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arxiv: 2605.11126 · v1 · submitted 2026-05-11 · 🧮 math.GR

Connections between the topology of the Morse boundary, the Morse local-to-global property and acylindrical hyperbolicity

Carolyn Abbott , Stefanie Zbinden This is my paper

Pith reviewed 2026-05-13 01:10 UTC · model grok-4.3

classification 🧮 math.GR
keywords Morse boundaryMorse local-to-globalacylindrical hyperbolicitysmall cancellationgeometric group theoryσ-compactgeodesicsfinitely generated groups
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The pith

A group has a σ-compact Morse boundary if and only if it is Morse local-to-global.

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

The paper proves that for finitely generated groups, the Morse boundary being σ-compact is equivalent to the group satisfying the Morse local-to-global property. This link connects the topology of the boundary directly to the group's ability to extend local Morse geodesics globally. The authors achieve this by generalizing intersection-of-relators techniques from small cancellation theory so they apply beyond hyperbolic groups. They also define a new class of groups where a geodesic counts as Morse exactly when its overlaps with relators grow sublinearly in relator length.

Core claim

A finitely generated group has σ-compact Morse boundary precisely when it is Morse local-to-global. The proof uses generalized small-cancellation arguments that work for arbitrary finitely generated groups. As a consequence, the authors produce the first example of a non-virtually-cyclic Morse local-to-global group that contains an infinite-order Morse element yet fails to be acylindrically hyperbolic.

What carries the argument

The Morse local-to-global property, which converts local control on Morse geodesics into global control and thereby forces the Morse boundary to be σ-compact.

If this is right

  • Any group whose Morse boundary is σ-compact must admit global extension of its local Morse geodesics.
  • In the new class of groups resembling graded small-cancellation groups, a geodesic is Morse exactly when its intersection with relators is sublinear in relator length.
  • The geodesic Morse local-to-global property supplies a practical test for deciding whether a given group is Morse local-to-global.
  • The constructed example separates the Morse local-to-global property from acylindrical hyperbolicity for groups that are not virtually cyclic.

Where Pith is reading between the lines

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

  • The same generalized small-cancellation techniques might apply to other boundaries or to quasi-geodesics.
  • Algorithmic questions about the Morse boundary, such as decidability of membership, could become tractable once the local-to-global property holds.
  • Further examples in the new class may separate additional geometric properties while preserving the Morse local-to-global feature.

Load-bearing premise

The usual definitions of the Morse boundary and the Morse local-to-global property apply to the finitely generated groups under study, so that the equivalence follows from the generalized small-cancellation tools.

What would settle it

A single finitely generated group that has σ-compact Morse boundary yet fails the Morse local-to-global property (or the converse) would disprove the claimed equivalence.

Figures

Figures reproduced from arXiv: 2605.11126 by Carolyn Abbott, Stefanie Zbinden.

Figure 1
Figure 1. Figure 1: Diagram of implications. The black solid arrows are implications already known [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: a projective loop. Right: A short subsegment [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The different cases represent different subsegments being long. [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The point ζpt 1 q is the point on ζ closest to ηpsq in Case 3b. Since p is a 3K–quasi￾geodesic, p, and specifically ζpt 2 q, is in the Mp3Kq–neighborhood of ηr0, ss [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: In this picture, t 1 1 ą t 1 2 and hence t 1 “ t 1 2 . If t 1 2 ą t 1 1 we instead have t 1 “ t 1 1 . 21 [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: If any quadrangles satisfies the properties written in black, then the properties [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Setup of Lemma 3.18. Finally, let L 1 ě |α| ` 101δ ` L ` K, where K “ 2pdiampβq ` |β| ` diampγq ` |γ|q and |β| denotes the domain length of β. Let g P G be such that α 1 “ rη1pL 1 q, g ¨ η2pL 1 qs is a translate of α. Such a g exists because G acts transitively on the vertices of X “ CaypG, Sq. Let η 1 1 “ η1 and η 1 2 “ g ¨ η2. This is depicted in [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The path η is a 4C2–quasi-geodesic. Since ηpsq lies on ηi´2, we have that |s´s0| ď |ηi´2| ď Li´2{2 ď Li´1{2. This is depicted in [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Proof of Claim 3.24. The paths λ1 and λ2 are p3Kq– and K1–quasi-geodeiscs respectively. to M1 p3K1 q, we have that λ2 and λrT 1 , Ts are contained in the NpKq–neighbourhood of ηrt, t2s. It remains to show that λr0, ss is contained in the NpKq–neighbourhood of ηrt1, t2s. In fact, it suffices to show that dpβi´1, λpsqq ď M1 p3Kq, as then λr0, ss is contained in the NpKq neighbourhood of ηrt1, t2s by inductio… view at source ↗
Figure 10
Figure 10. Figure 10: The contiguity diagram Γγ. (2) Let γ ˚ α1 ˚ β ˚ α2 be a loop in Gi such that γ is i–good and α1, α2 and β are geodesics in Gi´1, where G´1 is defined to be G0. Then either |γ| ď 5p|α1| ` |α2|q or there exists a subgeodesic γ 1 Ă γ X β of length |γ 1 | ě |γ| ´ 5 p|α1| ` |α2|q. Proof. We will prove both statements simultaneously by induction on i. For i “ 0, the statements follow because G0 is a free group.… view at source ↗
Figure 11
Figure 11. Figure 11: A short path p “ p1 ˚ p2 ˚ p3 from x to β. By Lemma 4.3, the contiguity degree of any contiguity diagram is at most 1{2`2εi{|BΠ|, and by Lemma 4.4, the sum of all contiguity degrees is at least 1 ´ 23µ. Together with Claim 4.10, this implies pΠ, Γα1 , α1q ` pΠ, Γα2 , α2q ě 1 2 ´ 1 10 ´ 23µ ´ 2 εi |BΠ| ě 1 4 ` 6 εi |BΠ| . (4.13) Here we used that µ ď 1{230 by Definition 4.5 and εi ď |BΠ|{1000 by (P5). Clai… view at source ↗
Figure 12
Figure 12. Figure 12: A short path p “ α 1 1 ˚ p1 ˚ q ˚ p2 ˚ α 1 2 from γ ` to γ ´. Finally, we prove (1) for i. Let γ be i-good, and let β be a geodesic in Gi from γ ` to γ ´. Applying (2) for i to the loop γ ˚ β yields that γ is a subsegment of β. Since β is a geodesic, this implies γ “ β and concludes the proof. We next turn our attention to contiguity diagrams in a group with an expanding graded small cancellation presenta… view at source ↗
Figure 13
Figure 13. Figure 13: The εj–continguity diagram Γk. by the triangle inequality, these points have to lie on β. Thus, in any case, we can write β “ β1 ˚β 1˚β2, where for ℓ “ 1, 2, the endpoints of βℓ and αℓ have distance at most 4εm. See [PITH_FULL_IMAGE:figures/full_fig_p033_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The embedded relator γ subdivided into γ1 and γ2 is isometrically embedded. Let Π be any cell of ∆. Since Greendlinger cells have maximal rank, the rank of Π is at most the rank of Π1 . If Π has the same rank as Π1 , then by assumption |Π| ď C|Π1 |. If the rank of Π is less than the rank of Π1 , then |Π| ď |Π1 | by (P2). In either case, the statement follows from (4.16). We next show that images of relato… view at source ↗
Figure 15
Figure 15. Figure 15: One possible configuration of subdiagrams ∆ [PITH_FULL_IMAGE:figures/full_fig_p037_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The different possibilities for the diagram [PITH_FULL_IMAGE:figures/full_fig_p039_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The different possibilities for the diagram [PITH_FULL_IMAGE:figures/full_fig_p039_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Definition of the path p in Claim 4.22. The dotted lines are there to suggest that BΠ1 X γ and BΠ1 X γ do not necessarily need to be contiguous. If BΠ1 does not intersect β 1 1 , the statement follows from γ being a geodesic, which implies |BΠ1 X γ| ď |BΠ1 |{2. If BΠ1 does intersect β 1 1 , let z P β 1 1 be the point on BΠ1 furthest away from p1, let Lβ “ |β 1 1 X BΠ1 | and let Lγ “ |γ X BΠ1 |. There is a… view at source ↗
read the original abstract

We relate the topology of the Morse boundary of a group to geometric and algorithmic properties of the group. In particular, we show that a group has $\sigma$-compact Morse boundary if and only if it is Morse local-to-global. We also provide tools such as the geodesic Morse local-to-global property to show that groups are (not) Morse local-to-global. Our strategy generalizes tools from small cancellation theory, such as the intersection of relators, to arbitrary finitely generated groups. Further, we introduce a class of groups akin to graded small-cancellation groups and show that, for groups in this class, a geodesic is Morse if and only if its intersection with relators grows sublinearly in the length of the relators. We use this to construct the first example of a non-virtually cyclic Morse local-to-global group with an infinite-order Morse element that is not acylindrically hyperbolic.

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

2 major / 2 minor

Summary. The manuscript proves that a finitely generated group has σ-compact Morse boundary if and only if it satisfies the Morse local-to-global (MLTG) property. It generalizes intersection-of-relators estimates from small cancellation theory to arbitrary finitely generated groups, introduces a class of groups analogous to graded small cancellation groups in which a geodesic is Morse precisely when its intersections with relators grow sublinearly, and constructs the first non-virtually cyclic MLTG group possessing an infinite-order Morse element that is not acylindrically hyperbolic.

Significance. If the central equivalence and the supporting generalization hold, the work supplies a topological characterization of the MLTG property and new criteria for verifying it, while furnishing a concrete example that separates MLTG from acylindrical hyperbolicity. The extension of small cancellation techniques beyond classical presentations, together with the explicit construction, would constitute a substantive advance in the study of Morse boundaries and quasi-isometry invariants.

major comments (2)
  1. [§3] §3: The claimed equivalence and the subsequent example rest on a generalization of the 'intersection of relators' condition to arbitrary finitely generated groups. The manuscript must supply an explicit, presentation-independent definition of this intersection growth (or of the 'generalized relators') that is invariant under change of finite generating set; without it the predicate is not known to be a quasi-isometry invariant, undermining both the iff statement and the construction of the non-AH MLTG example.
  2. [Abstract, §4] Abstract and §4 (construction): The example group is asserted to be MLTG, to possess an infinite-order Morse element, and to fail acylindrical hyperbolicity. The verification that this group lies in the newly introduced class and satisfies the sublinear-intersection characterization must be checked against the same presentation-independent definition; any dependence on a specific presentation would render the separation from AH non-canonical.
minor comments (2)
  1. [§3] Notation for the generalized intersection function should be introduced with a displayed definition and a short paragraph confirming its independence from the choice of generators.
  2. [§2] The statement of the geodesic MLTG property (used as a tool) would benefit from an explicit comparison with the standard MLTG property, including a short proof or reference that the two coincide under the paper's hypotheses.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comments point by point below, outlining the revisions we will undertake to ensure all definitions are presentation-independent and the results are robust.

read point-by-point responses

  1. Referee: [§3] §3: The claimed equivalence and the subsequent example rest on a generalization of the 'intersection of relators' condition to arbitrary finitely generated groups. The manuscript must supply an explicit, presentation-independent definition of this intersection growth (or of the 'generalized relators') that is invariant under change of finite generating set; without it the predicate is not known to be a quasi-isometry invariant, undermining both the iff statement and the construction of the non-AH MLTG example.

    Authors: We agree that an explicit, presentation-independent formulation is essential for the equivalence to be a quasi-isometry invariant. In §3 the intersection growth is defined via the word metric on the Cayley graph with respect to a fixed finite generating set S. We will add a new lemma (Lemma 3.12) establishing that sublinear intersection growth with respect to S implies the same (up to a multiplicative constant adjustment to the sublinear function) with respect to any other finite generating set T. The proof relies on the bi-Lipschitz equivalence of the word metrics d_S and d_T. With this lemma in place, the generalized relator condition, the Morse local-to-global property, and the equivalence with σ-compactness of the Morse boundary are all well-defined and invariant. We will also revise the statement of Theorem 3.1 and the surrounding discussion to reference this invariance explicitly. revision: yes

  2. Referee: [Abstract, §4] Abstract and §4 (construction): The example group is asserted to be MLTG, to possess an infinite-order Morse element, and to fail acylindrical hyperbolicity. The verification that this group lies in the newly introduced class and satisfies the sublinear-intersection characterization must be checked against the same presentation-independent definition; any dependence on a specific presentation would render the separation from AH non-canonical.

    Authors: We accept that the verification for the example in §4 must be performed with the invariant definition. The group is constructed from an explicit (infinite) presentation, but we will add a new subsection (4.3) that first invokes the invariance lemma from §3 and then verifies sublinear intersection growth directly with respect to the standard generating set, showing that the same bounds hold (up to constants) for any finite generating set. This confirms that the group belongs to the generalized class, satisfies the Morse local-to-global property, contains an infinite-order Morse element, and is not acylindrically hyperbolic, all in a presentation-independent manner. We will also update the abstract to reflect that the separation is canonical. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalence derived from independent standard definitions and new generalization

full rationale

The claimed iff between σ-compact Morse boundary and Morse local-to-global property rests on standard quasi-isometry-invariant definitions of both notions plus a generalization of small-cancellation intersection estimates to arbitrary finitely generated groups. No step reduces the target equivalence to a fitted parameter, self-definition, or self-citation chain; the generalization is introduced as novel content rather than presupposed. The example construction follows from the new tools without circular reduction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review prevents exhaustive identification of all free parameters or axioms; relies on standard domain definitions of Morse boundary and geodesics.

axioms (1)
  • domain assumption Standard definitions of Morse boundary, Morse geodesics, and acylindrical hyperbolicity from geometric group theory
    Paper invokes these as background for the equivalence and example construction.
invented entities (1)
  • Class of groups akin to graded small-cancellation groups no independent evidence
    purpose: To prove that a geodesic is Morse iff its intersection with relators grows sublinearly
    New class introduced to generalize small cancellation tools to arbitrary finitely generated groups.

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