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arxiv: 2606.07430 · v1 · pith:FQPSXJ34new · submitted 2026-06-05 · 🧮 math.GR

Spectral properties of the Schreier graphs of the basilica group

Kyle Ambrose , Noah Dunham , Michael Morris , Luke G. Rogers , Alexander Teplyaev This is my paper

Pith reviewed 2026-06-27 20:23 UTC · model grok-4.3

classification 🧮 math.GR
keywords basilica groupSchreier graphsLaplacian spectracharacteristic polynomialsKNS spectral measuredynamical systemsautomaton groupsself-similar graphs
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The pith

A recursive framework for characteristic polynomials of basilica Schreier graph Laplacians reveals an underlying dynamical system and approximates the KNS spectral measure.

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

The paper develops a recursive method to find the characteristic polynomials of the Laplacians on the sequence of Schreier graphs for the basilica group. This recursion is built by extending known decompositions of certain subgraphs. The resulting relations correspond to iteration of a simple dynamical system on the polynomials. From this the authors derive that the associated spectral measures approximate the Kesten-von Neumann-Serre measure in the limit of large graphs. Readers care because the basilica group is a standard example of an amenable but not elementarily amenable automaton group, so concrete spectral information helps test broader conjectures about its geometry and random walks.

Core claim

By building a new recursive framework for the characteristic polynomials of the Laplacians on the Schreier graphs Γ_n of the basilica group, the analysis shows that these polynomials are generated by iteration of a simple dynamical system; this framework in turn establishes approximation results for the Kesten-von Neumann-Serre spectral measure of the limiting infinite graph.

What carries the argument

The recursive framework for characteristic polynomials, obtained by extending subgraph decompositions of the basilica graphs.

If this is right

  • The eigenvalues of the Laplacian on Γ_n become computable for arbitrary n by iterating the recursion from a small number of base cases.
  • The distribution of eigenvalues converges to the KNS measure at a rate controlled by the dynamical system.
  • Spectral properties such as the density of states on the infinite Schreier graph can be read off from the attractor of the dynamical system.
  • The same recursion supplies a concrete algorithm for approximating the integrated density of states for finite-level graphs.

Where Pith is reading between the lines

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

  • The same style of recursion may exist for Schreier graphs of other iterated monodromy groups of quadratic polynomials.
  • Numerical iteration of the dynamical system could produce high-resolution plots of the KNS measure without diagonalizing large matrices.
  • The dynamical system may admit an invariant measure whose support describes the spectrum exactly rather than approximately.

Load-bearing premise

The recursive relations for the characteristic polynomials can be written down at every level by direct use of the known subgraph results, with no new obstructions appearing.

What would settle it

Explicit computation of the characteristic polynomial for Γ_4 or Γ_5 by another method and direct comparison against the polynomial produced by applying the claimed recursion to the polynomial for Γ_3.

Figures

Figures reproduced from arXiv: 2606.07430 by Alexander Teplyaev, Kyle Ambrose, Luke G. Rogers, Michael Morris, Noah Dunham.

Figure 1
Figure 1. Figure 1: The graphs G0, G1, G2. Gn−2 Gn−2 Gn−1 v [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The graph Gn constructed from a copy of Gn−2 and two copies of Gn−1 at a gluing point v. particular we give a formula for the KNS measure as a sum of point masses (Theorem 5.7), estimate the rate of convergence of the spectral counting measures to the KNS measure (Theorem 5.8, but see also Remark 5.9), and establish that the eigenvalues of the basilica Schreier graphs accumulate precisely on the support of… view at source ↗
Figure 3
Figure 3. Figure 3: The graph Γn constructed from a copy of Gn, and a copy of Gn−1 at a gluing point u. Gn−1 Gn−1 Gn−2 Gn−2 u v [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The graph Γn constructed from two copies of Gn−1, and two copies of Gn−2, attached at gluing points u and v. Note that the labelling in the figures is consistent: viewing the copy of Gn in [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 2
Figure 2. Figure 2: We consider f that is a solution of ∆(Gn)f = λf on Gn \ ∂Gn. If f = 0 on ∂Gn it is a Dirichlet eigenfunction and λ ∈ DirSpec(∆(Gn)). If, in addition, ∂f = 0 on ∂Gn we call f a Dirichlet-Neumann eigenfunction. Theorem 3.1 (From [5] Section 3). Suppose ∆(Gn)f = λf on Gn \ ∂Gn. (1) If f = 0 on ∂Gn, so f is a Dirichlet eigenfunction and λ ∈ DirSpec ∆(Gn), let m ≤ n be the level at which λ is born. Then: (i) If… view at source ↗
Figure 5
Figure 5. Figure 5: Γm+2 as a gluing of copies of Gm−1 (in red) and of Gm−2 (in black) with (m − 1, m − 2)-gluing points in blue. Lemma 3.12. In the preceding decomposition, any 2-series eigenfunction on Γn born at level m ≤ n vanishes at the (m − 1, m − 2)-gluing points. It therefore vanishes identically on the copies of Gm−1 in the decomposition and its restriction to the level m − 2 vertices is a scalar multiple of the fun… view at source ↗
Figure 4
Figure 4. Figure 4: Since the copies of Gn have both boundary points identified, just as they were in Γn in [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Graphs B3, C3, D3, and Γ3 with one vertex removed. Shaded vertices are removed in the corresponding matrices. Proposition 4.9. For n ≥ 1 the characteristic polynomial Pn of Γn satisfies (4.2) Pn = (λ − 4)cncn−1 − 2dn−1cn − 2dncn−1 − 2cngn−1 − 2cn−1gn, where (4.3) gn−1 = Y 1≤j< n 2 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The 2-series eigenvalues of Γ10, arranged so that the roots of γn−2 occur on the horizontal line at height n for n = 0, . . . , 10 (the first three such lines being empty). Using the recursion in Theorem 4.11 it is fairly quick to generate the eigenvalues in the 2-series [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The 0-series eigenvalues of Γ10, arranged so that the roots of ψn occur on the horizontal line at height n for n = 0, . . . , 10. whereupon moving the γ factors to the left side and combining them with the ψ factors yields the product Zn−1Zn−2, while Lemma 4.13 gives that the remaining term on the right is ζ −1 n−2 so we have (4.11) Zn−1Zn−2 □  Zn − 1 Zn−2 − 1  = 1 ζn  ζn − 2 ζn−2 − 2  . Corollary 4.15… view at source ↗
Figure 9
Figure 9. Figure 9: The eigenvalues of Γ10 with 2-series in grey and 0-series in black, arranged so that the horizontal line at height n shows the eigenvalues born at level n for n = 0, 1, . . . , 10. There is considerable overlap of points on the lines for larger n. Definition 5.2. The KNS spectral measure is the weak limit of the sequence of spectral counting measures. µ = w-lim n→∞ νn. Remark 5.3. It is a standard result t… view at source ↗
Figure 10
Figure 10. Figure 10: Spectral counting functions for the 0-series eigenvalues (left) and 2-series eigenvalues (right) of ∆(Γ9). KNS measure, and that this KNS measure is given by ([5] Corollary 4.3) (5.3) χ = X∞ k=1 X {λ:γk(λ)=0} 1 6 · 2 −k · δλ Corollary 5.10 (Equivalence of KNS Spectral Measures). The KNS measures for the se￾quences ∆(Γn) and the Dirichlet Laplacian ∆(Gn) are the same. Proof. This is simply a matter of chan… view at source ↗
Figure 11
Figure 11. Figure 11: Spectral counting function for ∆(Γn) with n = 5 (left) and n = 9 (right). the 0-series eigenvalues are simple (Lemma 3.8) but the number of 0-series eigenvalues has power-law growth, as established in Corollary 4.6, but a proof would require information about the distribution of these eigenvalues, so as to ensure there is no point where a positive proportion of their spectral counting measure accumulates.… view at source ↗
read the original abstract

We study the spectral properties of Laplacians on the Schreier graphs $\Gamma_n$ of the basilica group, the iterated monodromy group of the polynomial $z^2 - 1$, which is an important example in the theory of self-similar, amenable but not elementarily amenable, automaton groups. Building heavily on results by Brzoska, Jarvis, George, Rogers and Teplyaev about certain subgraphs of the basilica graphs, we develop a new recursive framework for computing the characteristic polynomials of these Laplacians. Our analysis reveals a simple underlying dynamical system and proves approximation results for the Kesten-von Neumann-Serre (KNS) spectral measure.

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 manuscript studies spectral properties of Laplacians on the Schreier graphs Γ_n of the basilica group (iterated monodromy group of z²-1). Building on subgraph results of Brzoska, Jarvis, George, Rogers and Teplyaev, it develops a recursive framework for the characteristic polynomials, identifies an underlying dynamical system, and proves approximation results for the Kesten-von Neumann-Serre spectral measure.

Significance. If the recursive construction succeeds without new obstructions, the work would supply an explicit dynamical system whose iterates approximate the KNS measure for this canonical example of a self-similar amenable but not elementarily amenable group, extending prior subgraph analyses to the full Schreier graphs.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (recursive framework): the central claim that the recursive framework for characteristic polynomials of Laplacians on Γ_n extends the subgraph results without additional obstructions at each iteration is load-bearing for the approximation theorems, yet the abstract supplies neither the explicit recursion formula, base cases, nor inductive step; verification that the iterated monodromy action preserves the necessary algebraic relations therefore cannot be checked from the given text.
  2. [§4] §4 (dynamical system and KNS approximation): the identification of the 'simple underlying dynamical system' and the proof of approximation results for the KNS measure rest on the recursion being obstruction-free; without the explicit construction, it remains open whether the extension from subgraphs to full graphs at successive n introduces new relations that would invalidate the claimed convergence.
minor comments (1)
  1. [§2] Notation for the graphs Γ_n and the precise definition of the KNS measure should be stated once in §2 before being used in later sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below, agreeing that greater explicitness will strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (recursive framework): the central claim that the recursive framework for characteristic polynomials of Laplacians on Γ_n extends the subgraph results without additional obstructions at each iteration is load-bearing for the approximation theorems, yet the abstract supplies neither the explicit recursion formula, base cases, nor inductive step; verification that the iterated monodromy action preserves the necessary algebraic relations therefore cannot be checked from the given text.

    Authors: Section 3 presents the recursive framework, including base cases for small n derived from the subgraph results and the inductive step using the self-similar structure of the basilica group. We agree, however, that the abstract is too brief and does not display the recursion formula or confirm the absence of new obstructions. In the revision we will add a concise statement of the recursion to the abstract and insert an explicit lemma in §3 verifying that the iterated monodromy action preserves the required algebraic relations at each step. revision: yes

  2. Referee: [§4] §4 (dynamical system and KNS approximation): the identification of the 'simple underlying dynamical system' and the proof of approximation results for the KNS measure rest on the recursion being obstruction-free; without the explicit construction, it remains open whether the extension from subgraphs to full graphs at successive n introduces new relations that would invalidate the claimed convergence.

    Authors: The dynamical system and KNS approximation theorems in §4 are obtained directly by iterating the recursion constructed in §3. To make the obstruction-free extension explicit, the revised manuscript will include a dedicated paragraph or short lemma showing that the passage from the Brzoska–Jarvis–George–Rogers–Teplyaev subgraphs to the full Schreier graphs Γ_n introduces no new relations that alter the spectral convergence. This will render the argument self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity; new recursive framework is independent of cited subgraph results

full rationale

The paper cites prior subgraph results from Brzoska et al. (including overlapping authors Rogers and Teplyaev) as a foundation for extending to full Schreier graphs Γ_n, but explicitly develops a new recursive framework for characteristic polynomials, identifies a dynamical system, and proves KNS approximation results as original contributions. No equation or claim reduces by construction to the cited inputs, no fitted parameters are renamed as predictions, and no uniqueness theorem or ansatz is smuggled via self-citation. The derivation chain remains self-contained against external benchmarks once the subgraph base is granted; self-citation here is ordinary supporting context rather than load-bearing reduction. This matches the most common honest non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the ledger is therefore empty.

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