Linear independence of generalized Poincar\'{e} series for anti-de Sitter $3$-manifolds

Let $\Gamma$ be a discrete group acting properly discontinuously and isometrically on the three-dimensional anti-de Sitter space $\mathrm{AdS}^{3}$, and $\square$ the Laplacian which is a second-order hyperbolic differential operator. We study linear independence of a family of generalized Poincar\'{e} series introduced by Kassel-Kobayashi [Adv. Math. 2016], which are defined by the $\Gamma$-average of certain eigenfunctions on $\mathrm{AdS}^{3}$. We prove that the multiplicities of $L^{2}$-eigenvalues of the hyperbolic Laplacian $\square$ on $\Gamma\backslash\mathrm{AdS}^{3}$ are unbounded when $\Gamma$ is finitely generated. Moreover, we prove that the multiplicities of \textit{stable $L^{2}$-eigenvalues} for compact anti-de Sitter $3$-manifolds are unbounded.


Introduction
A pseudo-Riemannian manifold is a smooth manifold M equipped with a smooth non-degenerate symmetric bilinear tensor g of signature (p, q) on M . It is called Riemannian if q = 0, and Lorentzian if q = 1. As in the Riemannian case, the Laplacian M := div M • grad M is defined as a second-order differential operator on M . We note that it is a hyperbolic differential operator if M is Lorentzian. We write L 2 (M ) for the Hilbert space of square-integrable functions on M with respect to the Radon measure induced by the pseudo-Riemannian structure.
For λ ∈ C, we denote by In the Riemannian case, the Laplacian is an elliptic differential operator and the distribution of its discrete spectrum has been investigated extensively, such as the Weyl law for compact Riemannian manifolds. However, it is not the case for non-Riemannian manifolds. Kobayashi [18], and later Fox-Strichartz [4], investigated the distribution of discrete spectrum of the Laplacian M of some pseudo-Riemannian manifolds, i.e., when M is the flat pseudo-Riemannian manifold R p,q /Z p+q and is the Lorentzian manifold S 1 × S q , respectively. Let us recall some basic notions. A discontinuous group for a homogeneous manifold X = G/H is a discrete subgroup Γ of G acting properly discontinuously and freely on X (Kobayashi [17,Def. 1.3]). In this case, the quotient space X Γ := Γ\X carries a C ∞ -manifold structure such that the quotient map p Γ : X → X Γ is a covering of C ∞ class, hence X Γ has a (G, X)-structure induced by p Γ . If we drop the assumption of freeness, X Γ is not always a manifold but carries a nice structure called an orbifold or V -manifold. Proper discontinuity is a more serious assumption which assures X Γ to be Hausdorff in the quotient topology. We remark that the Γ-action on X may fail to be properly discontinuous when H is noncompact. In order to overcome this difficulty, Kobayashi [15] and Benoist [1] established the properness criterion for reductive G generalizing the original criterion by Kobayashi [14]. Whereas discontinuous groups for the de Sitter space dS n := SO 0 (n, 1)/SO 0 (n − 1, 1) are always finite groups (the Calabi-Markus phenomenon, see [3], [14]), there are a rich family of discontinuous groups for the anti-de Sitter space, see e.g. [5], [16], [22]. We treat, in this article, the three-dimensional anti-de Sitter space AdS 3 := SO 0 (2, 2)/({±1} × SO 0 (2, 1)).
(2) The assumption that Γ is finitely generated could be relaxed. In fact, the exponential growth condition (see (2.8)) for Γ-orbits is essential in the proof of Theorem 1.1, and there exist infinitely generated discontinuous groups Γ satisfying (2.8) and the conclusion of Theorem 1.1 holds for such Γ (see Theorem 3.1 which is proved without finitely generated assumption).
(3) An analogous statement to Theorem 1.1 also holds when Γ\AdS 3 is an orbifold. See Section 2.3 for the argument when we drop the assumption that the Γ-action is free. Now we consider a small deformation of a discrete subgroup. The study of stability for properness was intiated by Kobayashi [16] and Kobayashi-Nasrin [19] and has been developed by Kassel [9] and others. Moreover, Kassel-Kobayashi [11] proved the existence of infinite stable L 2 -eigenvalues under any small deformation of discontinuous groups. In this article, we also consider the multiplicities of stable L 2 -eigenvalues (Definition 1.3) and prove that they are unbounded.
To be precise, let X n be the n-fold covering of X 1 := AdS 3 for 1 ≤ n ≤ ∞, and G n the Lie group of its isometries. Every compact anti-de Sitter 3-manifold M is of the form M ∼ = Γ\X n for some finite n where Γ(⊂ G n ) is a discontinuous group for X n by Thm. 7.2] and Klingler [13]. We take n to be the smallest integer of this property.
Let Hom(Γ, G n ) be the set of group homomorphisms with compact-open topology, and U Γ the set of neighborhoods W in Hom(Γ, G n ) of the natural inclusion Γ ⊂ G n such that for any ϕ ∈ W , the map ϕ is injective and ϕ(Γ) acts properly discontinuously on X n . One knows U Γ = ∅ ( [16], [13]). By definition, λ is a stable L 2 -eigenvalue if min ϕ∈W N ϕ(Γ)\Xn (λ) = 0 for some W ∈ U Γ . Moreover, for any λ ∈ C and any inclusion W ′ ⊂ W in U Γ , we have an obvious inequality min

Definition 1.3. For a compact anti-de Sitter 3-manifold M , we say that
is the multiplicity of a stable L 2 -eigenvalue λ.
There exist infinitely many m ∈ N such that N M (λ m ) ≥ 1, namely λ m is a stable L 2 -eigenvalue for sufficiently large m ([11, Cor. 9.10]). However, to the best knowledge of the author, it is not known whether N M (λ) is finite. We prove: The organization of this article is as follows. A key step to our proof is to find a family of L 2 -eigenfunctions of AdS 3 with eigenvalue λ m on AdS 3 for which the corresponding "generalized Poincaré series" are linearly independent, see Proposition 3.2. In Section 2, we recall some facts about L 2 -eigenfunctions of AdS 3 and their generalized Poincaré series which were introduced in [11] as the Γ-average of these eigenfunctions. We then give a uniform estimate of the "pseudo-distance" between the origin and the second closest point of each Γ-orbit (see Section 2.4). In Section 3, we complete a proof of Proposition 3.2. In Section 4, we prove a generalization of Theorem 1.4 to the case of convex cocompact groups (Definition 4.3).

Preliminaries about the anti-de Sitter space
In this section, we collect some preliminary results about AdS 3 . We refer to [11,Sect. 9] where they illustrate their general theory for reductive symmetric spaces X = G/H in details in the special setting where X = AdS 3 . See also [7].
Let Q be a quadratic form on R 4 defined by The tangent space T x (H 2,1 ) at x ∈ H 2,1 is isomorphic to the orthogonal complement (Rx) ⊥ with respect to Q. Then −Q| (Rx) ⊥ is a quadratic form of signature (2, 1) on T x (H 2,1 ) ∼ = (Rx) ⊥ and thus −Q induces a Lorentzian structure on H 2,1 with constant sectional curvature −1. The 3-dimensional anti-de Sitter space inherits a Lorentzian structure through the double covering π : H 2,1 → AdS 3 .

Some coordinates and "pseudo-balls"
In this subsection, we work with coordinates on H 2,1 and consider "pseudo-balls" in AdS 3 . We identify H 2,1 with SL(2, R) using the isomorphism For t ≥ 0 and θ ∈ R, we use the notations (2.2) We embed H 2,1 into C 2 by We note that z 1 = 0 if x ∈ H 2,1 . Via the identification (2.1), we have if x = k(θ 1 )a(t)k(θ 2 ) ∈ SL(2, R) (a "polar coordinate"). In particular, we have Next, we consider pseudo-balls on AdS 3 , as a special case of Kassel-Kobayashi [11] for reductive symmetric spaces.
This function is invariant under x → −x, hence defines a function on AdS 3 , to be also denoted by · (a "pseudo-distance" from the origin). The compact set is called a pseudo-ball of radius R.

Square-integrable eigenfunctions of the Laplacian on the anti-de Sitter space
In this subsection, we consider the square-integrable eigenfunctions ψ m,k of Let m be a positive integer and k be a non-negative integer. In the coordinates (2.3), the function z , hence defines a real analytic function on AdS 3 , to be denoted by ψ m,k . Then we have Discrete spectrum Spec d ( AdS 3 ) coincides with {λ m | m ∈ N} and L 2 λm (AdS 3 ) is generated by ψ m,0 and its complex conjugate ψ m,0 as a representation of SO 0 (2, 2). By (2.4) We refer to ψ m,k as a spherical function of type (−m, m + k) in accordance with the action of SO(2) × SO(2).

Convergence of generalized Poincaré series
In this subsection, we explain the fact about discrete spectrum of locally symmetric spaces by Kassel-Kobayashi [11] in our AdS 3 setting. We use the following notation.
• Let E and`E be respectively the identity elements of G and`G.
. From now on, we consider only discontinuous groups Γ for AdS 3 which are discrete subgroups of G. This is enough for our purpose.
In order to study Spec d ( Γ\AdS 3 ), Kassel-Kobayashi [11] considered the convergence and non-vanishing of generalized Poincaré series for K-finite square-integrable eigenfunctions ϕ of AdS 3 . For this, they used an analytic estimate of ϕ and a geometric estimate of the number of Γ-orbits The convergence of generalized Poincaré series is proved by [11] as follows. For g ∈ G and a function f on AdS [11], see also [7, Sect. 8.1] for the proof). Let Γ ⊂ G be a discontinuous group for AdS 3 satisfying the exponential growth condition
(3) There exist discontinuous groups which do not satisfy the exponential growth condition (2.8). Indeed, for any increasing function f : for sufficiently large R > 0 in [7].
The conclusion of Fact 2.4 still holds if we drop the assumption that Γ acts freely on X = AdS 3 . In this case, the quotient space X Γ = Γ\X is an orbifold. To formulate more precisely in the orbifold case, we observe that the quotient space X Γ is Hausdorff, and carries a natural Radon measure (see e.g. Discrete spectrum Spec d ( X Γ ) and its multiplicity N X Γ are defined similarly to the case where Γ acts also freely.

"Injectivity radii" of anti-de Sitter 3-manifolds
Let Γ be a discontinuous group for AdS 3 . In this subsection, we give a uniform estimate of the pseudo-distance between the origin and the second closest point of each Γ-orbit. We recall that Γ(⊂`G×`G) acts isometrically on AdS By the inequality (see e.g. [7, Lem. 8.7]) (g 1 , g 2 )x ≥ | g 1 − g 2 | − x for (g 1 , g 2 ) ∈ G and x ∈ AdS 3 , we get: Proposition 2.7. Let Γ be a discrete subgroup of G acting properly discontinuously on AdS 3 . Then there exists g ∈ G satisfying ε g −1 Γg > 0.
Remark 2.8. One sees in the proof below that the set of such g is dense in G.

Proof of Theorem 1.1
In this section, we prove Theorem 1.1.
More generally, without finitely generated assumption of Γ, we study linear independence of the generalized Poincaré series of the spherical functions ψ m,k of type (−m, m + k) defined in Section 2.2. By choosing k = 3 j (j = 0, 1, 2, . . .), we prove:  Proof of Theorem 3.1. We have an obvious equality of the multiplicity of L 2eigenvalues, N Γ\AdS 3 = N (g −1 Γg)\AdS 3 for any g ∈ G through the natural isomorphism Γ\AdS 3 ∼ = (g −1 Γg)\AdS 3 as Lorentzian manifolds. By replacing Γ with g −1 Γg if necessary, we may and do assume ε Γ > 0 by Proposition 2.7. Then Proposition 3.2 implies that L 2 λm (Γ\AdS 3 ) contains at least k linearly independent elements if m > m Γ (k) for any fixed k ∈ N, which means dim C L 2 λm (Γ\AdS 3 ) ≥ k. Hence Theorem 3.1 follows. Kassel-Kobayashi [11] proved the non-vanishing of the generalized Poincaré series (ψ m,0 ) Γ for sufficiently large m ∈ N by showing that the first term in the generalized Poincaré series is larger at the origin than the sum of the remaining terms. For this, they utilized the fact that ψ m,0 (`E) = 1. Our strategy for the proof of Proposition 3.2 is along the same line, however, there are some technical difficulties since ψ m,k for k ≥ 1 vanishes at the origin. We then make use of an observation that ψ m,k decays more slowly at the origin than at infinity, to be precise, by the following formula, see (2.5): Actually, we use an analytic lemma (Lemma 3.3) to prove that the first term in the generalized Poincaré series (ψ m,k ) Γ is larger at points sufficiently close to the origin than the sum of the remaining terms if m ≫ 0. Moreover, we use a combinatorial lemma (Lemma 3.4) to find points at which leading terms of (Re(ψ m,k )) Γ do not cancel each other for any linear combination.
For C, a, ε > 0 and s ∈ N, we set
(3.4) By (2.5), for any y ∈ AdS 3 , we get We define a = (a j ) k−1 j=0 by a j = 1 for b j ≥ 0 and a j = −1 for b j < 0, and set We note that all the coefficients of f b are non-negative by Lemma 3.4. Moreover, we get cos(3 j θ a,3 ) −1 ≤ 3 k−1 for all j = 0, 1, . . . , k − 1 by using the inequality and, for any x ∈ B(4ε), we have The third and forth inequalities respectively follow from the exponential growth condition (2.8) and Lemma 3.3. On the other hand, we set x a,ε := k( θ a, 3 2 )a(ε)k( θ a, 3 2 ) −1 ∈ B(4ε).
Then it follows from (2.5) that

Proof of Theorem 1.4
In this section, we prove Theorem 1.4 by applying Proposition 3.2. We work in the following setting. We allow ∆ to have torsion.
• j, ρ : ∆ →`G are two group homomorphisms with j injective and discrete.
We use the following structural results of discontinuous groups for the proof of Theorem 1.4. . Let Γ be a finitely generated discrete subgroup of G acting properly discontinuously on AdS 3 . Then Γ is of either type (i) or (ii) as follows: type (i) Γ is of the form ∆ j,ρ up to switching the two factors.
A non-elementary discrete subgroup Γ of a connected linear real reductive Lie group L of real rank 1 is called convex cocompact if Γ acts cocompactly on the convex hull of its limit set in the Riemannian symmetric space associated to L. For example, cocompact lattices and Schottky groups are convex cocompact. More generally, one may think of the notion of convex cocompactness of discontinuous groups for AdS 3 :

Definition 4.3 ([11, Def. 9.1]). A discontinuous group Γ for AdS 3 is called convex cocompact if Γ is of the form ∆ j,ρ up to finite index and switching the two factors, where ∆ is torsion-free and j(∆) is convex cocompact in`G.
We note that a discontinuous group ∆ j,ρ acts cocompactly on AdS 3 if and only if j(∆) is cocompact in`G because ∆ j,ρ is isomorphic to j(∆) as abstract groups. By Fact 4.2, discontinuous groups acting cocompactly on AdS 3 are convex cocompact.

Proof of Theorem 1.4 for Γ of type (i)
In this subsection, we prove Theorem 1.4 for Γ of type (i). For this, we use the constant C Lip (j, ρ) introduced by Kassel [8] and Guéritaud-Kassel [6], which quantifies the properness of the action of ∆ j,ρ on AdS 3 .

Proof of Theorem 1.4 for Γ of type (ii)
In this subsection, we prove Theorem 1.4 for the case where Γ is standard. For this, we use the following fact by Kobayashi [16] and Kassel [9] applied to our AdS 3 setting, which gives the stability for properness under any small deformation of standard convex cocompact discontinuous groups.
Let r > 0 be the constant in Fact 4.9. For an integer n ≥ 2, we define a positive number η n by cosh η n := 1 + 2(sinh r 4 sin π n ) 2 .
We get the following by easy computations: Lemma 4.13. By an abuse of notation, we regard k(θ), a(t) in (2.2) as elements of`G = PSL(2, R). Then We give a uniform estimate of ε Γ in (2.9) and N Γ (x, R) in (2.7) for standard discrete subgroups Γ of class n after taking a conjugation of Γ. Lemma 4.14. Let Γ be a standard discrete subgroup of class n ≥ 2. There exists g ∈ G such that ε g −1 Γg ≥ min{η n /3, r/6} and N g −1 Γg (x, R) < ce 16R for any x ∈ AdS 3 and any R > 0.

Theorem 4.15.
There exists a constant µ n > 0 depending only on n such that for any convex cocompact standard discrete subgroup Γ of class n and any m, k ∈ N with m > 3 k µ n , N Γ\AdS 3 (λ m ) ≥ k.
Proof. If n = 1, then this follows from Theorem 4.7 since convex cocompact discontinuous groups are finitely generated, hence we assume that n ≥ 2. In this case, we shall prove that Γ and its small deformation are standard of class n. When n ≥ 2, the group Γ 1 = ker(pr 1 | Γ ) is a cyclic group of order n. By Fact 4.9, replacing Γ by some conjugate under`G × {`E}, we may and do assume γ 1 ≥ r for any (γ 1 , γ 2 ) ∈ Γ \ Γ 1 . By Fact 4.10, there exists a neighborhood W of the natural inclusion Γ ⊂ G such that for any ϕ ∈ W , the restriction of ϕ to the finite subgroup Γ 1 is injective and the inequalities hold where ϕ i = pr i •ϕ for i = 1, 2. Then ϕ is injective and discrete. We claim ϕ 1 (Γ 1 ) is trivial. Indeed, if there exists γ ∈ Γ 1 \ {E} such that ϕ 1 (γ) =`E, then the normalizer of ϕ(Γ 1 ) in G is contained in`K 1 ×`G where `K 1 is the maximal compact subgroup of`G containing ϕ 1 (Γ 1 ). Hence ϕ(Γ) ⊂ K 1 ×`G. By the inequalities (4.2), ϕ(Γ) is finite, hence Γ is also finite. This contradicts the assumption that Γ is non-elementary. Thus ϕ 1 (Γ 1 ) is trivial and ϕ 2 (Γ 1 ) is non-trivial. Hence the normalizer of ϕ(Γ 1 ) in G is contained in`G ×`K 2 , where`K 2 is the maximal compact subgroup of`G containing ϕ 2 (Γ 1 ). Therefore pr 2 (ϕ(Γ)) is bounded. Moreover ϕ(Γ) 1 = ϕ(Γ 1 ) by the inequalities (4.2), hence the discrete subgroup ϕ(Γ) is standard of class n. By the explicit description (3.3) of m Γ (k) and Lemma 4.14, Theorem 4.15 follows from Proposition 3.2.
Remark 4.16. In the above proof, we have shown that a convex cocompact standard discrete subgroup Γ of class n ≥ 2 and its small deformation are standard of class n. Therefore we obtain a stronger result that for any convex cocompact standard discrete subgroup Γ of class n ≥ 2 and any integer m > 3µ n if the following statement holds,: N Γ\AdS 3 (λ m ) = ∞ for any standard discrete subgroup Γ and any m ∈ N such that N Γ\AdS 3 (λ m ) ≥ 1. The latter statement is discussed in [10] by using discretely decomposable blanching laws of unitary representations (cf. [12]).
Thus the proof of Theorem 1.4 is completed.