Apollonian random manifolds and their bass notes (2512.13139v1)

Abstract: We study the spectrum of the Laplacian on two models of random hyperbolic 3-orbifolds, related to the Apollonian group and the super Apollonian group. We determine explicit spectral gaps for these random orbifolds. Moreover, we use our model to investigate the bass note spectrum of the set of hyperbolic 3-orbifolds.

• The paper introduces two probabilistic models of random hyperbolic 3-orbifolds using Apollonian group actions, providing explicit estimates for their Laplacian spectra.
• It utilizes random covering constructions and advanced spectral theory to achieve precise control over spectral gaps and the first nonzero eigenvalue.
• The work bridges combinatorial group methods with analytical spectral analysis, offering new insights into nonarithmetic manifolds and progress on Sarnak's bass note conjecture.

Apollonian Random Manifolds and Their Bass Notes: A Technical Summary

Introduction and Motivation

This paper introduces and investigates two probabilistic models of random hyperbolic 3-orbifolds, structurally indexed by the Apollonian group and the super Apollonian group. The study focuses primarily on the spectral properties of their Laplacians, specifically spectral gaps and the so-called "bass note" spectrum—the set of possible values for the first nonzero eigenvalue (after 0, if present) of the Laplacian across the landscape of hyperbolic 3-orbifolds. The authors bridge group-theoretic constructions with analytical spectral invariants, contributing new explicit estimates in the context where little was previously known, especially in dimension 3.

Models and Constructions

The central objects are twofold:

1. Random Covers of the Apollonian Orbifold: Here, the Apollonian group $$\Gamma_{\mathrm{Ap}} \simeq (\mathbb{Z}/2\mathbb{Z})^{*4}$$, realized as a Kleinian reflection group via reflections in four non-adjacent faces of a right-angled, regular, ideal hyperbolic octahedron, acts on $\mathbb{H}^3$. Random degree $2n$ covering spaces $S_n$ are constructed by factoring through random homomorphisms into the symmetric group, sending standard generators to random perfect matchings.
2. Random Covers via the Super Apollonian Group: The super Apollonian group $\Gamma_{\mathrm{SA}}$ arises as the right-angled Coxeter group associated to the 1-skeleton of the 3-cube, generated by all face reflections. This group admits a surjection onto $(\mathbb{Z}/2\mathbb{Z})^{*4}$ by trivializing the reflection generators corresponding to four non-adjacent faces. The kernel $\Gamma_\infty$ (dubbed the "infilonian group") plays a significant role, as do the random degree $2n$ covers $M_n$ of $\Gamma_{\mathrm{SA}} \backslash\mathbb{H}^3$ arising from this combinatorics.

The paper provides a detailed construction for both models, including their geometric and spectral framework, with further elaboration on the structure of their boundaries and the use of the associated "octa-tree"—an infinite aggregation of such octahedra glued according to the free product combinatorics.

Spectral Gap and Laplacian Analysis

The main achievements pertain to explicit control of spectral gaps on these random 3-orbifolds and their finite covers:

• Persistence of Spectral Gaps: For random $2n$-fold covers of the Apollonian orbifold, the spectrum below $1-\epsilon$ in probability consists only of the (well-known and numerically precise) Patterson–Sullivan eigenvalue; no new spectrum appears in this range as $n \to \infty$.
• Spectral Gaps in Lattice Quotients: For the more intricate super Apollonian setting, random covers $M_n$ exhibit with high probability that the first eigenvalue of the Laplacian converges, in probability, to $\lambda_0(\Gamma_\infty \backslash \mathbb{H}^3)$. For the doubles of these orbifolds (manifold covers without orbifold points), Neumann and Dirichlet spectra are analyzed separately, showing the Neumann ground state converges to the same limit, while the Dirichlet gap is bounded below by $1 - \epsilon$, reflecting the underlying geometry.
• Comparison with Known Arithmetic Gaps: The new upper and lower bounds on $\lambda_0(\Gamma_\infty \backslash \mathbb{H}^3)$ are compared to the previously best-known small eigenvalues for arithmetic 3-manifolds. The current congruence arithmetic record is $\frac{77}{81} \approx 0.9506$, while the Apollonian constructions provide concrete examples with smaller spectral gaps but in random, nonarithmetic settings.

Group-Theoretic and Analytical Methods

The proofs are built on several foundational advances:

• Strong Convergence of Permutation Representations: The paper leverages the structural properties of $(\mathbb{Z}/2\mathbb{Z})^{*4}$, recent results on strong convergence of random permutation representations of finitely generated free products, and their stability under random quotients. The results of Bordenave–Collins are exploited regarding convergence to regular representations.
• Spectral Theory of Nonuniform Lattices: The analysis is deeply rooted in the spectral decomposition for noncompact quotients of $\mathbb{H}^3$. Careful estimates are established for Eisenstein series, residual spectrum, and scattering matrices, particularly for congruence subgroups of $\mathrm{PGL}(2,\mathbb{Z}[i])$.
• Explicit Bass Note Estimates: For specific groups and geometric models, explicit lower bounds on the smallest eigenvalue are derived via the Brooks–Burger transfer principle, and critical exponents are computed or bounded using, in part, recent work by Coulon on limit sets of reflection groups.

The Bass Note Spectrum and Approximation

Motivated by conjectures of Sarnak concerning the closure of the set of first eigenvalues for the Laplacian on all finite-volume hyperbolic 3-orbifolds (the "bass note" spectrum), the authors provide constructive advances:

• Approximation Intervals for Bass Notes: They show that the interval $$[0, \lambda_0(\Gamma_\infty \backslash \mathbb{H}^3)]$$ is contained within the closure of the set of $\lambda_1(X)$, where $X$ ranges over arithmetic hyperbolic 3-orbifolds; similarly, this lies within the closure for all finite-volume such orbifolds.
• Dense Coverage via Random Covers: By cleverly switching within the space of degree-2 covers and proving delocalization of eigenfunctions (utilizing diffusion and tangle-freeness adapted from random regular graph theory), they show that bass notes can be densely approximated in the interval, making substantial progress toward Sarnak's conjecture.

Technical Innovations

Noteworthy technical contributions include:

• A generalization of the notion of tangle-freeness to settings where the local model is not the universal cover $\mathbb{H}^3$ but a quotient (the octa-tree), and the subsequent derivation of $L^\infty$ eigenfunction bounds in this context.
• Precise analysis of residual spectrum for various arithmetic lattices, including explicit scattering matrix calculations for congruence subgroups of $\mathrm{PGL}(2,\mathbb{Z}[i])$.
• Careful combinatorics of random covers (using perfect matchings) and their correspondence to underlying group actions, making the probabilistic models explicit and tractable for spectral estimation.

Implications and Future Directions

This work produces explicit new results for the distribution of Laplacian eigenvalues on random and arithmetic finite-volume hyperbolic 3-orbifolds. The approach establishes a foundation for the statistical spectral theory of higher-dimensional random and arithmetic manifolds, and the explicit geometric and group-theoretic constructions suggest paths for further numerical and analytic work—particularly toward the full understanding of the closure of the set of first eigenvalues (bass notes) in dimension 3.

On the theoretical side, the methods indicate a promising avenue for analyzing the effect of random covering constructions on spectrum in non-compact and infinite-volume settings, and for investigating quantum ergodicity and eigenfunction localization in random 3-manifolds.

Conclusion

The paper provides a rigorous spectral analysis of two classes of random hyperbolic 3-orbifolds rooted in the Apollonian and super Apollonian group structures, culminating in sharp control of their Laplacian spectral gaps and the approximation of the bass note spectrum. It achieves this via an overview of modern advances in geometric group theory, probabilistic constructions, and spectral analysis, opening new directions in the study of random 3-manifolds and their spectral invariants (2512.13139).