Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds
Let Γ be a discrete group acting properly discontinuously and isometrically on the three-dimensional anti-de Sitter space AdS³, and □ the Laplacian, which is a second-order hyperbolic differential operator. We study the linear independence of a family of generalized Poincaré series introduced by Kas...
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| description | Let Γ be a discrete group acting properly discontinuously and isometrically on the three-dimensional anti-de Sitter space AdS³, and □ the Laplacian, which is a second-order hyperbolic differential operator. We study the linear independence of a family of generalized Poincaré series introduced by Kassel-Kobayashi [Adv. Math. 287 (2016), 123-236, arXiv:1209.4075], which are defined by the Γ-average of certain eigenfunctions on AdS³. We prove that the multiplicities of 𝐿²-eigenvalues of the hyperbolic Laplacian □ on Γ∖AdS³ are unbounded when Γ is finitely generated. Moreover, we prove that the multiplicities of stable 𝐿²-eigenvalues for compact anti-de Sitter 3-manifolds are unbounded.
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| first_indexed | 2026-03-14T13:22:12Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 042, 15 pages
Linear Independence of Generalized Poincaré Series
for Anti-de Sitter 3-Manifolds
Kazuki KANNAKA
RIKEN iTHEMS, Wako, Saitama 351-0198, Japan
E-mail: kazuki.kannaka@riken.jp
Received May 13, 2020, in final form April 13, 2021; Published online April 23, 2021
https://doi.org/10.3842/SIGMA.2021.042
Abstract. Let Γ be a discrete group acting properly discontinuously and isometrically
on the three-dimensional anti-de Sitter space AdS3, and � the Laplacian which is a second-
order hyperbolic differential operator. We study linear independence of a family of gene-
ralized Poincaré series introduced by Kassel–Kobayashi [Adv. Math. 287 (2016), 123–236,
arXiv:1209.4075], which are defined by the Γ-average of certain eigenfunctions on AdS3.
We prove that the multiplicities of L2-eigenvalues of the hyperbolic Laplacian � on Γ\AdS3
are unbounded when Γ is finitely generated. Moreover, we prove that the multiplicities of
stable L2-eigenvalues for compact anti-de Sitter 3-manifolds are unbounded.
Key words: anti-de Sitter 3-manifold; Laplacian; stable L2-eigenvalue
2020 Mathematics Subject Classification: 58J50; 53C50; 22E40
1 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 := divM ◦ gradM 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 L2(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
L2
λ(M) :=
{
f ∈ L2(M) | �Mf = λf in the weak sense
}
.
The set of L2-eigenvalues Specd(�M ) :=
{
λ ∈ C | L2
λ(M) 6= 0
}
is called the discrete spectrum
of �M .
Our interest is the multiplicities of L2-eigenvalues λ of �M , denoted by
NM (λ) := dimC L
2
λ(M) ∈ N ∪ {∞}.
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 Rie-
mannian manifolds. However, it is not the case for non-Riemannian manifolds. Kobayashi [19],
and later Fox–Strichartz [4], investigated the distribution of the discrete spectrum of the Lapla-
cian �M of some pseudo-Riemannian manifolds, i.e., when M is the flat pseudo-Riemannian
manifold Rp,q/Zp+q and is the Lorentzian manifold S1 × Sq, 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 [18,
Definition 1.3]). In this case, the quotient space XΓ := Γ\X carries a C∞-manifold structure
mailto:kazuki.kannaka@riken.jp
https://doi.org/10.3842/SIGMA.2021.042
2 K. Kannaka
such that the quotient map pΓ : X → XΓ is a covering of C∞ class, hence XΓ has a (G,X)-
structure induced by pΓ. If we drop the assumption of freeness, XΓ 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 XΓ to be Hausdorff in the quotient topology. We remark
that the action of a discrete subgroup Γ on X may fail to be properly discontinuous when H
is noncompact. In order to overcome this difficulty, Kobayashi [16] and Benoist [1] established
the properness criterion for reductive G generalizing the original criterion by Kobayashi [15].
Whereas discontinuous groups for the de Sitter space dSn := SO0(n, 1)/SO0(n−1, 1) are always
finite groups (the Calabi–Markus phenomenon, see [3, 15]), there are a rich family of discon-
tinuous groups for the anti-de Sitter space, see, e.g., [5, 17, 23]. We treat, in this article, the
three-dimensional anti-de Sitter space AdS3 := SO0(2, 2)/({±1} × SO0(2, 1)).
For m ∈ N, we set
λm := 4m(m− 1).
We prove:
Theorem 1.1. For any finitely generated discontinuous group Γ for AdS3,
lim
m→∞
NΓ\AdS3(λm) =∞.
Remark 1.2.
(1) A discontinuous group Γ for AdS3 is called standard [10, Definition 1.4] if it is contained in
a reductive subgroup of SO0(2, 2) which acts properly on AdS3 such as SU(1, 1). When Γ
is torsion-free and standard, Kassel–Kobayashi [11, 12] established the theory of spectral
decomposition of L2(Γ\AdS3) into eigenfunctions of the (hyperbolic) Laplacian. Moreover,
a stronger result than Theorem 1.1 holds in this case: NΓ\AdS3(λm) = ∞ for sufficiently
large m ∈ N (Kassel–Kobayashi [13]). On the other hand, a full spectral decomposition is
not known. The construction of L2-eigenfunctions by generalized Poincaré series still works
for the non-standard case, showing that λm is an L2-eigenvalue on Γ\AdS3 for sufficiently
large m ∈ N [10]. Theorem 1.1 is also applicable to non-standard Γ, for example, in the
case where Γ is Zariski dense in SO(2, 2).
(2) The assumption that Γ is finitely generated could be relaxed. In fact, the exponential
growth condition (see (2.9)) for Γ-orbits is essential in the proof of Theorem 1.1, and
there exist infinitely generated discontinuous groups Γ satisfying (2.9) 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\AdS3 is an orbifold. See Sec-
tion 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 pro-
perness was intiated by Kobayashi [17] and Kobayashi–Nasrin [20] and has been developed
by Kassel [9] and others. Moreover, Kassel–Kobayashi [10] proved the existence of infinite stable
L2-eigenvalues under any small deformation of discontinuous groups. In this article, we also
consider the multiplicities of stable L2-eigenvalues (Definition 1.3) and prove that they are
unbounded.
To be precise, let Xn be the n-fold covering of X1 := AdS3 for 1 ≤ n ≤ ∞, and Gn the Lie
group of its isometries. Every compact anti-de Sitter 3-manifold M is of the form M ∼= Γ\Xn
for some finite n, where Γ(⊂ Gn) is a discontinuous group for Xn by Kulkarni–Raymond [21,
Theorem 7.2] and Klingler [14]. We take n to be the smallest integer of this property.
Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds 3
Let Hom(Γ, Gn) be the set of group homomorphisms with compact-open topology, and UΓ
the set of neighborhoods W in Hom(Γ, Gn) of the natural inclusion Γ ⊂ Gn such that for
any ϕ ∈ W , the map ϕ is injective and ϕ(Γ) acts properly discontinuously on Xn. One knows
UΓ 6= ∅ [14, 17]. By definition, λ is a stable L2-eigenvalue if minϕ∈W Nϕ(Γ)\Xn(λ) 6= 0 for some
W ∈ UΓ. Moreover, for any λ ∈ C and any inclusion W ′ ⊂ W in UΓ, we have an obvious
inequality
min
ϕ∈W ′
Nϕ(Γ)\Xn(λ) ≥ min
ϕ∈W
Nϕ(Γ)\Xn(λ).
Definition 1.3. For a compact anti-de Sitter 3-manifold M , we say that
ÑM (λ) := sup
W∈UΓ
min
ϕ∈W
Nϕ(Γ)\Xn(λ)
is the multiplicity of a stable L2-eigenvalue λ.
There exist infinitely many m ∈ N such that ÑM (λm) ≥ 1, namely λm is a stable L2-eigen-
value for sufficiently large m [10, Corollary 9.10]. However, to the best knowledge of the author,
it is not known whether ÑM (λm) is finite. We prove:
Theorem 1.4. For any compact anti-de Sitter 3-manifold M ,
lim
m→∞
ÑM (λm) =∞.
The organization of this article is as follows. A key step to our proof is to find a family of L2-
eigenfunctions of �AdS3 with eigenvalue λm on AdS3 for which the corresponding “generalized
Poincaré series” are linearly independent, see Proposition 3.2. In Section 2, we recall some facts
about L2-eigenfunctions of �AdS3 and their generalized Poincaré series which were introduced
in [10] 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).
2 Preliminaries about the anti-de Sitter space
In this section, we collect some preliminary results about AdS3. We refer to [10, Section 9]
where they illustrate their general theory for reductive symmetric spaces X = G/H in details
in the special setting where X = AdS3. See also [7].
Let Q be a quadratic form on R4 defined by Q(x) = x2
1 + x2
2− x2
3− x2
4 for x = (x1, x2, x3, x4)
and we set
H2,1 :=
{
x = (x1, x2, x3, x4) ∈ R4 | Q(x) = 1
} ∼= SO0(2, 2)/SO0(2, 1).
The tangent space Tx(H2,1) at x ∈ H2,1 is isomorphic to the orthogonal complement (Rx)⊥
with respect to Q. Then −Q|(Rx)⊥ is a quadratic form of signature (2, 1) on Tx(H2,1) ∼= (Rx)⊥
and thus −Q induces a Lorentzian structure on H2,1 with constant sectional curvature −1.
The 3-dimensional anti-de Sitter space
AdS3 := H2,1/{±1} ∼= SO0(2, 2)/({±1} × SO0(2, 1)),
inherits a Lorentzian structure through the double covering π : H2,1 → AdS3.
4 K. Kannaka
2.1 Some coordinates and “pseudo-balls”
In this subsection, we work with coordinates on H2,1 and consider “pseudo-balls” in AdS3.
We identify H2,1 with SL(2,R) using the isomorphism
H2,1
∼=−→ SL(2,R),
x = (x1, x2, x3, x4) 7−→
(
x1 + x4 −x2 + x3
x2 + x3 x1 − x4
)
.
(2.1)
For t ≥ 0 and θ ∈ R, we use the notations
k(θ) =
(
cos θ − sin θ
sin θ cos θ
)
, a(t) =
(
et 0
0 e−t
)
. (2.2)
We embed H2,1 into C2 by
x 7→ (z1, z2) =
(
x1 +
√
−1x2, x3 +
√
−1x4
)
. (2.3)
We note that z1 6= 0 if x ∈ H2,1. Via the identification (2.1), we have
(z1, z2) =
(
(cosh t)e
√
−1(θ1+θ2), (sinh t)e
√
−1(θ1−θ2)
)
, (2.4)
if x = k(θ1)a(t)k(θ2) ∈ SL(2,R) (a “polar coordinate”). In particular, we have
cosh 2t = x2
1 + x2
2 + x2
3 + x2
4.
Next, we consider pseudo-balls on AdS3, as a special case of Kassel–Kobayashi [10] for reductive
symmetric spaces.
Definition 2.1. For x = (x1, x2, x3, x4) ∈ H2,1, ‖x‖ ∈ R≥0 is defined by
cosh ‖x‖ := x2
1 + x2
2 + x2
3 + x2
4 (= cosh(2t)).
This function is invariant under x 7→ −x, hence defines a function on AdS3, to be also denoted
by ‖ · ‖ (a “pseudo-distance” from the origin). The compact set
B(R) :=
{
y ∈ AdS3 | ‖y‖ ≤ R
}
is called a pseudo-ball of radius R.
2.2 Square-integrable eigenfunctions of the Laplacian
on the anti-de Sitter space
In this subsection, we consider square-integrable eigenfunctions of �AdS3 with eigenvalues λm =
4m(m − 1). We recall from [10, Section 9] the following decomposition of the open subset
{Q > 0} of the flat pseudo-Riemannian manifold R2,2 =
(
R4, Q(dx)
)
:
{Q > 0}
∼=−→ R>0 ×H2,1,
x 7−→
(√
Q(x), x/
√
Q(x)
)
.
Let r =
√
Q(x). Then one has, see [10, p. 215],
−r2�R2,2 = −
(
r
∂
∂r
)2
− 2r
∂
∂r
+ �H2,1 . (2.5)
Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds 5
Let m be a positive integer and k be a non-negative integer. In the coordinates (2.3), the homoge-
neous function z
−(k+2m)
1 zk2 of degree −2m is harmonic with respect to �R2,2 , hence its restriction
to the submanifold H2,1 is an eigenfuction of �H2,1 with eigenvalue λm = 4m(m− 1) by the for-
mula (2.5). Moreover, it is square-integrable with respect to the measure sinh(2t)dθ1dtdθ2 in the
polar coordinate (2.4) induced from the Lorentzian metric on H2,1, as in the k = 0 case [10,
Section 9]. This L2-eigenfunction is invariant under (z1, z2) 7→ (−z1,−z2), hence defines a real
analytic L2-eigenfunction on AdS3 with eigenvalue λm, to be denoted by ψm,k. The discrete
spectrum Specd(�AdS3) coincides with {λm | m ∈ N} and L2
λm
(
AdS3
)
is generated by ψm,0 and
its complex conjugate ψm,0 as a representation of SO0(2, 2) (see [10, Claim 9.12]). By (2.4),
we have
ψm,k(π(x)) = e−2
√
−1(mθ1+(m+k)θ2) tanhk t cosh−2m t (2.6)
for x = k(θ1) a(t) k(θ2) ∈ H2,1. We refer to ψm,k as a spherical function of type (−m,m + k)
in accordance with the action of SO(2)× SO(2).
2.3 Convergence of generalized Poincaré series
In this subsection, we explain the fact about the discrete spectrum of locally symmetric spaces
by Kassel–Kobayashi [10] in our AdS3 setting. We use the following notation.
Notation 2.2.
� Let `G = PSL(2,R) = SL(2,R)/{±1} and G = `G×`G.
� Let `K = PSO(2) = SO(2)/{±1} and K = `K ×`K.
� Let E and `E be respectively the identity elements of G and `G.
Remark 2.3. The double covering SO0(2, 2) → G induces an isomorphism AdS3 ∼= G/diag`G
(∼= `G). From now on, we consider only discontinuous groups Γ for AdS3 which are discrete
subgroups of G. This is enough for our purpose.
In order to study Specd
(
�Γ\AdS3
)
, Kassel–Kobayashi [10] considered the convergence and
non-vanishing of generalized Poincaré series
ϕΓ(Γx) :=
∑
γ∈Γ
ϕ
(
γ−1x
)
(2.7)
for K-finite square-integrable eigenfunctions ϕ of �AdS3 . For this, they used an analytic estimate
of ϕ and a geometric estimate of the number of Γ-orbits
NΓ(x,R) := #{γ ∈ Γ | γx ∈ B(R)} (2.8)
in the pseudo-ball B(R) for R > 0. Since the Γ-action is properly discontinuous and B(R) is
compact, we have NΓ(x,R) <∞.
The convergence of generalized Poincaré series is proved by [10] as follows. For g ∈ G and
a function f on AdS3, `∗gf is defined by `∗gf(x) = f
(
g−1x
)
.
Fact 2.4 (Kassel–Kobayashi [10]). Let Γ ⊂ G be a discontinuous group for AdS3 satisfying the
exponential growth condition
∃A, a > 0, ∀x ∈ AdS3, ∀R > 0, NΓ(x,R) < AeaR. (2.9)
Then, for any K-finite eigenfunction ϕ of �AdS3 with eigenvalue λm and any g ∈ G, if m > a,
then (`∗gϕ)Γ (see (2.7)) is continuous and square-integrable on Γ\AdS3 and an eigenfunction
of �Γ\AdS3 with eigenvalue λm.
6 K. Kannaka
Remark 2.5.
(1) Fact 2.4 does not assert the non-vanishing of the series (`∗gϕ)Γ which is more involved.
Kassel–Kobayashi [10] proved that there exists g ∈ G such that (`∗gψm,0)Γ 6= 0 for suffi-
ciently large m ∈ N.
(2) By [10, Lemma 4.6.4], if a discontinuous group Γ is sharp in the sense of [10, Defini-
tion 4.2], then Γ satisfies the exponential growth condition (2.9). Moreover, Kassel [8] and
Guériataud–Kassel [6] proved that finitely generated discontinuous groups for AdS3 are
always sharp (see Fact 4.5 below).
(3) There exist discontinuous groups which do not satisfy the exponential growth condi-
tion (2.9). Indeed, for any increasing function f : R → R>0 and any x ∈ AdS3, we const-
ructed a discontinuous group Γf,x for AdS3 satisfying NΓf,x(x,R) > f(R) 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 =
AdS3. In this case, the quotient space XΓ = Γ\X is an orbifold. To formulate more precisely
in the orbifold case, we observe that the quotient space XΓ is Hausdorff, and carries a natural
Radon measure (see, e.g., [2, Chapter VII, Section 2, No. 2, Proposition 4]). A continuous
function g on XΓ is smooth if the pull-back p∗Γg is a smooth function on X, where pΓ : X → XΓ
is the natural quotient map. We write C∞c (XΓ) for the set of smooth functions on XΓ with
compact support. For g ∈ C∞c (XΓ), we define �XΓ
g ∈ C∞c (XΓ) by identifying it with the
Γ-invariant function �X(p∗Γg). For λ ∈ C, we define
L2
λ(XΓ) :=
{
f ∈ L2(XΓ) | ∀g ∈ C∞c (XΓ), 〈f,�XΓ
g〉XΓ
= λ〈f, g〉XΓ
}
.
The discrete spectrum Specd(�XΓ
) and its multiplicity NXΓ
are defined similarly to the case
where Γ acts also freely.
2.4 “Injectivity radii” of anti-de Sitter 3-manifolds
Let Γ be a discontinuous group for AdS3. 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 AdS3(∼= `G) by (γ1, γ2)x = γ1xγ
−1
2
for (γ1, γ2) ∈ Γ and x ∈ `G. We set
εΓ := inf
(γ1,γ2)∈Γ\{E}
1
3
∣∣‖γ1‖ − ‖γ2‖
∣∣. (2.10)
By the inequality (see, e.g., [7, Lemma 5.5])
‖(g1, g2)x‖ ≥
∣∣‖g1‖ − ‖g2‖
∣∣− ‖x‖ for (g1, g2) ∈ G and x ∈ AdS3,
we get:
Lemma 2.6. If εΓ > 0, then γB(εΓ) ∩B(εΓ) = ∅ for all γ ∈ Γ \ {E}.
Proposition 2.7. Let Γ be a discrete subgroup of G acting properly discontinuously on AdS3.
Then there exists g ∈ G satisfying εg−1Γg > 0.
Remark 2.8. One sees in the proof below that the set of such g is dense in G.
Proposition 2.7 follows obviously from the proper discontinuity of the Γ-action and the fol-
lowing lemma:
Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds 7
Lemma 2.9. For any countable subset Γ of G, there exists g ∈ G such that ‖γ1‖ 6= ‖γ2‖ for all
γ = (γ1, γ2) ∈ g−1Γg \ {E}.
Proof of Lemma 2.9. For γ ∈ Γ, the map fγ : G→ G defined by g 7→ g−1γg is real analytic.
For the analytic subset F = {(g1, g2) ∈ G | ‖g1‖ = ‖g2‖} of G, we claim that the set f−1
γ (F )
is a proper subset of G if γ 6= E. For this, we may assume γ1 6= `E without loss of generality.
Then there exists g1 ∈ `G satisfying ‖g−1
1 γ1g1‖ 6= ‖γ1‖ as one can find g1 depending on the three
cases where γ1 is hyperbolic, parabolic, or elliptic. Hence (g1,`E) /∈ f−1
γ (F ) if ‖γ1‖ = ‖γ2‖, and
E /∈ f−1
γ (F ) if not. Thus f−1
γ (F ) is a proper subset of G.
Therefore the analytic set f−1
γ (F ) has no interior point, and thus so does the countable union⋃
γ∈Γ\{E} f
−1
γ (F ) by the Baire category theorem (see, e.g., [22, Theorem 2.2]). Hence there
exists an element g of G \
⋃
γ∈Γ\{E} f
−1
γ (F ) and we have ‖γ1‖ 6= ‖γ2‖ for all γ = (γ1, γ2) ∈
g−1Γg \ {E}. �
3 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 ψm,k of type (−m,m + k) defined
in Section 2.2. By choosing k = 3j (j = 0, 1, 2, . . .), we prove:
Theorem 3.1. If Γ is a discontinuous group for AdS3 satisfying the exponential growth condi-
tion (2.9), then
lim
m→∞
NΓ\AdS3(λm) =∞.
Theorem 1.1 is a direct consequence of Theorem 3.1 by Remark 2.5(2).
Proposition 3.2. Let Γ be a discrete subgroup of G acting properly discontinuously on AdS3
and satisfying the exponential growth condition (2.9). If εΓ > 0, then there exists a real number
mΓ(k) (given explicitly by (3.1)) for k ∈ N such that {(Re(ψm,3j ))
Γ}k−1
j=0 ⊂ L2
λm
(Γ\AdS3) are
linearly independent for all integers m > mΓ(k).
Postponing the proof of Proposition 3.2 until the end of this section, we prove Theorem 3.1.
Proof of Theorem 3.1. We have an obvious equality of the multiplicity of L2-eigenvalues,
NΓ\AdS3 = N(g−1Γg)\AdS3 for any g ∈ G through the natural isomorphism Γ\AdS3 ∼=
(
g−1Γg
)
\AdS3 as Lorentzian manifolds. By replacing Γ with g−1Γg if necessary, we may and do as-
sume εΓ > 0 by Proposition 2.7. Then Proposition 3.2 implies that L2
λm
(
Γ\AdS3
)
contains
at least k linearly independent elements if m > mΓ(k) for any fixed k ∈ N, which means
dimC L
2
λm
(Γ\AdS3) ≥ k. Hence Theorem 3.1 follows. �
Kassel–Kobayashi [10] proved the non-vanishing of the generalized Poincaré series (ψm,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
ψm,0(̀ E) = 1. Our strategy for the proof of Proposition 3.2 is along the same line, however,
there are some technical difficulties since ψm,k for k ≥ 1 vanishes at the origin. We then make
use of an observation that ψm,k decays more slowly at the origin than at infinity, to be precise,
by the following formula, see (2.6):
|ψm,k(x)| = cosh−2m(‖x‖/2) tanhk(‖x‖/2).
8 K. Kannaka
Actually, we use an analytic lemma (Lemma 3.3) to prove that the first term in the generalized
Poincaré series (ψm,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(ψm,k))
Γ do not cancel each other for any linear combination.
For C, a, ε > 0 and s ∈ N, we set
m(C, a, ε, s) :=
(log 2)s+ 2aε+ log
(
1 + 2sCe6aε
)
log cosh ε
and
m̃(C, a, δ, s) := inf
0<ε<δ
m(C, a, ε, s).
Note that m̃(C, a, δ, s) = O
(
δ−2
)
as δ → 0 and = O(1) as δ →∞.
Lemma 3.3. For any integer m > m(C, a, ε, s) and any one-variable polynomial f of degree
≤ s with non-negative coefficients,
C
∞∑
n=1
e4a(n+1)ε(cosh 2nε)−mf(tanh 2(n+ 1)ε) < (cosh ε)−mf(tanh ε).
Proof. We may assume that f(x) = xj for j = 0, 1, . . . , s. Since
1 ≤ tanhnx
tanhx
≤ n, (coshx)n ≤ coshnx
for x ∈ R, we have
(LHS)/(RHS) = C
∞∑
n=1
e4a(n+1)ε
(
cosh 2nε
cosh ε
)−m(tanh 2(n+ 1)ε
tanh ε
)j
≤ Ce6aε
∞∑
n=1
(
e2aε(cosh ε)−m
)2n−1
(2(n+ 1))s.
We set d := e2aε(cosh ε)−m. Then d < 1 by m > m(C, a, ε, s). Since n + 1 ≤ 2n for all n ∈ N,
we have
(LHS)/(RHS) ≤ 2sCe6aε
∞∑
n=1
(2sd)n = 2sCe6aε 2sd
1− 2sd
.
Again by m > m(C, a, ε, s), we have 2sd <
(
1 + 2sCe6aε
)−1
. Therefore we obtain
(LHS)/(RHS) < 1. �
Let χ : {±1} → {0, 1} be the map defined by χ(1) = 0 and χ(−1) = 1. For a = (aj)
k−1
j=0 ∈ {±1}k
and an odd integer N ≥ 3, we set
θa,N := π
k−1∑
i=0
(χ(ai)− χ(ai−1))N−i.
Here we use the convention a−1 = 1.
Lemma 3.4. For any a = (a0, . . . , ak−1) ∈ {±1}k and any odd integer N , we have
aj cos
(
N jθa,N
)
> 0 for j = 0, 1, . . . , k − 1.
Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds 9
Proof. Since Nk−1θa,N ≡ πχ(ak−1) (mod 2π), we have cos
(
Nk−1θa,N
)
= ak−1. It is easy
to check that |N jθa,N − N jθ(a0,··· ,aj),N | < π/2 for j = 0, 1, . . . , k − 1, hence the signature
of cos(N jθa,N ) is equal to that of cos(N jθ(a0,··· ,aj),N ) = aj . �
Remark 3.5. We have used the geometric progression (N j)k−1
j=0 in Lemma 3.4. On the other
hand, an analogous statement does not hold if we use arithmetic progressions. For example,
there does not exist θ ∈ R satisfying aj cosmjθ > 0 for all j = 0, 1, 2, 3, 4 if we choose (aj)
4
j=0 =
(1, 1, 1,−1, 1) and an arithmetic progression (mj)
4
j=0.
For a discontinuous group Γ and k ∈ N, one can take mΓ(k) in Proposition 3.1 by
mΓ(k) = inf
(A,a)∈Cexp(Γ)
max
{
m̃
(
3k−1A, a, εΓ/4, 3
k−1
)
/2, a
}
, (3.1)
where Cexp(Γ) :=
{
(A, a) ∈ R2 | ∀x ∈ AdS3,∀R > 0, NΓ(x,R) < AeaR
}
. Here, we adopt
the convention that inf∅ f = ∞ for a real-valued function f . In particular, mΓ(k) = ∞ when
Cexp(Γ) = ∅ or εΓ = 0.
Proof of Proposition 3.2. By the exponential growth condition (2.9), Cexp(Γ) 6= ∅ and thus
mΓ(k) < ∞. We take an integer m > mΓ(k). Then there exist ε with 0 < ε < εΓ/4 and
(A, a) ∈ Cexp(Γ) satisfying the inequality m > max
{
m
(
3k−1A, a, ε, 3k−1
)
/2, a
}
.
To see C-linear independence of the real-valued functions
{
(Re(ψm,3j ))
Γ
}k−1
j=0
, it is enough to
prove the non-vanishing of the real part Re
(
ψΓ
m,b
)
= (Re(ψm,b))
Γ of the generalized Poincaré
series of a linear combination
ψm,b :=
k−1∑
j=0
bjψm,3j
for any b = (b0, b1, . . . , bk−1) ∈ Rk \ {0}. By Lemma 2.6, for x ∈ B(4ε), we have
ψΓ
m,b(Γx) = ψm,b(x) +
∑
γ∈Γ
‖γ−1x‖>4ε
ψm,b
(
γ−1x
)
. (3.2)
By (2.6), for any y ∈ AdS3, we get
|ψm,b(y)| ≤
(
cosh
‖y‖
2
)−2m k−1∑
j=0
|bj |
(
tanh
‖y‖
2
)3j
.
We define a = (aj)
k−1
j=0 by aj = 1 for bj ≥ 0 and aj = −1 for bj < 0, and set
fb(u) :=
k−1∑
j=0
bj cos
(
3jθa,3
)
u3j .
We note that all the coefficients of fb are non-negative by Lemma 3.4. Moreover, we get∣∣ cos
(
3jθa,3
)∣∣−1 ≤ 3k−1 for all j = 0, 1, . . . , k − 1 by using the inequality sin(πx/2) ≥ x
for 0 ≤ x ≤ 1. Thus
|ψm,b(y)| ≤ 3k−1
(
cosh
‖y‖
2
)−2m
fb
(
tanh
‖y‖
2
)
10 K. Kannaka
and, for any x ∈ B(4ε), we have∣∣∣∣ ∑
γ∈Γ
‖γ−1x‖>4ε
Re
(
ψm,b
(
γ−1x
))∣∣∣∣ ≤ ∞∑
n=1
∑
γ∈Γ
4εn<‖γ−1x‖≤4ε(n+1)
|ψm,b(γ−1x)|
≤ 3k−1
∞∑
n=1
NΓ(x, 4ε(n+ 1)) (cosh 2εn)−2m fb (tanh 2ε(n+ 1))
≤ 3k−1A
∞∑
n=1
e4aε(n+1) (cosh 2εn)−2m fb (tanh 2ε(n+ 1))
< (cosh ε)−2m fb (tanh ε) . (3.3)
The third and forth inequalities respectively follow from the exponential growth condition (2.9)
and Lemma 3.3. On the other hand, we set
xa,ε := k
(
θa,3
2
)
a(ε)k
(
θa,3
2
)−1
∈ B(4ε).
Then it follows from (2.6) that
Reψm,b(xa,ε) = (cosh ε)−2m fb (tanh ε) . (3.4)
By (3.2), (3.3), and (3.4), we obtain (Re(ψm,b))
Γ(Γxa,ε) 6= 0. Hence we complete the proof by
the continuity of ψΓ
m,b (Fact 2.4). �
4 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.
Setting 4.1.
� ∆ is a discrete subgroup of `G = PSL(2,R).
� j, ρ : ∆→ `G are two group homomorphisms with j injective and discrete.
� ∆j,ρ is a discrete subgroup of G = `G×`G given by {(j(γ), ρ(γ)) | γ ∈ ∆}.
We use the following structural results of discontinuous groups for the proof of Theorem 1.4.
Fact 4.2 ([10, Lemma 9.2]). Let Γ be a finitely generated discrete subgroup of G acting properly
discontinuously on AdS3. 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,
type (ii) Γ is contained in a conjugate of `G× `K or `K × `G.
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 AdS3:
Definition 4.3 ([10, Definition 9.1]). A discontinuous group Γ for AdS3 is called convex cocom-
pact 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 AdS3 if and only if j(∆) is co-
compact in `G because ∆j,ρ is isomorphic to j(∆) as abstract groups. By Fact 4.2, discontinuous
groups acting cocompactly on AdS3 are convex cocompact.
Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds 11
4.1 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 CLip(j, ρ)
introduced by Kassel [8] and Guéritaud–Kassel [6], which quantifies the properness of the action
of ∆j,ρ on AdS3.
Definition 4.4. Let dH2 be the hyperbolic distance of the 2-dimensional hyperbolic space
H2(∼= `G/̀ K). In Setting 4.1, we denote by CLip(j, ρ) the infimum of Lipschitz constants
Lip(f) = sup
y 6=y′
dH2(f(y), f(y′))
dH2(y, y′)
of maps f : H2 → H2 that are (j, ρ)-equivariant.
The map (j, ρ) 7→ CLip(j, ρ) is continuous over the set of (j, ρ) ∈ Hom(∆,`G)2 such that j is
injective and j(∆) is convex cocompact in `G [6, Proposition 1.5].
Fact 4.5 ([6, 8]). Assume that ∆ is finitely generated. Then the action of ∆j,ρ on AdS3 is
properly discontinuous if and only if min
{
CLip(j, ρ), CLip(ρ, j)
}
< 1.
Remark 4.6. In the setting of Fact 4.5, if CLip(ρ, j) < 1, then ρ is injective and discrete.
Moreover, if j(∆) is convex cocompact, then so is ρ(∆).
Therefore, Theorem 1.4 for Γ of type (i) reduces to the following:
Theorem 4.7. In Setting 4.1, we assume that ∆ is finitely generated and that CLip(j, ρ) < 1.
Then there exists a constant µ1 > 0 independent of j, ρ and ∆ such that for any m, k ∈ N with
m > 3kµ1(1− CLip(j, ρ))−2,
N∆j,ρ\AdS3(λm) ≥ k.
For the proof of Theorem 4.7, we need two results from Kassel–Kobayashi [10] applied to
our setting G = `G×`G. If a discontinuous group Γ satisfies the assumption of Fact 4.8 below,
then it is ((1 − α)/2, 0)-sharp in the sense of [10, Definition 4.2]. Hence we get the following
by applying [10, Lemma 4.6.4]:
Fact 4.8 ([10]). Let Γ ⊂ G be a discontinuous group for AdS3. We assume that there exists
0 ≤ α < 1 such that ‖γ2‖ ≤ α‖γ1‖ or ‖γ1‖ ≤ α‖γ2‖ for any (γ1, γ2) ∈ Γ. Then there exists
c > 0 independent of α and Γ such that for any x ∈ AdS3 and any R > 0,
NΓ(x,R) ≤ #(Γ ∩K)ce8R(1−α)−1
.
The following theorem traces back to the Kazhdan–Margulis theorem for discrete subgroups
of semisimple groups.
Fact 4.9 ([10, Proposition 8.14]). There exists a constant r > 0 satisfying the following property:
for any discrete subgroup `Γ of `G, there exists `g ∈ `G such that ‖`γ‖ ≥ r for all `γ ∈ `g−1`Γ`g\
{`E}.
In the following, we use the upper half plane model
{
z = x+
√
−1y ∈ C | Im z > 0
}
equipped
with the metric tensor ds2 =
(
dx2 + dy2
)
/y2 for the hyperbolic space H2. Then ‖̀ g‖ is equal to
the hyperbolic distance dH2
(̀
g
√
−1,
√
−1
)
for g̀ ∈ AdS3 ∼= `G (see, e.g., [6, equation (A.1)]).
12 K. Kannaka
Proof of Theorem 4.7. The idea of the proof is similar to [10, Theorem 9.9], however, we
give a proof for the sake of completeness. By Fact 4.9, replacing j by some conjugate under `G,
we may assume ‖j(γ)‖ ≥ r for any γ ∈ ∆ \ {̀ E}. In particular, Γ∩K = {E} for such j and for
any ρ. We fix δ > 0 such that
α := CLip(j, ρ) + δ < 1.
Then, replacing ρ by some conjugate under `G, we may assume
‖ρ(γ)‖ ≤ α‖j(γ)‖ for any γ ∈ ∆. (4.1)
Indeed, by Definition 4.4, there exists a (j, ρ)-equivariant map fδ : H2 → H2 satisfying Lip(fδ)
< α. We take gδ ∈ `G such that gδ
√
−1 = fδ
(√
−1
)
. Then, for any γ ∈ ∆, we have∥∥g−1
δ ρ(γ)gδ
∥∥ = dH2
(
fδ(
√
−1), ρ(γ)fδ
(√
−1
))
< αdH2
(√
−1, j(γ)
√
−1
)
= α‖j(γ)‖.
Hence (4.1) holds by replacing ρ with g−1
δ ρ(·)gδ, and therefore we get
NΓ(x,R) ≤ ce8R(1−(CLip(j,ρ)+δ))−1
by Fact 4.8. Then the constant εΓ in (2.10) has the following lower bound:
3εΓ = inf
γ∈∆\{`E}
|‖j(γ)‖ − ‖ρ(γ)‖| ≥ inf
γ∈∆\{`E}
(1− α)‖j(γ)‖ ≥ r(1− α).
Note that log cosh t = O
(
t2
)
as t→ 0. By the explicit description (3.1) of mΓ(k), Theorem 4.7
follows from Proposition 3.2. �
4.2 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 [17] and Kassel [9] applied to our AdS3 setting, which gives the sta-
bility for properness under any small deformation of standard convex cocompact discontinuous
groups.
Fact 4.10 ([9, Theorem 1.4]). Let Γ be a convex cocompact discrete subgroup of `G×`K. Then
for any α, β > 0, there exists a neighborhood W ⊂ Hom(Γ, G) of the natural inclusion Γ ⊂ G
such that for any ϕ ∈W ,
|µ(ϕ(γ))− µ(γ)| ≤
{
α |µ(γ)| if γ ∈ Γ \K,
β if γ ∈ Γ ∩K,
where µ(g1, g2) := (‖g1‖, ‖g2‖) ∈ R2 for (g1, g2) ∈ G, ‖ · ‖ is given in Definition 2.1, and
|(x1, x2)| :=
√
x2
1 + x2
2 for (x1, x2) ∈ R2.
We introduce the following terminology for the estimate of the discrete spectrum since a dis-
continuous group Γ is not necessarily torsion-free. Let prj : G = `G × `G → `G be the j-th
projection (j = 1, 2). In the following definition, we assume that pr2(Γ) is bounded. Then the
group Γ1 := ker(pr1 |Γ) is cyclic since Γ1 is a discrete subgroup of a conjugate of the product
group {̀ E} ×`K (∼= R/Z).
Definition 4.11. A discrete subgroup Γ of G is said to be standard of class n if pr2(Γ) is
bounded and the cyclic group Γ1 = ker(pr1 |Γ) is of order n.
Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds 13
Remark 4.12.
(1) If Γ is torsion-free, then it is of class 1.
(2) If pr2(Γ) is bounded for a discrete subgroup Γ of G, then the group pr1(Γ) is discrete
in `G. Moreover, if Γ is of class 1, then it is of the form ∆j,ρ such that ∆ = pr1(Γ)
and CLip(j, ρ) = 0.
Let r > 0 be the constant in Fact 4.9. For an integer n ≥ 2, we define a positive number ηn by
cosh ηn := 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∥∥∥∥a(r8
)−1
k
(
jπ
n
)
a
(
r
8
)∥∥∥∥ ≥ ηn for j = 1, . . . , n− 1.
We give a uniform estimate of εΓ in (2.10) and NΓ(x,R) in (2.8) for standard discrete sub-
groups 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 εg−1Γg ≥ min{ηn/3, r/6} and Ng−1Γg(x,R) < ce16R for any x ∈ AdS3 and any R > 0.
Proof. Let Γ1 = ker(pr1 |Γ) as in Definition 4.11. Since Γ is of class n, the group pr2(Γ1) is
generated by k(π/n) ∈ `G = PSL(2,R). We take g̀ ∈ `G in Fact 4.9 applied to `Γ = pr1(Γ)
and set g := (̀ g, a(r/8)) ∈ G. Replacing Γ by g−1Γg, we get ‖γ1‖ ≥ r for (γ1, γ2) ∈ Γ \ Γ1 by
Fact 4.9 and ‖γ2‖ ≥ ηn for (γ1, γ2) ∈ Γ1 \ {E} by Lemma 4.13. Moreover, if (γ1, γ2) ∈ Γ, then
‖γ2‖ = ‖a(r/8)−1ka(r/8)‖ for some k ∈ `K, hence ‖γ2‖ ≤ r/2 because ‖g1g2‖ ≤ ‖g1‖+ ‖g2‖ for
g1, g2 ∈ `G and since ‖a(t)‖ = 2t for t ≥ 0 and ‖k‖ = 0 for k ∈ `K. To summarize,‖γ2‖ ≤ r
2 ≤
‖γ1‖
2 if (γ1, γ2) ∈ Γ \ Γ1,
‖γ2‖ ≥ ηn if (γ1, γ2) ∈ Γ1 \ {E}.
Then εΓ ≥ min{ηn/3, r/6} and Γ ∩K = {E}. Moreover, ‖γ1‖ ≤ ‖γ2‖/2 or ‖γ2‖ ≤ ‖γ1‖/2 for
any (γ1, γ2) ∈ Γ and thus NΓ(x,R) < ce16R for any x ∈ AdS3 and any R > 0 by Fact 4.8. �
Theorem 4.15. There exists a constant µn > 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 > 3kµn,
ÑΓ\AdS3(λm) ≥ 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 Γ1 = ker(pr1 |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 ‖γ1‖ ≥ r for any (γ1, γ2) ∈ Γ \ Γ1. 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 Γ1 is injective and the inequalities‖ϕ1(γ)‖ ≥ 1
2r, ‖ϕ2(γ)‖ ≤ 1
2‖ϕ1(γ)‖ if γ ∈ Γ \ Γ1,
|µ(ϕ(γ))| < 1
2r if γ ∈ Γ1
(4.2)
hold where ϕi = pri ◦ϕ for i = 1, 2. Then ϕ is injective and discrete.
14 K. Kannaka
We claim ϕ1(Γ1) is trivial. Indeed, if there exists γ ∈ Γ1\{E} such that ϕ1(γ) 6= `E, then the
normalizer of ϕ(Γ1) in G is contained in `K1×`G, where `K1 is the maximal compact subgroup
of `G containing ϕ1(Γ1). Hence ϕ(Γ) ⊂ `K1 × `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 ϕ1(Γ1) is
trivial and ϕ2(Γ1) is non-trivial. Hence the normalizer of ϕ(Γ1) in G is contained in `G×`K2,
where `K2 is the maximal compact subgroup of `G containing ϕ2(Γ1). Therefore pr2(ϕ(Γ)) is
bounded. Moreover ϕ(Γ)1 = ϕ(Γ1) by the inequalities (4.2), hence the discrete subgroup ϕ(Γ) is
standard of class n. By the explicit description (3.1) of mΓ(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
ÑΓ\AdS3(λm) =∞
for any convex cocompact standard discrete subgroup Γ of class n ≥ 2 and any integer m > 3µn
if the following statement holds: NΓ\AdS3(λm) = ∞ for any standard discrete subgroup Γ and
any m ∈ N such that NΓ\AdS3(λm) ≥ 1. The latter statement is discussed in [13] by using
discretely decomposable blanching laws of unitary representations (cf. [11]).
Thus the proof of Theorem 1.4 is completed.
Acknowledgements
The author would like to express his sincere gratitude to Professor Toshiyuki Kobayashi whose
suggestions led him to study the multiplicities of L2-eigenvalues for anti-de Sitter manifolds.
He also would like to show his appreciation to Dr. Hiroyoshi Tamori whose comments led him
to an explicit description of m(C, a, ε, s) in Lemma 3.3. Thanks are also due to the anonymous
referees for their helpful comments to improve the paper. This work was supported by JSPS
KAKENHI Grant Number 18J20157 and the Program for Leading Graduate Schools, MEXT,
Japan.
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https://arxiv.org/abs/1912.12601
https://doi.org/10.3792/pjaa.96.013
https://doi.org/10.3792/pjaa.96.013
https://arxiv.org/abs/2001.03292
https://doi.org/10.1007/BF01445255
https://doi.org/10.1007/BF01443517
https://doi.org/10.1007/s002080050153
https://doi.org/10.1007/978-3-642-56478-9_37
https://doi.org/10.1007/978-981-10-2636-2_6
https://arxiv.org/abs/1609.05986
https://doi.org/10.1142/S0129167X06003862
https://arxiv.org/abs/math.DG/0603318
https://doi.org/10.4310/jdg/1214439564
https://doi.org/10.4310/jdg/1214439564
https://doi.org/10.5802/aif.1754
1 Introduction
2 Preliminaries about the anti-de Sitter space
2.1 Some coordinates and ``pseudo-balls''
2.2 Square-integrable eigenfunctions of the Laplacian on the anti-de Sitter space
2.3 Convergence of generalized Poincaré series
2.4 ``Injectivity radii'' of anti-de Sitter 3-manifolds
3 Proof of Theorem 1.1
4 Proof of Theorem 1.4
4.1 Proof of Theorem 1.4 for Gamma of type (i)
4.2 Proof of Theorem 1.4 for Gamma of type (ii)
References
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| id | nasplib_isofts_kiev_ua-123456789-211307 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T13:22:12Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kannaka, Kazuki 2025-12-29T11:07:58Z 2021 Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds. Kazuki Kannaka. SIGMA 17 (2021), 042, 15 pages 1815-0659 2020 Mathematics Subject Classification: 58J50; 53C50; 22E40 arXiv:2005.03308 https://nasplib.isofts.kiev.ua/handle/123456789/211307 https://doi.org/10.3842/SIGMA.2021.042 Let Γ be a discrete group acting properly discontinuously and isometrically on the three-dimensional anti-de Sitter space AdS³, and □ the Laplacian, which is a second-order hyperbolic differential operator. We study the linear independence of a family of generalized Poincaré series introduced by Kassel-Kobayashi [Adv. Math. 287 (2016), 123-236, arXiv:1209.4075], which are defined by the Γ-average of certain eigenfunctions on AdS³. We prove that the multiplicities of 𝐿²-eigenvalues of the hyperbolic Laplacian □ on Γ∖AdS³ are unbounded when Γ is finitely generated. Moreover, we prove that the multiplicities of stable 𝐿²-eigenvalues for compact anti-de Sitter 3-manifolds are unbounded. The author would like to express his sincere gratitude to Professor Toshiyuki Kobayashi, whose suggestions led him to study the multiplicities of 𝐿²-eigenvalues for anti-de Sitter manifolds. He also would like to show his appreciation to Dr. Hiroyoshi Tamori, whose comments led him to an explicit description of 𝑚(𝐶, 𝑎, ε, s) in Lemma 3.3. Thanks are also due to the anonymous referees for their helpful comments to improve the paper. This work was supported by JSPS KAKENHI Grant Number 18J20157 and the Program for Leading Graduate Schools, MEXT, Japan. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds Article published earlier |
| spellingShingle | Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds Kannaka, Kazuki |
| title | Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds |
| title_full | Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds |
| title_fullStr | Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds |
| title_full_unstemmed | Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds |
| title_short | Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds |
| title_sort | linear independence of generalized poincaré series for anti-de sitter 3-manifolds |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211307 |
| work_keys_str_mv | AT kannakakazuki linearindependenceofgeneralizedpoincareseriesforantidesitter3manifolds |