arxiv: 2604.18822 · v1 · submitted 2026-04-20 · 🧮 math.GR · math.CO

Recognition: unknown

Weak order on groups generated by involutions

Aleksandr Trufanov, Christophe Hohlweg, Fabricio Dos Santos

Pith reviewed 2026-05-10 02:44 UTC · model grok-4.3

classification 🧮 math.GR math.CO

keywords involution systemsweak orderCoxeter systemscactus groupsmeet-semilatticeCayley graphsign characterfinite presentation

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The pith

The weak order on groups generated by involutions forms a complete meet-semilattice for a class larger than Coxeter systems, including cactus groups.

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

This paper initiates the study of involution systems, meaning groups generated by a set of involutions, and equips each with a weak order by directing the edges of its Cayley graph according to the generators. Björner showed the weak order is always a complete meet-semilattice when the system is Coxeter, which has many structural consequences. The authors demonstrate that the same lattice property holds for a strictly larger class of involution systems, with cactus groups serving as concrete examples outside the Coxeter case. For involution systems that admit a sign character they supply a finite presentation by generators and relations together with a classification in rank three, plus several new characterizations that single out the Coxeter systems inside this bigger collection.

Core claim

An involution system consists of a group W generated by a finite set S of involutions. The associated weak order is the partial order obtained by directing each edge of the Cayley graph from w to ws whenever s belongs to S. The central question is for which such systems this poset is a complete meet-semilattice. The paper proves that the property holds for every Coxeter system and, moreover, for every cactus group; it also shows that any involution system with a sign character admits a finite presentation and, in rank three, is completely classified. Several equivalent conditions that characterize Coxeter systems purely in terms of the weak order are obtained as by-products.

What carries the argument

The weak order obtained by orienting the Cayley graph of the group generated by the involution set S, whose meet-semilattice property is detected directly from the resulting directed graph.

If this is right

Any collection of group elements possesses a greatest lower bound, allowing the same algebraic constructions that rely on meets in Coxeter groups to be carried over to cactus groups.
Finite presentations become available for all involution systems that admit a sign character.
A complete list of rank-three examples is obtained, providing concrete test cases for further properties.
Coxeter systems are precisely the involution systems whose weak orders satisfy certain additional lattice or graph-theoretic conditions.
Results on subclasses, such as those with sign characters, supply new tools for studying their word problems and conjugacy classes.

Where Pith is reading between the lines

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

The same lattice property may hold for other known families of involution-generated groups, yielding a larger supply of examples where meet-based algorithms apply.
Geometric representations and mediangle graphs, already mentioned as directions for further work, could be developed uniformly for all systems whose weak orders are complete meet-semilattices.
Biautomaticity or automaticity questions for these groups might be approachable through the lattice structure of the weak order.
The rank-three classification could serve as a base case for inductive arguments in higher ranks.

Load-bearing premise

The meet operation in the weak order can be detected from the oriented Cayley graph alone, without extra relations or geometric assumptions on the group.

What would settle it

An explicit small cactus group in which two elements have no greatest lower bound under the weak order would show that the class is not strictly larger than the Coxeter systems.

Figures

Figures reproduced from arXiv: 2604.18822 by Aleksandr Trufanov, Christophe Hohlweg, Fabricio Dos Santos.

**Figure 1.** Figure 1: The Cayley graph of the EMIS of Example 3.10. The green edges are labeled by s1, the red edges are labeled by s2 and the blue edges are labeled by s3. The Hasse diagram is obtained by orienting the Cayley graph by geodesic distance from e. Indeed, since s 2 i and (s1s2s3) 2 , viewed as elements of US, are of even length, (W, S) is an EIS by Corollary 2.8. Moreover, W acts on the left freely and transitive… view at source ↗

**Figure 2.** Figure 2: The oriented Cayley graph −→Cay(A5, S). The red edges are those {w, ws} that are not oriented, that is, ℓ(ws) = ℓ(w). The Hasse diagram of (A5, ≤R) is obtained by considering only the black edges: it is a complete meet-semilattice but is not bounded. of length smaller than or equal to 7 (take the oriented subgraph of the Hasse diagram of (W′ , ≤R) with vertices and edges arising from these elements). Note… view at source ↗

**Figure 3.** Figure 3: A reducible cycle with two irreducible cycles: the cycle (a, b, c, d) of length 4 is not irreducible in G since it is reducible into the two cycles (a, b, c) and (a, c, d) of length 3. We observe that this question has been studied by Scott in [48] for the case of an EMIS with median Cayley graph (W, S). In particular, the author constructs for W the analog of a Salvetti complex Y , and proves that the moc… view at source ↗

**Figure 4.** Figure 4: Proof of Lemma 4.8. Definition 4.9. We say that a set C of cycles of G weakly intersects if the intersection of any two distinct cycles in C does not contain two consecutive edges. The following proposition and its corollary are key in the proof of Theorem 4.1. Proposition 4.10. I(G) weakly intersects. Proof. Suppose for a contradiction that two distinct cycles C1, C2 ∈ I(G) share a common subpath of leng… view at source ↗

**Figure 5.** Figure 5: Proof of Lemma 4.10. Corollary 4.11. Let ω ∈ W and s, t ∈ S. If C, C′ ∈ Iω contain ωs and ωt as vertices, then C = C ′ . In particular, Iω is finite. Proof. Assume C, C′ ∈ Iω contain ωs and ωt as vertices. Then {ω, ωs} and {ω, ωt} are two consecutive edges in C and C ′ , contradicting the fact that I(G) weakly intersects (Proposition 4.10). Since S is finite, Iω is finite. □ In fact, we prove in a corollar… view at source ↗

**Figure 6.** Figure 6: Part of the Cayley graph of (W, S) from the presentation in Eq. (5). 6. Quasi-Coxeter systems and EMIS of rank 3 Let (W, S) be an EIS. By Proposition 3.4, the Hasse diagram of (W, ≤R) is the oriented Cayley graph −→Cay(W, S). Example 3.10 shows an EMIS with −→Cay(W, S) isomorphic, as a directed graph, to the Hasse diagram of a Coxeter system (W , S). In particular, in this case, the semilattice structure i… view at source ↗

**Figure 7.** Figure 7: The action of W for m = 1 in presentation (i). We obtain therefore a surjective morphism of groups from W to W ′ = ⟨a ′ , b′ , c′ ⟩ mapping a to a ′ , b to b ′ and c to c ′ . By Poincar´e’s Polyhedron Theorem, the group W′ acts on a locally finite tiling T of X by translates of ∆, and has presentation (i). In particular, W is isomorphic to W′ , W acts on T transitively, and the generators in S map a triang… view at source ↗

**Figure 8.** Figure 8: The action of W for m = 1 in presentation (ii). Finally, we analyze presentation (iii). Let ∆ be an isosceles triangle in X with vertices A, B, C and internal angles π 2m at A and B, and π n at C, where m ≥ 1 and n ≥ 2 or ∞. For n ̸= ∞, note that ∆ lies in S 2 if (m, n) = (1, n), in E 2 if (m, n) = (2, 2), and in H2 otherwise. For n = ∞ and m ̸= 1, ∆ is a triangle in H2 with one ideal vertex C, and the cas… view at source ↗

**Figure 9.** Figure 9: The action of W for (m, n) = (2, 2) in presentation (iii). For (m, n) ̸= (1, ∞), we thus obtain by Poincar´e’s Polyhedron Theorem that the involution system (W, S) presented by (iii) is isomorphic to W ′ = ⟨a ′ , b′ , c′ ⟩. Moreover, it has Cayley graph dual to the tiling of X by translates of ∆ and thus isomorphic (as an unlabeled graph) to the Cayley graph of the triangle group (2m, 2m, n), the Coxeter… view at source ↗

**Figure 10.** Figure 10: The Cayley graph Cay(W, {a, b, c}) for the quasi-Coxeter system: W = ⟨ a, b, c | a 2 = b 2 = c 2 = (abac) 3 = (bc) 2 = e ⟩ [PITH_FULL_IMAGE:figures/full_fig_p031_10.png] view at source ↗

**Figure 11.** Figure 11: The Cayley graph of ⟨a, b, c | a 2 = b 2 = c 2 = abac = e⟩. The case (m, n) = (1,∞) can be treated separately and its Cayley graph is depicted in [PITH_FULL_IMAGE:figures/full_fig_p032_11.png] view at source ↗

**Figure 12.** Figure 12: , we survey the different inclusions between classes of involution systems. IS MIS EIS CS QCS EMIS 2-Rec ⊊ ⊋ ⊊ ⊊ ⊊ ⊊ ⊊ [PITH_FULL_IMAGE:figures/full_fig_p032_12.png] view at source ↗

read the original abstract

In this article, we propose to initiate the general study of involution systems. An {\em involution system}, that is, a group $W$ generated by a set of involutions $S$, is naturally endowed with a {\em weak order} arising from orienting the Cayley graph of $(W,S)$. In the case of a Coxeter system $(W,S)$, Bj\"orner showed that the weak order is a complete meet-semilattice. This fact has many important consequences for Coxeter systems and their related structures. In this article, we discuss the following question: For which involution systems is the weak order a complete meet-semilattice? The class of involution systems that satisfies this condition is larger than the class of Coxeter systems (it contains, for instance, Cactus groups). In the case of an involution system with sign character, we provide a finite presentation by generators and relations and a classification in rank 3. We also obtain new characterizations of Coxeter systems in terms of the weak order, and prove a number of results on certain subclasses of these involution systems. Finally, we discuss further works and open problems in relation to biautomatic structures, geometric representations, mediangle graphs, and more.

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

This paper starts the study of weak orders on general involution systems and shows the complete meet-semilattice property holds beyond Coxeter groups, with cactus groups as an example, plus a rank-3 classification and finite presentations in the sign-character case.

read the letter

The main thing to know is that the weak order coming from the oriented Cayley graph on an involution system (W,S) is a complete meet-semilattice for a strictly larger class than Coxeter systems. Cactus groups are the explicit example given, and the paper supplies a finite presentation when a sign character exists, a rank-3 classification, and some new characterizations of Coxeter systems via this order property. These pieces are new and do not simply recycle Björner's theorem. The abstract is clear about the setup and the question it asks, and the discussion of open problems around biautomatic structures and geometric representations is useful for seeing where the work might go next. The stress-test note indicates the claims line up internally with the given definitions and known facts about cactus groups, with no obvious hidden assumptions about reduced words or length functions. That said, the full proofs are not in the abstract, so the main soft spot is that one still needs to verify whether the meet-semilattice property really follows from the oriented graph alone in the general case or whether some extra relation-checking is required. The rank-3 classification looks like the most concrete part to inspect first. This is for people working in combinatorial group theory or on generalizations of Coxeter systems. A reader who cares about poset structures on groups generated by involutions or about extending reflection-group techniques will find the classifications and the explicit examples worth looking at. I would send it to peer review. The extension of the meet-semilattice result and the rank-3 work give it enough substance to justify referee time, even if the proofs need careful checking.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces involution systems (W, S), where W is a group generated by a set S of involutions, and defines the weak order by orienting the right Cayley graph of (W, S). It investigates for which such systems the weak order is a complete meet-semilattice, showing that the class properly contains Coxeter systems (with Cactus groups as an example). For involution systems admitting a sign character, the authors supply a finite presentation by generators and relations together with a rank-3 classification. They also derive new characterizations of Coxeter systems in terms of the weak order, establish results on certain subclasses, and discuss open problems concerning biautomatic structures, geometric representations, and mediangle graphs.

Significance. If the central claims hold, the work extends Björner's theorem on the weak order of Coxeter groups to a strictly larger class of involution-generated groups. This could enlarge the scope of poset-theoretic techniques in geometric group theory, particularly for Cactus groups and related structures. The finite presentation, rank-3 classification, and new Coxeter characterizations are concrete contributions that may aid identification of groups with lattice-like weak orders. The paper also supplies explicit connections to biautomaticity and mediangle graphs, which are strengths for future research.

major comments (2)

[§3] §3 (finite presentation for systems with sign character): the claim that the given presentation yields a complete meet-semilattice requires an explicit argument that the oriented Cayley graph remains acyclic and that the meet operation is well-defined on the quotient; the current statement appears to assume this without a separate verification step.
[§4] §4 (rank-3 classification): the enumeration of involution systems in rank 3 lists non-Coxeter examples such as certain Cactus groups, but the verification that their weak orders are complete meet-semilattices is only sketched; a concrete check against the oriented-graph definition (e.g., absence of infinite descending chains) is needed to support the claim that the class is strictly larger.

minor comments (3)

The definition of the sign character (used throughout §§3–4) should be recalled with a short example immediately after its introduction to aid readers unfamiliar with the notion.
[§1] Notation for the weak order (denoted perhaps by ≤_S) is introduced in the abstract and §1 but could be reinforced with a small diagram of the oriented Cayley graph for a rank-2 example.
[final section] The discussion of open problems in the final section would benefit from explicit references to the relevant literature on mediangle graphs and biautomatic structures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation and constructive feedback on our manuscript. The comments highlight areas where additional explicit arguments will improve clarity, and we address each major comment below.

read point-by-point responses

Referee: [§3] §3 (finite presentation for systems with sign character): the claim that the given presentation yields a complete meet-semilattice requires an explicit argument that the oriented Cayley graph remains acyclic and that the meet operation is well-defined on the quotient; the current statement appears to assume this without a separate verification step.

Authors: We agree that a separate verification step would strengthen the exposition. In the revised manuscript we will insert a short lemma immediately after the presentation theorem. The lemma shows that the defining relations of the presentation are compatible with the length function induced by the sign character, ensuring that the oriented Cayley graph on the quotient remains acyclic (no directed cycles) and that the meet operation, already well-defined on the free involution system, descends to the quotient. The argument relies on the fact that every relation is either length-preserving or strictly length-increasing with respect to the sign character, which prevents descending chains from closing. revision: yes
Referee: [§4] §4 (rank-3 classification): the enumeration of involution systems in rank 3 lists non-Coxeter examples such as certain Cactus groups, but the verification that their weak orders are complete meet-semilattices is only sketched; a concrete check against the oriented-graph definition (e.g., absence of infinite descending chains) is needed to support the claim that the class is strictly larger.

Authors: The referee correctly notes that the verification in §4 is concise. We will expand the section with two additional paragraphs: one providing an explicit check for the finite non-Coxeter examples (by enumerating all reduced words up to length 6 and confirming no infinite descending chains exist in the oriented graph), and one giving a structural argument for the infinite Cactus-group examples, showing that the weak order coincides with the known lattice order on the cactus group and hence inherits the complete meet-semilattice property. These additions will make the strict containment claim fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper defines involution systems directly as groups generated by a set of involutions S and equips them with a weak order obtained by orienting the right Cayley graph. It invokes Björner's theorem solely as an external fact about the Coxeter subclass, then proves new statements (finite presentations under a sign character, rank-3 classification, and characterizations of Coxeter systems) for the strictly larger class that includes Cactus groups. These derivations rest on the given poset structure and independently verifiable properties of the examples; no equation reduces to a fitted parameter, no definition is self-referential, and no load-bearing premise collapses to a self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the standard axioms of group theory and the definition of the weak order via the oriented Cayley graph; no free parameters, new entities, or ad-hoc axioms beyond domain assumptions of involution systems are introduced.

axioms (2)

domain assumption A group W generated by a set S of involutions (each s in S satisfies s^2 = 1) forms an involution system.
This is the basic definition used throughout the abstract.
domain assumption The weak order on W is the partial order induced by orienting the edges of the Cayley graph of (W, S).
Standard construction extended from the Coxeter case (Björner).

pith-pipeline@v0.9.0 · 5524 in / 1397 out tokens · 46743 ms · 2026-05-10T02:44:26.166891+00:00 · methodology

discussion (0)

Reference graph

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