Divergence, thickness and hypergraph index for general Coxeter groups
Pith reviewed 2026-05-24 10:47 UTC · model grok-4.3
The pith
Coxeter groups with finite hypergraph index h are thick of order at most h with divergence bounded above by a polynomial of degree h+1
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For a general Coxeter system (W,S), the hypergraph index is a computable combinatorial number h. If h is finite then W is strongly algebraically thick of order at most h, hence its divergence is bounded above by a polynomial of degree h+1. The paper proves this implication for all Coxeter systems and establishes equality in certain families obtained by a new construction that starts from any right-angled Coxeter group and produces infinitely many non-right-angled systems with the same hypergraph index. An upper bound on the index itself is given in terms of the topology of the associated Dynkin diagram.
What carries the argument
The hypergraph index of a Coxeter system (W,S), a combinatorial invariant that generalizes Levcovitz's right-angled definition and directly controls the order of algebraic thickness.
If this is right
- Linear divergence is fully characterized for general Coxeter groups.
- Any superlinear divergence must be at least quadratic.
- The order of thickness is at most the hypergraph index h whenever the index is finite.
- Divergence is bounded above by a polynomial of degree h+1.
- The hypergraph index of any Coxeter system is bounded above by a quantity determined by the topology of its Dynkin diagram.
Where Pith is reading between the lines
- The construction that preserves hypergraph index while moving from right-angled to non-right-angled systems could be iterated to produce dense families inside the space of all Coxeter systems.
- If the conjecture that thickness order equals hypergraph index holds in full generality, then divergence degree would become a computable invariant for all Coxeter groups.
- The link between hypergraph index and Dynkin-diagram topology raises the possibility that divergence bounds can be read off from representation-theoretic data associated to the diagram.
Load-bearing premise
The new combinatorial definition of hypergraph index for non-right-angled Coxeter systems correctly measures the same geometric thickness and divergence properties previously established only in the right-angled case.
What would settle it
A single Coxeter system whose hypergraph index equals some finite h but whose thickness order exceeds h, or whose divergence function grows faster than any polynomial of degree h+1, would falsify the central implication.
read the original abstract
We study divergence and thickness for general Coxeter groups $W$. We first characterise linear divergence, and show that if $W$ has superlinear divergence then its divergence is at least quadratic. We then formulate a computable combinatorial invariant, hypergraph index, for arbitrary Coxeter systems $(W,S)$. This generalises Levcovitz's definition for the right-angled case. We prove that if $(W,S)$ has finite hypergraph index $h$, then $W$ is (strongly algebraically) thick of order at most $h$, hence has divergence bounded above by a polynomial of degree $h+1$. We conjecture that these upper bounds on the order of thickness and divergence are in fact equalities, and we prove our conjecture for certain families of Coxeter groups. These families are obtained by a new construction which, given any right-angled Coxeter group, produces infinitely many examples of non-right-angled Coxeter systems with the same hypergraph index. Finally, we give an upper bound on the hypergraph index of any Coxeter system $(W,S)$, and hence on the divergence of $W$, in terms of, unexpectedly, the topology of its associated Dynkin diagram.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper characterizes linear divergence in Coxeter groups and proves that superlinear divergence is at least quadratic. It introduces the hypergraph index, a computable combinatorial invariant for general Coxeter systems that generalizes Levcovitz's right-angled definition. The main theorem states that finite hypergraph index h implies the group is strongly algebraically thick of order at most h, hence has divergence bounded above by a polynomial of degree h+1. The authors conjecture that the bounds are sharp, prove the conjecture for families obtained via a new construction from right-angled groups, and give an upper bound on the hypergraph index (and thus divergence) in terms of the topology of the associated Dynkin diagram.
Significance. If the central implication holds, the hypergraph index supplies an explicit combinatorial upper bound on thickness and divergence for arbitrary Coxeter groups, extending prior right-angled results to a much larger class. The new construction producing infinitely many non-right-angled examples with prescribed index and the unexpected Dynkin-diagram bound are concrete advances that could be used for explicit computations in low-rank cases.
major comments (2)
- [§3] §3 (definition of hypergraph index): the generalization from the right-angled case must be shown to coincide exactly with Levcovitz's definition on right-angled systems; without an explicit reduction or example computation, it is unclear whether the thickness implication in the main theorem carries over without additional hypotheses.
- [Theorem 5.3] Theorem 5.3 (thickness bound): the argument that finite hypergraph index h yields strong algebraic thickness of order ≤ h relies on the hypergraph index controlling the existence of thick subsets; a load-bearing step appears to be the inductive construction of the required subsets, which should be checked against the precise definition of algebraic thickness used in the paper.
minor comments (2)
- [§7] The statement that the Dynkin-diagram bound is 'unexpected' should be supported by a brief comparison with existing bounds in the literature on Coxeter divergence.
- [§3] Notation for the hypergraph index (e.g., the precise tuple or multiset used to record the index) is introduced without a dedicated notation paragraph; a short table summarizing the definition for small examples would improve readability.
Simulated Author's Rebuttal
We thank the referee for their careful reading and for identifying points that require explicit verification. We address each major comment below and will incorporate the necessary clarifications and checks into a revised manuscript.
read point-by-point responses
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Referee: [§3] §3 (definition of hypergraph index): the generalization from the right-angled case must be shown to coincide exactly with Levcovitz's definition on right-angled systems; without an explicit reduction or example computation, it is unclear whether the thickness implication in the main theorem carries over without additional hypotheses.
Authors: We agree that an explicit verification is required for clarity. In the revised version we will insert a new proposition (Proposition 3.4) immediately after the definition of hypergraph index. The proposition states that when (W,S) is right-angled, the hypergraph H(W,S) constructed in §3 is identical to the hypergraph used by Levcovitz, obtained by taking the same vertex set S and the same collection of edges corresponding to the non-commuting pairs. The proof is a direct comparison of the two constructions; no additional hypotheses are needed. This will confirm that the thickness implication of the main theorem applies verbatim to the right-angled case. revision: yes
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Referee: [Theorem 5.3] Theorem 5.3 (thickness bound): the argument that finite hypergraph index h yields strong algebraic thickness of order ≤ h relies on the hypergraph index controlling the existence of thick subsets; a load-bearing step appears to be the inductive construction of the required subsets, which should be checked against the precise definition of algebraic thickness used in the paper.
Authors: The inductive construction in the proof of Theorem 5.3 is already written to satisfy the axioms of strong algebraic thickness as defined in the paper (Definition 2.7, which follows the standard formulation of Behrstock–Hagen–Sisto). Nevertheless, to address the referee’s concern we will add a short verification subsection (5.3.1) that walks through each inductive step and explicitly records which clause of Definition 2.7 is satisfied by the subsets produced at that stage. This will make the load-bearing correspondence fully transparent without altering the argument. revision: yes
Circularity Check
No circularity: new combinatorial definition implies geometric bounds via independent proof
full rationale
The paper defines hypergraph index as a fresh computable combinatorial invariant for general Coxeter systems, explicitly generalizing (but not depending on) Levcovitz's right-angled version. It then proves the implication 'finite h implies algebraic thickness order ≤ h and divergence ≤ polynomial of degree h+1' without any equation or step that reduces the claimed bound back to the definition by construction. No fitted parameters are renamed as predictions, no uniqueness theorems are imported from the same authors, and no ansatz is smuggled via self-citation. The conjecture of equality and the Dynkin-diagram upper bound are separate statements. The derivation chain is therefore self-contained and does not collapse to its inputs.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard properties of Coxeter groups, their Davis complexes, and algebraic thickness as developed in prior geometric group theory literature.
invented entities (1)
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hypergraph index
no independent evidence
discussion (0)
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