SARNAKS SPECTRAL GAP QUESTION DUBI KELMER ALEX KONTOROVICH AND CHRISTOPHER LUTSKO Abstract. We answer in the armative a question of Sarnaks from 2007 conrming that

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SARNAK’S SPECTRAL GAP QUESTION

DUBI KELMER, ALEX KONTOROVICH, AND CHRISTOPHER LUTSKO

Abstract. We answer in the aﬃrmative a question of Sarnak’s from 2007, conﬁrming that

the Patterson-Sullivan base eigenfunction is the unique square-integrable eigenfunction

of the hyperbolic Laplacian invariant under the group of symmetries of the Apollonian

packing. Thus the latter has a maximal spectral gap. We prove further restrictions on

the spectrum of the Laplacian on a wide class of manifolds coming from Kleinian sphere

packings.

1. Introduction

In 2007, Sarnak [Sar07] asked whether the Patterson-Sullivan base eigenvalue (see §2for

deﬁnitions and background) exhausts the discrete spectrum of the hyperbolic Laplacian of

the Apollonian 3-fold, so that this manifold has maximal “spectral gap”; he suggested that

this may indeed be the case. The purpose of this paper is to answer this question in the

aﬃrmative:

Theorem 1 (Spectral Gap for the Apollonian Group).Let Γ<Isom(H3)be the symmetry

group of an Apollonian circle packing. Then the base eigenvalue λ0≈0.9065 . . . is the only

discrete eigenvalue of the Laplacian acting on L2(Γ\H3).

This result implies an improved counting estimate on the number of circles in an Apol-

lonian packing with curvature bounded by a growing parameter (see [KL22, Theorem 1]).

Corollary 2. For a ﬁxed bounded Apollonian circle packing, P, the number N(T)of circles

in Pwith curvature bounded by Thas the following asymptotic form:

N(T) = cPTδ+O(Tη(log T)2/5),

as T→ ∞.Here δ≈1.30 . . . is the Hausdorﬀ dimension of the residual set of P, the error

exponent is

η=3

5δ+2

5≈1.18 . . . ,

and cP>0is a constant depending on P.

In general, to each transitive Kleinian packing (see §2for deﬁnitions), one can associate

a lattice (the supergroup). Theorem 1is a consequence of the following more general

theorem.

Kelmer is partially supported by NSF CAREER grant DMS-1651563.

Kontorovich is partially supported by NSF grant DMS-1802119 and BSF grant 2020119.

arXiv:2210.13969v1 [math.SP] 25 Oct 2022

2 DUBI KELMER, ALEX KONTOROVICH, AND CHRISTOPHER LUTSKO

Theorem 3. Let Pbe a transitive Kleinian sphere packing with symmetry group Γ<

Isom(Hn+1)and associated supergroup e

Γ. Let k(resp. e

k) denote the number of discrete

eigenvalues of the Laplacian acting on L2(Γ\Hn+1)(resp. L2(e

Γ\Hn+1)) below n−1; then

k≤e

Theorem 1then follows from combining two facts: (a) that for circle packings (that is,

when n= 2), n−1 = (n/2)2= 1, and (b) that in the Apollonian case, e

Γ is a double cover of

PSL(Z[i]); the latter satisﬁes Selberg’s eigenvalue conjecture. The same argument applies

to several other packings appearing in the literature to prove maximal spectral gaps.

To illustrate the methodology, we give an alternate proof of the theorem of Phillips-

Sarnak [PS85] that inﬁnite volume Hecke “triangle” groups similarly have only the base

eigenvalue in their discrete Laplace spectrum.

Theorem 4 ([PS85, Theorem 6.1]).For µ > 2, let Γµ:= h1µ

0 1 ,0−1

1 0 i<Isom(H2)be

an inﬁnite co-volume Hecke triangle group. Then the base eigenvalue λ0(µ)is the unique

discrete eigenvalue of the Laplacian acting on L2(Γµ\H2).

One of the key ideas is to replace Dirichlet boundary conditions and nodal domains

with Neumann boundary conditions and carefully chosen reﬂective walls, and to appeal

to the fact that the Hecke triangle group with µ= 2 (which is a congruence subgroup

of the modular group) is known to satisfy Selberg’s eigenvalue conjecture. After some

preliminaries in §2, we give the proofs of these theorems in §3.

Acknowledgements. We thank Peter Sarnak for bringing this problem to our attention,

and comments on an earlier draft.

2. Preliminaries

2.1. Kleinian Packings. The classical Apollonian circle packing is shown in Figure 1.

More generally, by a sphere packing Pof the n-sphere, Sn, we mean an inﬁnite collection

of round balls Bin Snwith pairwise disjoint interiors, such that their closure is all of Sn.

We treat Snas the ideal boundary of the ball model for hyperbolic space Hn+1. A ball B

in Snis the boundary at inﬁnity of a half-space bounded by a hyperplane Hin Hn+1; we

denote by RB∈Isom(Hn+1) the isometry of Hn+1 corresponding to reﬂection through H.

Given any packing P, we deﬁne the reﬂection group

ΓP<Isom(Hn+1)

of P, to be the group (inﬁnitely) generated by the reﬂections RB, for all balls Bin P.

We are interested in those packings Pwhich admit a large group of symmetries. In

particular, a packing is deﬁned (in [KK21]) to be Kleinian if the residual set of Pagrees

with the limit set of some such discrete, geometrically ﬁnite group Γ; the latter is called a

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SARNAK'SSPECTRALGAPQUESTIONDUBIKELMER,ALEXKONTOROVICH,ANDCHRISTOPHERLUTSKOAbstract.WeanswerinthearmativeaquestionofSarnak'sfrom2007,conrmingthatthePatterson-Sullivanbaseeigenfunctionistheuniquesquare-integrableeigenfunctionofthehyperbolicLaplacianinvariantunderthegroupofsymmetriesoftheApollonianpa...

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