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Groups with a navigable path system satisfy the weak rank rigidity conjecture: they either have linear divergence or a Morse element.

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load-bearing objection This paper unifies weak rank rigidity for groups with navigable path systems via a new generalised contraction space.

arxiv 2605.29837 v1 pith:HT64DK7N submitted 2026-05-28 math.GR math.MG

Weak rank rigidity for groups with a navigable path system

Cornelia Drutu , Davide Spriano , Stefanie Zbinden This is my paper
classification math.GR math.MG
keywords geometric group theorynon-positive curvaturerank rigidityMorse elementdivergenceHelly groupsCoxeter groupshierarchically hyperbolic groups
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proves that groups equipped with a navigable path system, a mild form of non-positive curvature, fulfill the weak rank rigidity conjecture by either displaying linear divergence or containing a Morse element. This covers a range of examples including discrete groups of projective automorphisms of open convex cones, Helly groups, Coxeter groups, weak Garside groups, and hierarchically hyperbolic groups. The result comes with a unified proof of the Morse local-to-global property for the whole class. A sympathetic reader would care because the condition links curvature properties directly to either linear divergence or the existence of Morse elements across many group families. The work introduces the generalised contraction space as the main new tool to encode negative curvature and support the arguments.

Core claim

Groups with a navigable path system satisfy the weak rank rigidity conjecture: they either have linear divergence or a Morse element. This class includes discrete groups of projective automorphisms of open convex cones, Helly groups, Coxeter groups, weak Garside groups, hierarchically hyperbolic groups, and other examples. Along the way the groups satisfy the Morse local-to-global property, with the generalised contraction space serving as the main new tool that encodes the negative curvature of a given space. In the metric setting the same condition yields a local definition of rank one or Morse geodesics, allows deduction of linearity of divergence from linearity on a sequence, and produce

What carries the argument

The navigable path system, which acts as the structural input enabling the unified proof of weak rank rigidity and the Morse local-to-global property across the listed classes.

Load-bearing premise

The groups under consideration admit a navigable path system as the mild form of non-positive curvature.

What would settle it

A concrete counterexample would be any group that admits a navigable path system yet exhibits neither linear divergence nor a Morse element.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The listed groups, including Coxeter groups and hierarchically hyperbolic groups, satisfy weak rank rigidity.
  • The same groups satisfy the Morse local-to-global property.
  • A local definition of rank one or Morse geodesics becomes available in the metric setting.
  • Linearity of divergence follows from linearity on a sequence.
  • Morse geodesics are strongly contracting in additional cases.

Where Pith is reading between the lines

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

  • The navigable path system condition could apply to further group families beyond those explicitly listed.
  • The generalised contraction space may serve as a tool for studying other rigidity phenomena in spaces with comparable curvature.
  • The approach might extend to questions about stronger forms of rank rigidity under similar curvature assumptions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that groups admitting a navigable path system (a mild non-positive curvature condition) satisfy the weak rank rigidity conjecture: they either have linear divergence or a Morse element. The result applies to discrete groups of projective automorphisms of open convex cones, Helly groups, Coxeter groups, weak Garside groups, hierarchically hyperbolic groups, and other examples. As a byproduct, these groups satisfy the Morse local-to-global property. The main new tool is the generalised contraction space, a hyperbolic space encoding negative curvature; the work also gives a local definition of rank-one/Morse geodesics and new cases of strong contraction.

Significance. If the central claims hold, the paper unifies proofs of weak rank rigidity and the Morse local-to-global property across a broad collection of groups with non-positive curvature features, answering an open question of Genevois for Helly groups. The generalised contraction space supplies a new metric tool that yields local characterisations of Morse geodesics (mirroring Jacobi-field criteria) and allows deduction of global divergence linearity from local data. These contributions strengthen the toolkit for studying rank rigidity and contraction properties in geometric group theory.

minor comments (3)
  1. [Introduction] The precise axioms of a navigable path system are introduced in the body but would benefit from an early, self-contained statement in the introduction to improve accessibility.
  2. Notation for the generalised contraction space (e.g., the underlying metric and projection maps) is used before its full definition; a forward reference or brief preview would clarify the exposition.
  3. The list of covered classes in the abstract and introduction could include a short table or explicit citations to the sections where each class is verified to admit a navigable path system.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, the assessment of its significance, and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces the navigable path system as an independent structural assumption (a mild non-positive curvature condition) and constructs the generalised contraction space as a new hyperbolic space encoding negative curvature. The weak rank rigidity result and Morse local-to-global property are derived from this input via metric geometry arguments that apply uniformly to listed classes (Helly groups, Coxeter groups, HHGs, etc.). No step reduces by definition to its own output, no parameter is fitted then renamed as prediction, and no load-bearing premise rests solely on self-citation chains. The derivation is self-contained against external metric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the definition of navigable path system as mild non-positive curvature and on the construction of the generalised contraction space; no numerical free parameters appear.

axioms (1)
  • standard math Standard axioms and definitions of groups, metric spaces, divergence, and Morse elements from geometric group theory
    Invoked throughout as background for the rigidity statement.
invented entities (1)
  • generalised contraction space no independent evidence
    purpose: hyperbolic space that encodes the negative curvature of a given space
    New tool introduced to obtain local definitions of rank-one geodesics and other properties

pith-pipeline@v0.9.1-grok · 5729 in / 1271 out tokens · 28719 ms · 2026-06-28T23:57:54.679046+00:00 · methodology

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read the original abstract

We show that groups with a mild form of non-positive curvature (a navigable path system) satisfy the weak rank rigidity conjecture: they either have linear divergence or a Morse element. This class includes discrete groups of projective automorphisms of open convex cones, Helly groups (answering a question of Genevois), Coxeter groups, weak Garside groups (in particular Deligne's groups and fundamental groups of Salvetti complexes of oriented matroids), hierarchically hyperbolic groups, and other examples. Along the way, we show that those groups satisfy the Morse local-to-global property, providing a unified proof for the whole class. In the metric setting, the same condition of non-positive curvature allows to provide a local definition (that is, in a sense, optimal) of rank one/Morse geodesics, mirroring the one using parallel Jacobi fields from Riemannian geometry; to deduce linearity of divergence from linearity on a sequence; to obtain new cases in which Morse geodesics are strongly contracting. The main new tool introduced is the generalised contraction space, a hyperbolic space that encodes the negative curvature of a given space.

Figures

Figures reproduced from arXiv: 2605.29837 by Cornelia Drutu, Davide Spriano, Stefanie Zbinden.

Figure 1
Figure 1. Figure 1: The points γ(t0) and h(s0) are the R–lower and R–upper projection points of h onto γ, the point γ(t1) falsifies (3.1). We now show that after exiting a small neighbourhood, special paths diverge linearly from weakly polygonally Morse special paths. Proposition 3.18. Let X be a geodesic metric space equipped with an undirected path system P. There exist constants A ≥ 1, ϵ > 0 such that for all R ≥ DP (1) th… view at source ↗
Figure 2
Figure 2. Figure 2: Proof of Claim 2. The path p has to come close to z0. d(x, γ(i))/C + 4DP (R + 1), implying that |si − (i + 1)| ≥ R. Thus, for A large enough compared to C and the quasi-geodesic constants of P, we have that ∥p∥ ≤ Aℓ, where ℓ = d(γ(si), γ(i + 1)). Hence we can use that γ is (R; ϵ, A; 3,P)–weakly polygonally Morse to obtain d(z, p) ≤ ϵℓ for all z ∈ γ[si , i + 1]. Let z0 = γ(t) be a point on γ[si , i + 1] at … view at source ↗
Figure 3
Figure 3. Figure 3: The path α does not intersect the D–ball around z. Claim 3. We have that d(α, z) > D/4. Proof of Claim. For i = 3, 5, we have d(z, αi) ≥ D by the definition of the projection π. For i = 4, d(αi , z) ≥ d(β, π(x)) − d(π(x), z) > D/4. For i = 1 we have d(αi , z) ≥ d(γ ′ pref[z1], z) − C0 > D/4. A similar argument works for i = 7. Lastly, for i = 2, 6 it follows from diam(αi) ≤ DP (D), C ≫ D and the triangle i… view at source ↗
Figure 4
Figure 4. Figure 4: Proof of Theorem 4.9. The path q stays far from m. and hence r ′ = d(c, {x, y}) ≥ nk/6 ≫ r. Since Divϵ(nk, δ) ≤ Cnk, we can find a path p : [0, T] → X of length at most Cnk from x to y with d(p, c) ≥ δr′ − ϵ ≥ 4δ ′ rC0. Let M be an integer constant. For 1 ≤ i ≤ M, let αi ∈ P be a special path from p((i − 1)T /M) to p(iT /M). Define the (M,P)–polygonal line α as α1 ∗ . . . ∗ αM. For M large enough compared … view at source ↗
Figure 5
Figure 5. Figure 5: Left: we replace αi with two special paths that have the cor￾rect distance from m. Center: a depiction of the central slide. Right: a depiction of the side slide. Such paths βi , γi always exist: start with βi = αi and γi a trivial path from α + i to itself. With this definition, we satisfy all the requirements except perhaps the upper bound on d(βi , m) and d(γi , m). We now ‘move β + i = γ − i to m’, tha… view at source ↗
Figure 6
Figure 6. Figure 6: Situation in X [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Situation in Y . f(β ′−1 n ) respectively and are at Hausdorff distance at most Cf from f(β ′−1 0 ) and f(β ′−1 n ) respectively. Lastly, let q−1, q(2n+1)k+2 ∈ P± f be special paths from α − to f(β ′+ 0 ) and from f(β ′− n ) to α +. Both of those special paths have endpoints at most C away from each other. The ((2n+ 1)k + 4,P ± f )–polygonal line q = q−1 ∗ q0 ∗ . . . ∗ q(2n+1)k+2 has endpoints q − = α − an… view at source ↗
Figure 8
Figure 8. Figure 8: The arrows indicate which direction comes from a combing line. Let q1 = h ′ 1 , let q2 be a combing line from z to y, let q3 a combing line from x to z and let q4 a combing line from x to h − 2 . By quasi-consistency, we have that d(m, qi) ≥ R−4κ0 for 1 ≤ i ≤ 4. Applying Claim 6 to q1 and q2 and then to q3 and q4 and concatenating the two obtained paths, we get a path p form h − 1 to h − 2 with ∥p∥ ≤ 2C ′R… view at source ↗
Figure 9
Figure 9. Figure 9: Construction of the path p in the proof of Lemma 5.7. 5.3.2. Geodesic combings. We show that if a space admits a bounded geodesic combing then any geodesic path system that contains the combing heap is navigable. Combined with the results from Section 4, this yields that such spaces are Morse dichotomous, that is, all their Morse geodesics are strongly contracting. Lemma 5.7. Let P be the combing path syst… view at source ↗
Figure 10
Figure 10. Figure 10: The red path has Hausdorff distance at most 2 to h. Let h ′ ∈ P be a special path (which is a geodesic) from x to y ′ and let z ∈ h and z ′ ∈ h ′ be points such that d(x, z) = d(x, z′ ) and such that d(x, z)+d(x, z′ ) = d(x, y)+d(x, y′ )−1 [PITH_FULL_IMAGE:figures/full_fig_p037_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The red path avoids the R–ball around m. Since d(y, y′ ) ≤ 1 and d(x, y) + d(x, y′ ) ≥ 2, such points always exist. Further note that d(z, z′ ) ≤ 2. Let m be the median of the points (z, z′ , x). Since d(x, z) = d(x, z′ ) we have that d(m, z) = d(m, z′ ) ≤ 1. This is depicted in [PITH_FULL_IMAGE:figures/full_fig_p038_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Depictions of Lemma 6.5 (left) and Lemma 6.6 (right). The red path h2 is K–midthin with neck size r. Definition 6.3 (Anti-contracting). Let K be a contraction triple. We say that a special path h ∈ P is K–anti-contracting if none of its subsegments is K–midthin. We say that a pair of vertices (x, y) ∈ V (X) × V (X) is K-anti-contracting if there exists a special path h ∈ P from x to y which is K–anti-cont… view at source ↗
Figure 13
Figure 13. Figure 13: Upper bounding the distance d(c ′ , m). Hence, either both q1, q2 lie on γ or they both lie on p1 ∪ p2. The former still cannot happen, as it violates the triangle inequality. To ensure the latter cannot happen, we ‘cut off the tip’ of p1 ∪ p2 before defining x as the minimizer of Φ on p1 ∪ p2. Then q1, q2 cannot both be close to p1 ∪ p2 as such points do not exist any more. Of course, you need to be extr… view at source ↗
Figure 14
Figure 14. Figure 14: Depiction of the constructed paths. The point x might also lie on p1 instead of p2. (shaded pink and turquoise in [PITH_FULL_IMAGE:figures/full_fig_p045_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: In the proof of Proposition 6.18, the path h ′ ⊂ h has end￾points close to γ and thus inherits P–contraction from γ. x ′ , y′ on h in the C neighbourhood of τ (x) and τ (y) respectively. This is depicted in [PITH_FULL_IMAGE:figures/full_fig_p049_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: In the proof of Proposition 6.19, the path p has to intersect the r–neighbourhood of m. d(m, [π(y), τ (y)]P ) ≥ d(m, π(y)) − diam([π(y), τ (y)]P ) > r. Also d(β, m) > d(β, π(x)) − d(π(x), m) > r by the choice of β, x and y. If q = hypref[τ (y)]−1 or q = hxpref[τ (x)], then d(m, q) > D ≥ r by the definition of a D–upper projection point. Finally, if q ⊂ γ, then for largen enough K, d(q, m) > r by Lemma 6.5… view at source ↗
Figure 17
Figure 17. Figure 17: The blue paths γ and γ ′ are K–midthin. We contradict this constellation by applying Proposition 6.11 to the triangle shaded in pink. paths from x to y from gx to x and from y to gy respectively. Let zx, zy be the first respectively last vertex on h at distance ⌈ϵ + δ0⌉ + 4 from x respectively y. With this, ˆdK(zx, zy) ≥ 2. Thus there exists a K–midthin subpath γ of [zx, zy] h with necksize r for some r ≥… view at source ↗
Figure 18
Figure 18. Figure 18: Proof of the weak diameter dichotomy. We also obtain that, in the presence of a nice enough group action, having an un￾bounded contraction space implies having an P–contracting ray. Corollary 6.23. Let G be a group acting properly and coboundedly on X. Assume that P is G–invariant, K is large enough and diam [K(XˆK) = ∞. Then G has a P–contracting element. In particular, X has a P–contracting quasi-geodes… view at source ↗
Figure 19
Figure 19. Figure 19: Showing that γ has a subpath γ ′ whose midpoint is close to m [PITH_FULL_IMAGE:figures/full_fig_p055_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The path γ ′ is midthin, as the polygonal path q ′ can be extended which allows the use of h being midthin. Let a ′ , b′ be points on γ and let a, b be points on p1 − NC (m) respectively p2 − NC (m) with d(a, a′ ) ≤ ℓ and d(b, b′ ) ≤ ℓ. Let η1 and η2 be special paths from a ′ to a and from b to b ′ respectively. This is depicted in [PITH_FULL_IMAGE:figures/full_fig_p055_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Proof of Claim 19. We finish the proof differently depending on whether d(c ′ , m) > R or not. Analogously, there exist a special paths η2, ξ2 = [b, η− 2 ]P such that d(b, η− 2 ) ≤ ρ, |η2| ≥ M and d(η2 ∗ ξ, c′ ) ≥ R. Let M′ ≫ M be another constant to be determined later and let a ′ , b′ ∈ V (X) be vertices on η1 and η2 respectively such that d(a, a′ ) ≥ M′ and d(b, b′ ) ≥ M′ . Such points exist if M is la… view at source ↗

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