeter groupsbuildingsCoxeter complexesgeometric actionsnonpositive curvatureaffine Coxeter groups

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The pith

Tilde A_n Coxeter groups act geometrically on weakly modular graphs.

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

The tilde A_n Coxeter groups fail to be systolic or cocompactly cubulated for n at least 3. The paper establishes that they nonetheless admit geometric actions on weakly modular graphs. Weak modularity is presented as a relaxed curvature condition that encompasses the 1-skeleta of CAT(0) cube complexes and systolic complexes. The argument proceeds by exhibiting canonical embeddings of the 1-skeleta of the associated Coxeter complexes into Euclidean space R^{n+1}. The same conclusion is reached for buildings of type tilde A_3.

Core claim

The tilde A_n Coxeter groups act geometrically on weakly modular graphs. This is shown by describing the canonical embeddings of the 1-skeleta of tilde A_n Coxeter complexes into R^{n+1}; these embeddings are used to verify the weak modularity property of the action. The result is also proved for buildings of type tilde A_3.

What carries the argument

The canonical embeddings of the 1-skeleta of tilde A_n Coxeter complexes into Euclidean space R^{n+1} that verify the weak modularity condition for the group actions.

If this is right

These groups possess geometric actions on graphs satisfying weak modularity.
Weak modularity supplies a common generalization that includes the 1-skeleta of both CAT(0) cube complexes and systolic complexes.
The same geometric action property holds for all buildings of type tilde A_3.
For n at least 3 this supplies a curvature-type property in cases where the stronger systolic and cubulation conditions are known to fail.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Embedding techniques of this kind may extend to other affine Coxeter groups that resist cubulation or systolicity.
The result supplies new examples of groups whose geometry can be studied through weakly modular graphs rather than stronger curvature models.
One could test whether the actions are cocompact and whether the graphs satisfy additional metric properties such as hyperbolicity in low dimensions.

Load-bearing premise

The canonical embeddings of the 1-skeleta into R^{n+1} suffice to establish the weak modularity property for the groups' actions.

What would settle it

An explicit geometric action of some tilde A_n group on a graph that violates one of the distance or link conditions required for weak modularity would falsify the claim.

Figures

Figures reproduced from arXiv: 1906.10259 by Zachary Munro.

**Figure 2.** Figure 2: Traversing an edge Proof of Theorem B. We first prove the triangle property. If A and B are adjacent vertices both of height n, then there is some subset SAB of the coordinates so that shifting SAB up one rung moves from A to B. That A and B are the same height means that SAB either contains (i) not all of the bottom rung and none of the top rung or (ii) all of the bottom and some of the top rung. After po… view at source ↗

**Figure 3.** Figure 3: The quadrangle property 4. Weak Modularity of Ae3-Buildings In this section, we prove that buildings of type Ae3 are weakly modular. However, we first prove a lemma needed for the theorem. For the remainder of the section, we fix generators for the Coxeter group Ae3 as in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Ae3 Coxeter diagram Lemma 4.1. If a 4-cycle with no diagonals in the 1-skeleton of an Ae3 Coxeter complex has a type z vertex, then the three remaining vertices are two type y vertices and a type z vertex. Furthermore, there exists an edge of type xw whose endpoints are adjacent to all vertices of the 4-cycle. Proof. We use our description of the 1-skeleton in Theorem A. By symmetry of the 1-skeleton, we c… view at source ↗

**Figure 5.** Figure 5: A = (Sy ∩ Sy0 − C) ∪ ((Sy ∪ Sy0 ) c − D) We are always guaranteed an edge at the center of such a 4-cycle with endpoints joined to each vertex of the 4-cycle: The endpoints of the edge are the vertices reached from Lz by increasing one of Sy ∩ Sy0 or Sy ∪ Sy0 , as seen in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: The xw edge the property is a vertex a, with two adjacent vertices b and c at distance 2 from a ( [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

**Figure 7.** Figure 7: Triangle property set-up Now consider the set-up for 1.7.ii, the quadrangle property. We are given a vertex a, with vertices c and d distance two from a, and a common neighbor b of c and d at distance 3 from a ( [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 8.** Figure 8: Quadrangle property set-up w y w y w y w y x y x [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

**Figure 9.** Figure 9: The link of d The dihedral angle between two faces meeting at an edge e is π divided by the order of the product of reflections through the faces. For example, consider the two faces of a simplex containing an edge yw. The yw edge has opposite vertices x and z and the action of x and z reflect through the faces meeting along yw. Since xz has order three the dihedral angle between the 2-simplices is π/3. Th… view at source ↗

**Figure 10.** Figure 10: The link of f Now consider lk(f) in some apartment containing df and af ( [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗

**Figure 11.** Figure 11: 6-cycle with alternating types Now, if we invert the diagram (i.e. consider b to be our initial vertex and e, f, and a to be the vertices relevant for the quadrangle property), then we can repeat our above argument. There are two possible outcomes. We either complete the quadrangle property and find a vertex adjacent to b, e, and f; or we get that ecb and f db are CAT(0) geodesics. In the case that ecb an… view at source ↗

**Figure 12.** Figure 12: Inverted quadrangle property of an edge containing x3, we get that x3 ∈ St(x1). But this is a contradiction, since there cannot be adjacent vertices of the same type in a Coxeter complex. So x1, x2, and x3 must be distinct. Analogously, we can assume w1, w2, and w3 are distinct. y y z w x z x z w z x w y [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗

**Figure 13.** Figure 13: A locally geodesic loop Consider now the piecewise geodesic cycle joining each of the three y vertices in the outer 6-cycle to the midpoints of the xiwi edges, for i = 1, 2, 3. This cycle is made up of six segments. Each segment has length π/4, since eight such congruent segments make up a geodesic cycle in an A3 Coxeter complex. Any two adjacent geodesic segments are contained in an apartment also contai… view at source ↗

read the original abstract

The $\widetilde{A}_n$ Coxeter groups are known to not be systolic or cocompactly cubulated for $n\geq 3$. We prove that these groups act geometrically on weakly modular graphs, a weak notion of nonpositive curvature generalizing the 1-skeleta of $\mathrm{CAT}(0)$ cube complexes and systolic complexes. To prove weak modularity we describe the canonical emeddings of the 1-skeleta of $\widetilde{A}_n$ Coxeter complexes into the Euclidean spaces $\mathbb{R}^{n+1}$. We also prove weak modularity for buildings of type $\widetilde{A}_3$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Munro proves geometric actions on weakly modular graphs for Ã_n Coxeter groups via their canonical embeddings into R^{n+1}.

read the letter

The main takeaway is that these Ã_n groups, which lack systolic or cocompact cubical actions for n ≥ 3, get a geometric action on weakly modular graphs instead. The proof route is the explicit description of the 1-skeleta embeddings into Euclidean space, plus a separate argument for buildings of type Ã_3. That supplies a concrete model using a curvature condition weaker than CAT(0) cubes or systolic complexes but still useful for studying the groups geometrically. The construction looks direct and fills a specific gap without claiming to solve larger problems in the area. The embeddings are the load-bearing step, and the paper treats them as sufficient to check the weak modularity axioms on the resulting graphs. No obvious circularity or invented objects appear in the setup. The Ã_3 case is handled as an extra result rather than the core claim. Soft spots are limited to the usual verification burden in these arguments: confirming that the embedded metric satisfies the precise distance and link conditions for weak modularity across all n, which could use more explicit checks for small n to make the general case easier to follow. Nothing in the available material suggests the central claim fails. This is for geometric group theorists already working with Coxeter groups, buildings, or alternative nonpositive curvature notions. A reader who cares about actions on graphs with controlled curvature properties will find a usable new example here. The paper is coherent on its own terms and has enough technical content to deserve a serious referee rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The paper proves that the affine Coxeter groups of type Ã_n (n≥3), which are known not to be systolic or cocompactly cubulated, act geometrically on weakly modular graphs. The proof proceeds by describing the canonical embeddings of the 1-skeleta of the associated Ã_n Coxeter complexes into Euclidean space R^{n+1} and verifying that these embeddings witness the weak modularity condition. The result is also extended to buildings of type Ã_3.

Significance. If the embeddings are shown to induce the required distance and convexity properties, the work supplies a concrete family of examples lying strictly between the CAT(0) cube complexes and systolic complexes, thereby demonstrating that weak modularity is strictly more general. The explicit Euclidean embeddings provide a verifiable, coordinate-based method that may apply to other affine Coxeter groups.

minor comments (3)

Abstract, line 3: 'emeddings' is a typographical error for 'embeddings'.
The manuscript should include a short subsection (perhaps §2 or §3) that recalls the precise definition of weak modularity used in the paper, including the two inequalities that must be verified for every triple of vertices.
Figure captions and the statement of the main theorem should explicitly indicate the range of n for which the result holds (n≥3) and note the known negative results for systolicity and cubulation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, the assessment of significance, and the recommendation of minor revision. We will incorporate any minor editorial or presentational improvements in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds by explicitly describing the canonical embeddings of the 1-skeleta of the tilde A_n Coxeter complexes into R^{n+1} and using those embeddings to verify the weak modularity condition directly. This is a constructive geometric argument whose steps are independent of the target property; the embeddings are standard and not defined in terms of weak modularity. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the chain. The result is therefore self-contained against external geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no information on free parameters, axioms, or invented entities; the work appears to be a pure existence proof in geometric group theory.

pith-pipeline@v0.9.0 · 5620 in / 995 out tokens · 26463 ms · 2026-05-25T16:38:20.987771+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The 1-skeleton of the Ã_n Coxeter complex can be described as follows. The vertices are of the form (x0,...,xn) in Z^{n+1} where x0+...+xn=0 and xi-xj ≡0 mod (n+1) for all i,j. Two vertices are adjacent iff they differ by a vector ... max ei - min ej =n+1.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages · 1 internal anchor

[1]

Peter Ambramenko and Kenneth Brown, Buildings: Theory and applications, Graduate Texts in Mathematics, Springer, 2008

work page 2008
[2]

Martin Bridson and Andr\'e Haefliger, Metric spaces of non-positive curvature, A Series of Comprehensive Studies in Mathematics, Springer, 1999

work page 1999
[3]

J \'e r \'e mie Chalopin, Victor Chepoi, Hiroshi Hirai, and Damian Osajda, Weakly modular graphs and nonpositive curvature, to appear in Mem. Amer. Math. Soc. (2019), arXiv:1409.3892

work page arXiv 2019
[4]

John Conway and Neil Sloane, Sphere packings, lattices and groups, Springer, 1991

work page 1991
[5]

Tadeusz Januszkiewicz and Jacek \'Swi a tkowski, Filling invariants of systolic complexes and groups, Geom. Topol. 11 (2007), no. 2, 727--758

work page 2007
[6]

Annette Karrer, Petra Schwer, and Koen Struyve, The triangle groups (2,4,4), (2,4,5) and (2,5,5) are not systolic , preprint (2018), arXiv:1812.08567

work page arXiv 2018
[7]

thesis, Ohio State University, 1987

G\'abor Moussong, Hyperbolic coxeter groups, Ph.D. thesis, Ohio State University, 1987

work page 1987
[8]

Reeves, Coxeter groups act on CAT (0) cube complexes , Journal of Group Theory 3 (2003)

Graham Niblo and L.D. Reeves, Coxeter groups act on CAT (0) cube complexes , Journal of Group Theory 3 (2003)

work page 2003
[9]

Piotr Przytycki and Petra Schwer, Systolizing buildings, Groups, Geometry, and Dynamics 10 (2016)

work page 2016
[10]

Mark Ronan, Lectures on buildings, The University of Chicago Press, 2009

work page 2009
[11]

Adam Wilks, The (2,4,5) triangle Coxeter group is not systolic , preprint (2017), arXiv:1706.08019

work page internal anchor Pith review Pith/arXiv arXiv 2017

[1] [1]

Peter Ambramenko and Kenneth Brown, Buildings: Theory and applications, Graduate Texts in Mathematics, Springer, 2008

work page 2008

[2] [2]

Martin Bridson and Andr\'e Haefliger, Metric spaces of non-positive curvature, A Series of Comprehensive Studies in Mathematics, Springer, 1999

work page 1999

[3] [3]

J \'e r \'e mie Chalopin, Victor Chepoi, Hiroshi Hirai, and Damian Osajda, Weakly modular graphs and nonpositive curvature, to appear in Mem. Amer. Math. Soc. (2019), arXiv:1409.3892

work page arXiv 2019

[4] [4]

John Conway and Neil Sloane, Sphere packings, lattices and groups, Springer, 1991

work page 1991

[5] [5]

Tadeusz Januszkiewicz and Jacek \'Swi a tkowski, Filling invariants of systolic complexes and groups, Geom. Topol. 11 (2007), no. 2, 727--758

work page 2007

[6] [6]

Annette Karrer, Petra Schwer, and Koen Struyve, The triangle groups (2,4,4), (2,4,5) and (2,5,5) are not systolic , preprint (2018), arXiv:1812.08567

work page arXiv 2018

[7] [7]

thesis, Ohio State University, 1987

G\'abor Moussong, Hyperbolic coxeter groups, Ph.D. thesis, Ohio State University, 1987

work page 1987

[8] [8]

Reeves, Coxeter groups act on CAT (0) cube complexes , Journal of Group Theory 3 (2003)

Graham Niblo and L.D. Reeves, Coxeter groups act on CAT (0) cube complexes , Journal of Group Theory 3 (2003)

work page 2003

[9] [9]

Piotr Przytycki and Petra Schwer, Systolizing buildings, Groups, Geometry, and Dynamics 10 (2016)

work page 2016

[10] [10]

Mark Ronan, Lectures on buildings, The University of Chicago Press, 2009

work page 2009

[11] [11]

Adam Wilks, The (2,4,5) triangle Coxeter group is not systolic , preprint (2017), arXiv:1706.08019

work page internal anchor Pith review Pith/arXiv arXiv 2017