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arxiv: 2606.05089 · v1 · pith:RKU5W752new · submitted 2026-06-03 · 🧮 math.MG · math.GR· math.PR

Quasi-isometric rigidity for random subsets in products of trees

Zhiqiang Li , Ranfeng Yu , Tianyi Zheng This is my paper

Pith reviewed 2026-06-28 02:43 UTC · model grok-4.3

classification 🧮 math.MG math.GRmath.PR
keywords quasi-isometric rigidityrandom subsetsproduct of treeshigher-rank latticesquasi-isometriesgeometric group theorytree products
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The pith

Quasi-isometric embeddings from a random subset of the product of two regular trees back into that product are rigid.

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

The paper proves that when D is a random subset of the product X of two regular trees, any quasi-isometric embedding of D into X must stay at bounded distance from an isometry of the ambient space. This extends earlier rigidity theorems known for lattices to the random case. The result yields an explicit description of all self-quasi-isometries of such a D. It also shows that two independently chosen random subsets are almost surely not quasi-isometric to each other.

Core claim

We prove a rigidity result for quasi-isometric embeddings from a random subset D of the product X of two regular trees into X itself. This can be seen as an extension of Eskin's quasi-isometric rigidity of higher-rank nonuniform lattices to random subsets. As a consequence, we give a description of the self-quasi-isometric embeddings of a random sample. We also show that two independent samples are almost surely non-quasi-isometric, confirming that such a phenomenon occurs in the higher-rank setting.

What carries the argument

The random subset D of the product X of two regular trees, sampled under a measure with sufficient independence, together with the quasi-isometric embedding of D into X.

If this is right

  • Self-quasi-isometric embeddings of a single random sample admit an explicit description.
  • Two independent random samples from X are almost surely not quasi-isometric.
  • The rigidity phenomenon for random subsets appears in the higher-rank setting.
  • This stands in contrast to results showing quasi-isometric equivalence for certain random sequences.

Where Pith is reading between the lines

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

  • The same sampling-plus-rigidity approach might apply to products of more than two trees or to other non-positively curved spaces.
  • Random subsets could serve as test cases for rigidity questions that are currently open for deterministic subsets.
  • One could attempt to verify the non-quasi-isometry statement by direct computation on finite approximations of the trees.

Load-bearing premise

The probability measure used to sample the random subset D must supply enough independence or density so that the rigidity argument carries through.

What would settle it

Exhibit a quasi-isometric embedding of some random D into X whose image stays unbounded distance from every isometry of X.

Figures

Figures reproduced from arXiv: 2606.05089 by Ranfeng Yu, Tianyi Zheng, Zhiqiang Li.

Figure 3.1
Figure 3.1. Figure 3.1: The set U ϵ,D 0 (F) Proposition 3.3. Let D be a random subset of X. For each constant κ > 1, there exist constants M1 ∈ N and ϵ0 ∈ (0, 1/100) such that for each ϵ ∈ (0, ϵ0), each constant C > 0, and each constant ρ > 1/ϵ, there exists a constant ∆ > 0 such that the following hold for each flat F: (i) The sets U ϵ,D,ρ 1 (F) and U ϵ,D,ρ 2 (F) defined in (3.4) are almost surely nonempty. (ii) Almost surely,… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: The sets P ′ i , Int(P ′ i , r1), V ′ i , and Vi So since graded quasi-isometric embeddings are quasi-isometric embeddings on bounded sets, we get from direct calculation that ϕe F,ϵ restricted to ϕe−1 F,ϵ ϕe F,ϵ(F) ∩ BR+∆ is a (2κ, C′ (R) − 4∆)- quasi-isometric embedding to B Sm′ ℓ=1 P ′ ℓ , ∆  by Proposition 3.3 and (4.9). Step 2. We now choose a suitable set σ of indices i. In the following steps, w… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: One of the possible cases For j /∈ σ, since ϕ ′ j [PITH_FULL_IMAGE:figures/full_fig_p019_4_2.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
read the original abstract

In this article, we prove a rigidity result for quasi-isometric embeddings from a random subset $D$ of the product $\mathbb{X}$ of two regular trees into $\mathbb{X}$ itself. This can be seen as an extension of Eskin's quasi-isometric rigidity of higher-rank nonuniform lattices to random subsets. As a consequence, we give a description of the self-quasi-isometric embeddings of a random sample. We also show that two independent samples are almost surely non-quasi-isometric, confirming that such a phenomenon occurs in the higher-rank setting, as suggested by Ab\'ert. This result contrasts with the result on quasi-isometric equivalence between random sequences by Basu and Sly.

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 / 1 minor

Summary. The paper proves a quasi-isometric rigidity theorem for embeddings of a random subset D of the product X of two regular trees into X itself, extending Eskin's rigidity for higher-rank nonuniform lattices. As consequences, it describes the self-quasi-isometries of such a random D and shows that two independent random samples are almost surely not quasi-isometric to each other.

Significance. If the central claims hold, the work extends boundary rigidity techniques from lattices to random subsets while preserving higher-rank phenomena, confirming Abért's suggested contrast with the lower-rank case of Basu-Sly. The almost-sure properties derived from the sampling measure constitute a genuine technical extension when the measure assumptions are verified.

major comments (2)
  1. [§2] §2 (definition of the sampling measure on X): the conditions guaranteeing almost-sure positive density in large balls and sufficient independence across the two tree factors must be stated explicitly; without them the boundary maps used to recover isometries from Eskin's argument are not guaranteed to be well-defined or measurable on a set of full measure.
  2. [Theorem 1.1] Theorem 1.1 (main rigidity statement): the hypotheses on D must include the precise almost-sure properties of the measure that replace the discreteness and recurrence used for lattices; the current formulation leaves open whether the extension applies for the stated random model.
minor comments (1)
  1. [Introduction] The abstract and introduction should cite the exact statement of Eskin's theorem being extended so that the differences in the random setting are immediately visible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where greater explicitness is needed regarding the almost-sure properties of the sampling measure. We will revise the manuscript to state these properties clearly in §2 and in the hypotheses of Theorem 1.1, thereby making the application of Eskin's boundary rigidity fully rigorous for the random model.

read point-by-point responses

  1. Referee: [§2] §2 (definition of the sampling measure on X): the conditions guaranteeing almost-sure positive density in large balls and sufficient independence across the two tree factors must be stated explicitly; without them the boundary maps used to recover isometries from Eskin's argument are not guaranteed to be well-defined or measurable on a set of full measure.

    Authors: We agree that these almost-sure properties must be stated explicitly. In the revision we will insert a new proposition in §2 that records the following facts, which follow directly from the product Bernoulli sampling: (i) for almost every D the intersection D ∩ B(x,R) has cardinality at least c·vol(B(x,R)) for large R and a uniform c>0; (ii) the two tree factors are sampled independently, which supplies the required decorrelation for the boundary maps. With these statements the boundary maps are measurable and well-defined on a full-measure set, allowing Eskin's argument to proceed verbatim. revision: yes

  2. Referee: [Theorem 1.1] Theorem 1.1 (main rigidity statement): the hypotheses on D must include the precise almost-sure properties of the measure that replace the discreteness and recurrence used for lattices; the current formulation leaves open whether the extension applies for the stated random model.

    Authors: The statement of Theorem 1.1 will be updated to list explicitly the almost-sure properties (positive lower density in balls and factor-wise independence) that replace the lattice assumptions. We will add a sentence noting that these properties hold with probability one under the product Bernoulli measure used to construct D. This makes clear that the rigidity conclusion applies to the random model under consideration. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation extends external Eskin rigidity via measure-theoretic a.s. properties without self-referential reduction

full rationale

The paper presents a mathematical proof extending Eskin's quasi-isometric rigidity theorem for higher-rank lattices to random subsets D of tree products, using almost-sure properties of an unspecified sampling measure on X. The abstract and provided context cite external results (Eskin, Abért, Basu-Sly) as contrast or motivation but do not invoke self-citations as load-bearing premises, nor do they define quantities in terms of the target rigidity statement. No equations or steps reduce a 'prediction' to a fitted input by construction, smuggle an ansatz via prior author work, or rename known patterns. The central claim remains a non-trivial extension whose validity hinges on independent verification of the measure's density/independence conditions rather than tautological rephrasing of inputs. This is the expected non-finding for a self-contained proof paper in geometric group theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; no free parameters, invented entities, or non-standard axioms are visible. The result rests on standard properties of regular trees and quasi-isometries.

axioms (1)
  • standard math Product of two regular trees carries a natural quasi-isometric structure extending Eskin's higher-rank lattice setting.
    Invoked to frame the random subset rigidity as an extension.

pith-pipeline@v0.9.1-grok · 5644 in / 1124 out tokens · 49897 ms · 2026-06-28T02:43:05.468065+00:00 · methodology

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