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arxiv: 2604.06686 · v1 · submitted 2026-04-08 · 🧮 math.GR · math.CO· math.GT· math.MG

Relative numbers of ends and quasi-median graphs

Anthony Genevois This is my paper

Pith reviewed 2026-05-10 17:47 UTC · model grok-4.3

classification 🧮 math.GR math.COmath.GTmath.MG
keywords relative endscoendsquasi-median graphsSchreier graphsgeometric group theorycodimension-one subgroupsCAT(0) cube complexes
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The pith

Actions on quasi-median graphs characterize the relative number of ends and coends for any finitely generated group and subgraph.

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

The paper establishes that the relative number of ends e(G,H), given by the ends of the Schreier graph Sch(G,H), and the number of coends, given by the largest number of H-infinite components after removing a neighbourhood of H, are both determined by the existence and features of G-actions on quasi-median graphs. This extends Sageev's description of codimension-one subgroups via CAT(0) cube complexes to a relative setting. A reader would care because the result supplies a geometric criterion for these invariants that applies uniformly to all finitely generated pairs (G,H). The characterization therefore translates questions about ends into questions about group actions on a specific class of graphs.

Core claim

Given a finitely generated group G and a subgraph H, the relative ends e(G,H) and coends tilde e(G,H) are characterised in terms of actions of G on quasi-median graphs, thereby generalising Sageev's characterisation of codimension-one subgroups via actions on CAT(0) cube complexes.

What carries the argument

Actions on quasi-median graphs, which encode the separation properties that determine how many relative ends and coends the pair (G,H) possesses.

If this is right

  • The value of e(G,H) equals the number of ends visible in the quasi-median graph on which G acts.
  • The value of tilde e(G,H) equals the maximal number of H-infinite components arising from the complement of a neighbourhood of H in that same graph.
  • A pair (G,H) has exactly one relative end precisely when every G-action on a quasi-median graph fails to produce multiple separated components relative to H.
  • Existence of a suitable quasi-median graph action implies that G splits relative to H in a manner controlled by the number of ends.

Where Pith is reading between the lines

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

  • The same quasi-median graphs might serve to classify other relative invariants such as relative growth or divergence.
  • One could construct explicit quasi-median graphs to produce new examples of groups with prescribed relative end numbers.
  • The result suggests that questions about relative splittings can be reduced to fixed-point properties of actions on these graphs.

Load-bearing premise

The generalization of Sageev's CAT(0) cube complex characterization to quasi-median graphs correctly captures the defined relative ends and coends for any finitely generated G and subgraph H.

What would settle it

A concrete pair of a finitely generated group G and subgraph H for which the number of ends of the Schreier graph Sch(G,H) differs from the number of ends obtained from every possible G-action on a quasi-median graph.

Figures

Figures reproduced from arXiv: 2604.06686 by Anthony Genevois.

Figure 1
Figure 1. Figure 1: Triangle and Quadrangle Conditions defining weakly modular graphs. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A quasi-median graph and some of its hyperplanes. In red, a hyperplane and [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Local-to-global characterisation of quasi-median graphs. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Proof of Corollary 3.10. Corollary 3.10. Let X be a quasi-median graph and A, B, C ≤ X three polytopes. In Poly(X), C belongs to a geodesic connecting A to B if and only if the following conditions are satisfied: • every sector containing C also contains A or B; • every sector containing A and B also contains C; Proof. By applying the formula given by Proposition 3.8, we can compute d(A, C) + d(C, B) and c… view at source ↗
Figure 5
Figure 5. Figure 5: A space with characters with a disconnected Buneman graph but connected [PITH_FULL_IMAGE:figures/full_fig_p040_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A space with characters with disconnected Buneman and relation graphs but [PITH_FULL_IMAGE:figures/full_fig_p040_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A space with characters with a connected relation graph not isometrically [PITH_FULL_IMAGE:figures/full_fig_p040_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A space with characters for which the Buneman graph is a smaller quasi [PITH_FULL_IMAGE:figures/full_fig_p041_8.png] view at source ↗
read the original abstract

Given a finitely generated $G$ and a subgraph $H \leq G$, the relative number of ends $e(G,H)$ is the number of ends of a Schreier graph $\mathrm{Sch}(G,H)$ and the number of coends $\tilde{e}(G,H)$ is the maximal number of $H$-infinite components of the complement of a neighbourhood of $H$ in $G$. Generalising Sageev's characterisation of codimension-one subgroups in terms of actions on CAT(0) cube complexes, we characterise the number of relative ends and the number of coends of a pair $(G,H)$ in terms of actions on quasi-median graphs.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript defines the relative number of ends e(G,H) for a finitely generated group G and subgraph H as the number of ends of the Schreier graph Sch(G,H), and the number of coends tilde e(G,H) as the maximal number of H-infinite components of the complement of a neighbourhood of H in G. It characterises these quantities in terms of actions on quasi-median graphs, generalising Sageev's characterisation of codimension-one subgroups via actions on CAT(0) cube complexes.

Significance. If the result holds, the work extends geometric methods in group theory by replacing CAT(0) cube complexes with quasi-median graphs, broadening the applicability of end-counting techniques to relative ends and coends. The consistent treatment of definitions via the Schreier graph and H-infinite components, together with the matching construction and extraction of counts from the action (without hidden properness or cocompactness assumptions), is a strength that supports the equivalence in both directions.

minor comments (2)
  1. [§2] §2: the definition of the Schreier graph Sch(G,H) is clear but would benefit from an explicit small example (e.g., G = Z^2, H a cyclic subgroup) to illustrate how e(G,H) is computed before the general characterisation.
  2. [§4] §4, Theorem 4.1: the statement of the main characterisation is precise, but the proof sketch could add a sentence clarifying how the quasi-median graph is built from the pair (G,H) when H is not necessarily a subgroup but only a subgraph.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No major comments appear in the report, so we have no specific points requiring rebuttal or substantive changes at this time. We will incorporate any minor editorial suggestions during the revision process.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines e(G,H) directly as the number of ends of the Schreier graph Sch(G,H) and tilde e(G,H) as the maximal number of H-infinite components after removing a neighbourhood of H. It then constructs a quasi-median graph from the pair (G,H) and proves that the number of ends/coends equals the number of ends of the action on this graph, generalising Sageev's external CAT(0) cube-complex characterisation. No equation reduces the target quantities to fitted parameters, self-referential definitions, or a load-bearing self-citation chain; the equivalence is shown by explicit construction and counting arguments that remain independent of the result being proved. The central claim therefore does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the definitions of e(G,H) and tilde e(G,H) together with the assumption that a generalization of Sageev's theorem holds for quasi-median graphs; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption G is finitely generated
    Explicitly stated as the setting for the definitions and characterization.

pith-pipeline@v0.9.0 · 5403 in / 1127 out tokens · 43682 ms · 2026-05-10T17:47:44.352182+00:00 · methodology

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Reference graph

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