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Gromov Hyperbolicity of Substitution graphs

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Pith's one-line read A substitution graph is Gromov hyperbolic if and only if every sufficiently long shortest horizontal path has a zero path matrix.

arxiv 2608.22836 v1 pith:5XCFLUAR submitted 2026-08-24 math.CO

Qingcheng Zeng , Cheng Zeng , Yumei Xue , Huixia He This is my paper · ORCID
classification math.CO
keywords graphssubstitutionedgesfinitegraphdefineddirectedfollow
verification ladder T0 review T1 audit T2 compute T3 formal

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The reading

Imagine building an infinite graph by starting with a single root word and repeatedly appending letters from a finite alphabet. Vertical edges connect each word to its longer children. Horizontal edges are added by two rules: words with the same parent are connected according to a small graph G, and words whose parents are already connected horizontally are connected according to another small graph J. Repeating these rules builds a large graph that resembles a tree with extra horizontal links.
Extended reading notes

Core claim

Theorem 2: U is hyperbolic if and only if there exists a positive integer k such that M(p)=0 for every shortest horizontal path p with length greater than k.

Load-bearing premise

The proof of Theorem 2 relies on the claim that every substitution graph is an augmented tree and therefore expansive, so that Kong-Lau-Wang Theorem 1 applies. This expansive property, equivalent to dh(u,v) >= dh(u-,v-) for same-level vertices, is asserted in Section 4 without detailed proof, and it is the bridge that lets hyperbolicity be translated into a statement about lengths of horizontal geodesics.

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Formalized claims in Lean

  1. Claim #1: Theorem 2: U is hyperbolic if and only if there exists a positive integer k such that M(p)=0 for every shortest horizontal path p with length greater than k.

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Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central derivation rests on standard results from the cited literature and on the structural properties of the new construction. There are no fitted numerical parameters and no new physical entities.

assumptions (3)
  • standard math Theorem 1 of Kong-Lau-Wang: an expansive graph is hyperbolic iff it is (m,k)-departing iff horizontal geodesic lengths are bounded.
    Used in the proof of Theorem 2 and Proposition 2 to connect hyperbolicity to the departing property.
  • domain assumption Every substitution graph is an augmented tree and hence an expansive graph.
    Stated in Section 4; supports the application of Theorem 1.
  • domain assumption The recursive horizontal edge rules produce a well-defined, uniquely typed set of edges.
    Foundation of the construction in Definition 1; used to classify edges as G-type or J-type throughout.

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Pith. "Pith review of Gromov Hyperbolicity of Substitution graphs." pith.science (2026). https://pith.science/paper/5XCFLUAR

@misc{pith2026260822836,
  author       = {Pith},
  title        = {Pith review of: Gromov Hyperbolicity of Substitution graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5XCFLUAR}},
  note         = {Machine review of arXiv:2608.22836}
}
read the original abstract

In this paper, we construct a class of infinite graphs, called substitution graphs. The vertex set consists of all finite words over a finite alphabet. A directed graph is formed by adding vertical edges connecting each word to its children and horizontal edges defined recursively by two finite directed graphs G and J: edges among vertices with the same parent follow G, while edges between vertices whose parents are horizontally linked follow J. The substitution graph is defined as its underlying graph. Substitution graphs provide a purely combinatorial model of self-similar structures, independent of any underlying geometric structure. Furthermore, we establish a necessary and sufficient condition for substitution graphs to be hyperbolic, formulated in terms of the vanishing of path matrices associated with sufficiently long shortest horizontal paths. Based on this characterization, we further derive several conditions that are either necessary or sufficient for hyperbolicity, depending only on the generators G and J.

Figures

Figures reproduced from arXiv: 2608.22836 by the authors.

Figure 1
Figure 1. Examples of substitution graphs.parent [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Coupling graph for Example 1. Remark 2. The recursive construction of the horizontal graphs Gn preserves cer￾tain elementary graph–theoretic properties of the base graphs on the alphabet level. For instance, if the underlying graph G is triangle-free, then each horizontal graph Gn is triangle-free for all n ∈ N. This can be proved by a straightforward induction on n using the definition of substitution graph. Motiva… view at source ↗
Figure 3
Figure 3. Case A. 0 ≤ ℓ ≤ 1. Case B. ℓ ≥ 2 and there exist an index i (1 ≤ i ≤ ℓ − 1) and integers r < s such that x − r = x − r+1 = · · · = x − s = yi . (4.12) Case C. ℓ ≥ 2 and for every index i (1 ≤ i ≤ ℓ − 1) and every k (0 ≤ k ≤ t − 1), x − k = x − k+1 = yi never holds. (4.13) Claim 2. Suppose K is nilpotent and the nilpotency index of K is I. Let p be a shortest horizontal path in U with L(p) > L0. If p falls into Case … view at source ↗
Figures from the paper (1 more)
Figure 3
Figure 3. Figure 3: Classification of a shortest horizontal path p. Proof. By definitions of Φp and of horizontal edges, for each edge ei = (xi−1, xi) of p, the pair (Φp(xi−1), Φp(xi)) is an edge of G with the same orientation as ei . Since Σ and Σ ′ are disjoint and p is a shortest horiz…

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