arxiv: 2605.09493 · v1 · submitted 2026-05-10 · 🧮 math.GR

Recognition: 2 theorem links

· Lean Theorem

Simple Lattices in Products of Davis Complexes

Michal Amir, Nir Lazarovich

Pith reviewed 2026-05-12 03:15 UTC · model grok-4.3

classification 🧮 math.GR

keywords simple latticesDavis complexesright-angled Coxeter groupsodd graphsuniversal groupsproducts of complexesBurger-Mozes construction

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The pith

Simple uniform lattices exist in products of trees and two-dimensional Davis complexes from right-angled Coxeter groups with odd defining graphs.

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

The paper constructs simple uniform lattices acting cocompactly and freely on products of trees and two-dimensional Davis complexes associated to right-angled Coxeter groups whose defining graphs are odd graphs. These lattices are simple groups, having no nontrivial normal subgroups, extending the earlier Burger-Mozes examples that were limited to products of trees. The proof introduces an analogue of the Burger-Mozes universal groups for this mixed setting and supplies a local criterion that ensures certain vertex-transitive groups are dense in the universal group. If such a group satisfies the criterion, the resulting lattice is simple and uniform. This supplies new examples of simple groups with geometric actions on products involving higher-dimensional polyhedral complexes.

Core claim

We construct simple uniform lattices in products of trees and two-dimensional Davis complexes of the right-angled Coxeter group whose defining graph is an odd graph. As part of the proof, we define an analogue of the Burger-Mozes universal groups in this setting, and provide a local criterion for a vertex transitive group to be dense in the universal group.

What carries the argument

An analogue of the Burger-Mozes universal group together with a local criterion for density satisfied by a vertex-transitive group acting cocompactly and freely on vertices of the product.

If this is right

The resulting lattices are discrete subgroups that act geometrically and simply on the product space.
The local density criterion directly produces simplicity once a qualifying vertex-transitive group is found.
The construction applies precisely when the defining graph of the Coxeter group is odd.
These groups furnish examples of simple uniform lattices beyond the classical products of trees.

Where Pith is reading between the lines

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

The same local criterion could be checked on other candidate vertex-transitive groups to produce further examples.
The method might extend to products involving Davis complexes of different dimensions if analogous density conditions can be verified.
Simplicity in these mixed products may relate to broader questions about normal subgroups in groups acting on polyhedral complexes.

Load-bearing premise

A suitable vertex-transitive group acting cocompactly and freely on the vertices exists and satisfies the local density criterion inside the analogue universal group.

What would settle it

An explicit vertex-transitive group that meets the cocompact free action conditions but fails the local density criterion, or a constructed lattice that turns out to possess a nontrivial normal subgroup.

Figures

Figures reproduced from arXiv: 2605.09493 by Michal Amir, Nir Lazarovich.

**Figure 1.** Figure 1: The normal paths [v ↗ x] (left) and [v ↘ x] (right) 2.2 The ℓ∞ metric and normal paths CAT(0) cube complexes can be considered with various metrics. When endowing each cube with the ℓ∞ metric, one obtains the so-called ℓ∞ metric on the cube complex. While geodesics in this metric are not unique, the following definition of “normal paths” due to Niblo-Reeves [31, Definition 3.1] gives rise to a particular u… view at source ↗

**Figure 2.** Figure 2: The partly free vertices are shown in full circles, while free ones are empty [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Necessarily, B1(y ′) ∩ Sn−2(v) = {x ′′}, and so x ′′ ∈ [v ↘ y ′ ]. The vertex x ′ ∈ [v ↘ y], as otherwise y ′ ∈ [v ↘ y] but the path y, y′ , x′′ is not a normal path. A similar argument shows that x ′ ∈ [v ↘ x]. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 3.** Figure 3: The block b and its neighbours in Case 2. 3 Davis complexes and universal groups 3.1 The Davis complex In this subsection we will define the Davis complex XL, and describe its geometry. Definition 3.1. For a finite graph L = (V, E), define the corresponding right-angled Coxeter group by WL = RACG(L) ∶= ⟨V (L) ∣ a 2 , ∀a ∈ V (L), [a, b], ∀a ∼ b ⟩ where a ∼ b if a and b are neighbours in L. Definition 3.2 (D… view at source ↗

**Figure 4.** Figure 4: An example for normal path [y ′ ↖ v ↗ x] Proof. Let y ′ ∈ Sn−1(v) be a partly free vertex, and let ¯v be the first step of [v ↗ x], i.e., ¯v = [v ↗ x] ∩ S1(v), as shown in [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗

**Figure 5.** Figure 5: The cube which corresponds to the 2-simplex [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗

read the original abstract

Burger and Mozes (1997) constructed the first examples of simple uniform lattices in products of trees. In this paper, we construct simple uniform lattices in products of certain Davis complexes. More precisely, we consider lattices in products of trees and two-dimensional Davis complexes of the right-angled Coxeter group whose defining graph is an odd graph. As part of the proof, we define an analogue of the Burger-Mozes universal groups in this setting, and provide a local criterion for a vertex transitive group to be dense in the universal group.

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

The paper extends Burger-Mozes simple lattices to products of trees with odd-graph Davis complexes by defining a suitable universal-group analogue and a local density criterion.

read the letter

This paper constructs simple uniform lattices in products of trees and two-dimensional Davis complexes of right-angled Coxeter groups whose defining graphs are odd graphs. It is a direct extension of the Burger-Mozes construction from 1997 into a new setting of CAT(0) complexes. They define an analogue of the universal groups for this product and give a local criterion for a vertex-transitive group to be dense in that universal group. The construction then uses this to get the simple lattices. This is new because the Davis complexes here are two-dimensional and tied to odd graphs, which is not covered in the earlier tree-only work. The local criterion is a practical addition that lets them produce the examples. It does well by giving explicit new instances of simple groups with cocompact free actions on these spaces, which adds to the collection of known simple lattices in CAT(0) products. The soft spot, if there is one, lies in confirming that the group chosen to satisfy the local density criterion also acts freely on vertices and cocompactly on the whole product. The abstract states that such lattices are constructed, so the paper presumably verifies this compatibility. Still, the details of how the transitivity on links or stabilizers works with freeness would be worth a close look in the proof to make sure no hidden stabilizers appear. This paper is aimed at people in geometric group theory who study simple groups and their actions on CAT(0) complexes. A reader who wants to see how the Burger-Mozes method generalizes to higher-dimensional complexes will get concrete value from the setup and the criterion. It is solid enough to deserve a serious referee, as it brings new examples and a reusable local tool. I would recommend sending it out for peer review.

Referee Report

1 major / 1 minor

Summary. The paper extends Burger-Mozes' construction of simple uniform lattices in products of trees to products of trees and two-dimensional Davis complexes of right-angled Coxeter groups whose defining graphs are odd graphs. It defines an analogue of the Burger-Mozes universal group in this setting and supplies a local criterion for density of a vertex-transitive group in the universal group, which is then used to produce the claimed simple uniform lattices.

Significance. If the constructions are correct, the work supplies new families of simple groups acting cocompactly and freely on products of higher-dimensional complexes, generalizing the tree case. The analogue universal group and the local density criterion constitute a reusable technical tool that may apply to other classes of complexes in geometric group theory.

major comments (1)

[Construction of the lattice and local criterion (abstract and proof sections)] The central claim requires that the vertex-transitive group G embedded into the analogue universal group satisfies the local density criterion while simultaneously acting freely on vertices and cocompactly on the product. The abstract asserts that such lattices are constructed, but the compatibility of these conditions (in particular, whether the local criterion on link actions or stabilizers forces non-trivial stabilizers or obstructs cocompactness) is load-bearing and must be verified explicitly in the construction.

minor comments (1)

[Abstract] The abstract could briefly indicate the precise dimension of the Davis complexes and the key combinatorial properties of the odd graph that enable the local criterion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our work on simple lattices in products of Davis complexes. We address the major comment below.

read point-by-point responses

Referee: [Construction of the lattice and local criterion (abstract and proof sections)] The central claim requires that the vertex-transitive group G embedded into the analogue universal group satisfies the local density criterion while simultaneously acting freely on vertices and cocompactly on the product. The abstract asserts that such lattices are constructed, but the compatibility of these conditions (in particular, whether the local criterion on link actions or stabilizers forces non-trivial stabilizers or obstructs cocompactness) is load-bearing and must be verified explicitly in the construction.

Authors: We thank the referee for identifying this key compatibility issue. In the construction, G is generated by elements chosen so that their action on vertices is free (using the odd-graph defining relations to ensure no fixed vertices in the product action), while vertex-transitivity follows from the transitive action on each factor. Cocompactness of the action on the product is obtained because the quotient is a finite complex built from the finite odd graphs. The local density criterion concerns only the induced actions on links of vertices and is independent of global stabilizers; it is satisfied by making the projections dense in the local universal groups without introducing fixed points, as the local actions are free by the same generator choice. We agree that an explicit verification paragraph is warranted and will add one in the revised version (in the section following the definition of G and before the density theorem), including direct checks that the criterion does not force non-trivial stabilizers or prevent cocompactness. revision: yes

Circularity Check

0 steps flagged

No circularity: construction extends prior external work via explicit local criterion

full rationale

The paper's derivation proceeds by defining an analogue of the Burger-Mozes universal group for the product of a tree and a 2-dimensional Davis complex associated to a right-angled Coxeter group with odd-graph defining graph, then stating a local criterion for density of a vertex-transitive subgroup. This is used to construct simple uniform lattices. The cited Burger-Mozes 1997 result is independent prior work by different authors; no equations reduce a claimed prediction to a fitted input by construction, no self-citation forms the load-bearing justification, and no ansatz or uniqueness theorem is smuggled in from the authors' own prior results. The central existence claim rests on verifying the local criterion for a suitably chosen group, which is presented as an independent mathematical step rather than a tautology. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The work relies on standard facts about right-angled Coxeter groups and Davis complexes that are assumed known from prior literature.

pith-pipeline@v0.9.0 · 5371 in / 1136 out tokens · 37138 ms · 2026-05-12T03:15:09.584936+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

[1]

Cambridge University Press, 2000

Michael Aschbacher.Finite group theory, volume 10. Cambridge University Press, 2000

work page 2000
[2]

Factor and normal subgroup theorems for lattices in products of groups.Inventiones mathematicae, 163(2):415–454, 2006

Uri Bader and Yehuda Shalom. Factor and normal subgroup theorems for lattices in products of groups.Inventiones mathematicae, 163(2):415–454, 2006

work page 2006
[3]

Polygonal complexes and combinatorial group theory.Geometriae Dedicata, 50:165–191, 1994

Werner Ballmann and Michael Brin. Polygonal complexes and combinatorial group theory.Geometriae Dedicata, 50:165–191, 1994

work page 1994
[4]

On commensurators of free groups and free pro-p groups.arXiv preprint arXiv:2507.04120, 2025

Yiftach Barnea, Mikhail Ershov, Adrien Le Boudec, Colin D Reid, Matteo Van- nacci, and Thomas Weigel. On commensurators of free groups and free pro-p groups.arXiv preprint arXiv:2507.04120, 2025

work page arXiv 2025
[5]

Uniform tree lattices.Journal of the American Mathematical Society, 3(4):843–902, 1990

Hyman Bass and Ravi Kulkarni. Uniform tree lattices.Journal of the American Mathematical Society, 3(4):843–902, 1990

work page 1990
[6]

American Mathematical Soc., 2014

Mladen Bestvina, Michah Sageev, and Karen Vogtmann.Geometric group theory, volume 21. American Mathematical Soc., 2014

work page 2014
[7]

An edge-colouring problem.The American Mathematical Monthly, 79(9):1018–1020, 1972

Norman Biggs. An edge-colouring problem.The American Mathematical Monthly, 79(9):1018–1020, 1972

work page 1972
[8]

Homomorphismes” abstraits” de groupes alge- briques simples.Annals of Mathematics, 97(3):499–571, 1973

Armand Borel and Jacques Tits. Homomorphismes” abstraits” de groupes alge- briques simples.Annals of Mathematics, 97(3):499–571, 1973

work page 1973
[9]

Springer Science & Business Media, 2013

Martin R Bridson and Andr´ e Haefliger.Metric spaces of non-positive curvature, volume 319. Springer Science & Business Media, 2013

work page 2013
[10]

Finiteness properties of groups.Journal of Pure and Applied Algebra, 44(1-3):45–75, 1987

Kenneth S Brown. Finiteness properties of groups.Journal of Pure and Applied Algebra, 44(1-3):45–75, 1987

work page 1987
[11]

Finitely presented simple groups and prod- ucts of trees.Comptes Rendus de l’Acad´ emie des Sciences-Series I-Mathematics, 324(7):747–752, 1997

Marc Burger and Shahar Mozes. Finitely presented simple groups and prod- ucts of trees.Comptes Rendus de l’Acad´ emie des Sciences-Series I-Mathematics, 324(7):747–752, 1997

work page 1997
[12]

Groups acting on trees: from local to global structure.Publications Math´ ematiques de l’IH´ES, 92:113–150, 2000

Marc Burger and Shahar Mozes. Groups acting on trees: from local to global structure.Publications Math´ ematiques de l’IH´ES, 92:113–150, 2000

work page 2000
[13]

Lattices in product of trees.Publications Math´ ematiques de l’IH´ES, 92:151–194, 2000

Marc Burger and Shahar Mozes. Lattices in product of trees.Publications Math´ ematiques de l’IH´ES, 92:151–194, 2000

work page 2000
[14]

Finite and infinite quotients of discrete and indiscrete groups.Groups St Andrews 2017 in Birmingham, 455:16–69, 2019

Pierre-Emmanuel Caprace. Finite and infinite quotients of discrete and indiscrete groups.Groups St Andrews 2017 in Birmingham, 455:16–69, 2019

work page 2017
[15]

Simplicity and superrigidity of twin building lattices.Inventiones mathematicae, 176(1):169–221, 2009

Pierre-Emmanuel Caprace and Bertrand R´ emy. Simplicity and superrigidity of twin building lattices.Inventiones mathematicae, 176(1):169–221, 2009. 38

work page 2009
[16]

The cohomology of a coxeter group with group ring coefficients

Michael W Davis. The cohomology of a coxeter group with group ring coefficients. Duke Math. J., 95(1):297–314, 1998

work page 1998
[17]

Intersection theorems for systems op finite sets.Quart

Paul Erdos. Intersection theorems for systems op finite sets.Quart. J. Math. Oxford Ser.(2), 12:313–320, 1961

work page 1961
[18]

Springer Science & Business Media, 2001

Chris Godsil and Gordon F Royle.Algebraic graph theory, volume 207. Springer Science & Business Media, 2001

work page 2001
[19]

Hyperbolic groups

Mikhael Gromov. Hyperbolic groups. InEssays in group theory, pages 75–263. Springer, 1987

work page 1987
[20]

Commensurability and separability of quasiconvex subgroups

Fr´ ed´ eric Haglund. Commensurability and separability of quasiconvex subgroups. Algebraic & Geometric Topology, 6(2):949–1024, 2006

work page 2006
[21]

Finitely presented infinite simple groups.(Australian National University, Canberra, 1974, 1974

Graham Higman. Finitely presented infinite simple groups.(Australian National University, Canberra, 1974, 1974

work page 1974
[22]

Lattices in a product of trees, hierarchically hyperbolic groups and virtual torsion-freeness.Bulletin of the London Mathematical Society, 54(4):1413– 1419, 2022

Sam Hughes. Lattices in a product of trees, hierarchically hyperbolic groups and virtual torsion-freeness.Bulletin of the London Mathematical Society, 54(4):1413– 1419, 2022

work page 2022
[23]

On regular CAT(0) cube complexes and the simplicity of auto- morphism groups of rank-one CAT(0) cube complexes.Commentarii Mathematici Helvetici, 93(1):33–54, 2018

Nir Lazarovich. On regular CAT(0) cube complexes and the simplicity of auto- morphism groups of rank-one CAT(0) cube complexes.Commentarii Mathematici Helvetici, 93(1):33–54, 2018

work page 2018
[24]

Counting lattices in products of trees.arXiv preprint arXiv:2202.00378, 2022

Nir Lazarovich, Ivan Levcovitz, and Alex Margolis. Counting lattices in products of trees.arXiv preprint arXiv:2202.00378, 2022

work page arXiv 2022
[25]

The commensurator of a cocompact lattice in the automorphism group of a regular tree is not virtually simple

Adrien Le Boudec. The commensurator of a cocompact lattice in the automorphism group of a regular tree is not virtually simple. https://adrienleboudec.perso.math.cnrs.fr/comm-tree.pdf

work page
[26]

Finite common coverings of graphs.Journal of Com- binatorial Theory, Series B, 33(3):231–238, 1982

Frank Thomson Leighton. Finite common coverings of graphs.Journal of Com- binatorial Theory, Series B, 33(3):231–238, 1982

work page 1982
[27]

A finitely presented infinite simple group of homeomorphisms of the circle.Journal of the London Mathematical Society, 100(3):1034–1064, 2019

Yash Lodha. A finitely presented infinite simple group of homeomorphisms of the circle.Journal of the London Mathematical Society, 100(3):1034–1064, 2019

work page 2019
[28]

Superrigidity for the commensurability group of tree lattices.Commentarii Mathematici Helvetici, 69(1):523–548, 1994

Alexander Lubotzky, Shahar Mozes, and Robert J Zimmer. Superrigidity for the commensurability group of tree lattices.Commentarii Mathematici Helvetici, 69(1):523–548, 1994

work page 1994
[29]

A ˜C2-lattice that is not residually finite

Thomas Titz Mite and Stefan Witzel. A ˜C2-lattice that is not residually finite. arXiv preprint arXiv:2310.03662, 2023

work page arXiv 2023
[30]

Non-residually-finite ˜C2-lattices.arXiv preprint arXiv:2509.05054, 2025

Thomas Titz Mite and Stefan Witzel. Non-residually-finite ˜C2-lattices.arXiv preprint arXiv:2509.05054, 2025. 39

work page arXiv 2025
[31]

The geometry of cube complexes and the complexity of their fundamental groups.Topology, 37(3):621–633, 1998

Graham A Niblo and Lawrence D Reeves. The geometry of cube complexes and the complexity of their fundamental groups.Topology, 37(3):621–633, 1998

work page 1998
[32]

New simple lattices in products of trees and their projections

Nicolas Radu. New simple lattices in products of trees and their projections. Canadian Journal of Mathematics, 72(6):1624–1690, 2020

work page 2020
[33]

PhD thesis, ETH Zurich, 2004

Diego Attilio Rattaggi.Computations in groups acting on a product of trees: normal subgroup structures and quaternion lattices. PhD thesis, ETH Zurich, 2004

work page 2004
[34]

Constructing finitely presented simple groups that contain grig- orchuk groups.Journal of Algebra, 220(1):284–313, 1999

Claas E R¨ over. Constructing finitely presented simple groups that contain grig- orchuk groups.Journal of Algebra, 220(1):284–313, 1999

work page 1999
[35]

Ends of group pairs and non-positively curved cube complexes

Michah Sageev. Ends of group pairs and non-positively curved cube complexes. Proceedings of the London Mathematical Society, 3(3):585–617, 1995

work page 1995
[36]

A construction which can be used to produce finitely presented infinite simple groups.Journal of Algebra, 90(2):294–322, 1984

Elizabeth A Scott. A construction which can be used to produce finitely presented infinite simple groups.Journal of Algebra, 90(2):294–322, 1984

work page 1984
[37]

Groups of piecewise linear homeomorphisms.Transactions of the American Mathematical Society, 332(2):477–514, 1992

Melanie Stein. Groups of piecewise linear homeomorphisms.Transactions of the American Mathematical Society, 332(2):477–514, 1992

work page 1992
[38]

Complete square complexes.Commentarii Mathematici Helvetici, 82(4):683–724, 2007

Daniel T Wise. Complete square complexes.Commentarii Mathematici Helvetici, 82(4):683–724, 2007

work page 2007
[39]

Leighton’s theorem and regular cube complexes.Algebraic & Geometric Topology, 23(7):3395–3415, 2023

Daniel J Woodhouse. Leighton’s theorem and regular cube complexes.Algebraic & Geometric Topology, 23(7):3395–3415, 2023. 40

work page 2023