Linear Independence of Generalized Poincaré Series for Anti-de Sitter
3-Manifolds

Linear Independence of Generalized Poincaré Series
for Anti-de Sitter 3-Manifolds

Kazuki KANNAKA

K. Kannaka

RIKEN iTHEMS, Wako, Saitama 351-0198, Japan<EMAIL_ADDRESS>
Received May 13, 2020, in final form April 13, 2021; Published online April
23, 2021

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 Poincaré series introduced by Kassel–Kobayashi [Adv. Math. 287
(2016), 123–236, arXiv:1209.4075], 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 stable
$L^{2}$-eigenvalues for compact anti-de Sitter $3$-manifolds are unbounded.

anti-de Sitter $3$-manifold; Laplacian; stable $L^{2}$-eigenvalue

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
$\square_{M}:=\operatorname{div}_{M}\circ\operatorname{grad}_{M}$ 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
$\lambda\in\mathbb{C}$, we denote by

$\displaystyle L^{2}_{\lambda}(M):=\big{\\{}f\in
L^{2}(M)\mid\square_{M}f=\lambda f\text{ in the weak sense}\big{\\}}.$

The set of $L^{2}$-eigenvalues
$\mathrm{Spec}_{d}(\square_{M}):=\big{\\{}\lambda\in\mathbb{C}\mid
L^{2}_{\lambda}(M)\neq 0\big{\\}}$ is called the discrete spectrum of
$\square_{M}$.

Our interest is the multiplicities of $L^{2}$-eigenvalues $\lambda$ of
$\square_{M}$, denoted by

$\displaystyle\mathcal{N}_{M}(\lambda):=\operatorname{dim}_{\mathbb{C}}L^{2}_{\lambda}(M)\in\mathbb{N}\cup\\{\infty\\}.$

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 [19], and later Fox–Strichartz
[4], investigated the distribution of the discrete spectrum of the Laplacian
$\square_{M}$ of some pseudo-Riemannian manifolds, i.e., when $M$ is the flat
pseudo-Riemannian manifold $\mathbb{R}^{p,q}/\mathbb{Z}^{p+q}$ and is the
Lorentzian manifold $S^{1}\times S^{q}$, respectively.

Let us recall some basic notions. A discontinuous group for a homogeneous
manifold $X\\!=\\!G/H$ is a discrete subgroup $\Gamma$ of $G$ acting properly
discontinuously and freely on $X$ (Kobayashi [18, Definition 1.3]). In this
case, the quotient space $X_{\Gamma}:=\Gamma\backslash X$ carries a
$C^{\infty}$-manifold structure such that the quotient map $p_{\Gamma}\colon
X\rightarrow X_{\Gamma}$ is a covering of $C^{\infty}$ class, hence
$X_{\Gamma}$ has a $(G,X)$-structure induced by $p_{\Gamma}$. If we drop the
assumption of freeness, $X_{\Gamma}$ 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 $X_{\Gamma}$ to be Hausdorff in the
quotient topology. We remark that the action of a discrete subgroup $\Gamma$
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
$\mathrm{dS}^{n}:=\mathrm{SO}_{0}(n,1)/\mathrm{SO}_{0}(n-1,1)$ are always
finite groups (the Calabi–Markus phenomenon, see [3, 15]), there are a rich
family of discontinuous 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
$\mathrm{AdS}^{3}:=\mathrm{SO}_{0}(2,2)/(\\{\pm
1\\}\times\mathrm{SO}_{0}(2,1))$.

For $m\in\mathbb{N}$, we set

$\displaystyle\lambda_{m}:=4m(m-1).$

We prove:

######  Theorem 1.1.

For any finitely generated discontinuous group $\Gamma$ for
$\mathrm{AdS}^{3}$,

$\displaystyle\lim_{m\to\infty}\mathcal{N}_{\Gamma\backslash\operatorname{AdS}^{3}}(\lambda_{m})=\infty.$

######  Remark 1.2.

  1. (1)

A discontinuous group $\Gamma$ for $\mathrm{AdS}^{3}$ is called standard [10,
Definition 1.4] if it is contained in a reductive subgroup of
$\mathrm{SO}_{0}(2,2)$ which acts properly on $\mathrm{AdS}^{3}$ such as
$\mathrm{SU}(1,1)$. When $\Gamma$ is torsion-free and standard,
Kassel–Kobayashi [11, 12] established the theory of spectral decomposition of
$L^{2}(\Gamma\backslash\mathrm{AdS}^{3})$ into eigenfunctions of the
(hyperbolic) Laplacian. Moreover, a stronger result than Theorem 1.1 holds in
this case:
$\mathcal{N}_{\Gamma\backslash\mathrm{AdS}^{3}}(\lambda_{m})=\infty$ for
sufficiently large $m\in\mathbb{N}$ (Kassel–Kobayashi [13]). On the other
hand, a full spectral decomposition is not known. The construction of
$L^{2}$-eigenfunctions by generalized Poincaré series still works for the non-
standard case, showing that $\lambda_{m}$ is an $L^{2}$-eigenvalue on
$\Gamma\backslash\mathrm{AdS}^{3}$ for sufficiently large $m\in\mathbb{N}$
[10]. Theorem 1.1 is also applicable to non-standard $\Gamma$, for example, in
the case where $\Gamma$ is Zariski dense in $\mathrm{SO}(2,2)$.

  2. (2)

The assumption that $\Gamma$ is finitely generated could be relaxed. In fact,
the exponential growth condition (see (2.11)) for $\Gamma$-orbits is essential
in the proof of Theorem 1.1, and there exist infinitely generated
discontinuous groups $\Gamma$ satisfying (2.11) and the conclusion of Theorem
1.1 holds for such $\Gamma$ (see Theorem 3.1 which is proved without finitely
generated assumption).

  3. (3)

An analogous statement to Theorem 1.1 also holds when
$\Gamma\backslash\mathrm{AdS}^{3}$ is an orbifold. See Section 2.3 for the
argument when we drop the assumption that the $\Gamma$-action is free.

Now we consider a small deformation of a discrete subgroup. The study of
stability for properness 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
$L^{2}$-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
$X_{1}:=\mathrm{AdS}^{3}$ for $1\leq n\leq\infty$, and $G_{n}$ the Lie group
of its isometries. Every compact anti-de Sitter $3$-manifold $M$ is of the
form $M\cong\Gamma\backslash X_{n}$ for some finite $n$, where $\Gamma(\subset
G_{n})$ is a discontinuous group for $X_{n}$ by Kulkarni–Raymond [21, Theorem
7.2] and Klingler [14]. We take $n$ to be the smallest integer of this
property.

Let $\mathrm{Hom}(\Gamma,G_{n})$ be the set of group homomorphisms with
compact-open topology, and $\mathcal{U}_{\Gamma}$ the set of neighborhoods $W$
in $\mathrm{Hom}(\Gamma,G_{n})$ of the natural inclusion $\Gamma\subset G_{n}$
such that for any $\varphi\in W$, the map $\varphi$ is injective and
$\varphi(\Gamma)$ acts properly discontinuously on $X_{n}$. One knows
$\mathcal{U}_{\Gamma}\neq\varnothing$ [14, 17]. By definition, $\lambda$ is a
stable $L^{2}$-eigenvalue if $\min_{\varphi\in
W}\mathcal{N}_{\varphi(\Gamma)\backslash X_{n}}(\lambda)\neq 0$ for some
$W\in\mathcal{U}_{\Gamma}$. Moreover, for any $\lambda\in\mathbb{C}$ and any
inclusion $W^{\prime}\subset W$ in $\mathcal{U}_{\Gamma}$, we have an obvious
inequality

$\displaystyle\min_{\varphi\in
W^{\prime}}\mathcal{N}_{\varphi(\Gamma)\backslash
X_{n}}(\lambda)\geq\min_{\varphi\in W}\mathcal{N}_{\varphi(\Gamma)\backslash
X_{n}}(\lambda).$

######  Definition 1.3.

For a compact anti-de Sitter $3$-manifold $M$, we say that

$\displaystyle\widetilde{\mathcal{N}}_{M}(\lambda):=\sup_{W\in\mathcal{U}_{\Gamma}}\min_{\varphi\in
W}\mathcal{N}_{\varphi(\Gamma)\backslash X_{n}}(\lambda)$

is the multiplicity of a stable $L^{2}$-eigenvalue $\lambda$.

There exist infinitely many $m\in\mathbb{N}$ such that
$\widetilde{\mathcal{N}}_{M}(\lambda_{m})\geq 1$, namely $\lambda_{m}$ is a
stable $L^{2}$-eigenvalue for sufficiently large $m$ [10, Corollary 9.10].
However, to the best knowledge of the author, it is not known whether
$\widetilde{\mathcal{N}}_{M}(\lambda_{m})$ is finite. We prove:

######  Theorem 1.4.

For any compact anti-de Sitter $3$-manifold $M$,

$\displaystyle\lim_{m\to\infty}\widetilde{\mathcal{N}}_{M}(\lambda_{m})=\infty.$

The organization of this article is as follows. A key step to our proof is to
find a family of $L^{2}$-eigenfunctions of $\square_{\mathrm{AdS}^{3}}$ with
eigenvalue $\lambda_{m}$ on $\mathrm{AdS}^{3}$ 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
$\square_{\mathrm{AdS}^{3}}$ and their generalized Poincaré series which were
introduced in [10] as the $\Gamma$-average of these eigenfunctions. We then
give a uniform estimate of the “pseudo-distance” between the origin and the
second closest point of each $\Gamma$-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 $\mathrm{AdS}^{3}$.
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
$X=\mathrm{AdS}^{3}$. See also [7].

Let $Q$ be a quadratic form on $\mathbb{R}^{4}$ defined by
$Q(x)=x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2}$ for
$x=(x_{1},x_{2},x_{3},x_{4})$ and we set

$\displaystyle\mathbb{H}^{2,1}:=\big{\\{}x=(x_{1},x_{2},x_{3},x_{4})\in\mathbb{R}^{4}\mid
Q(x)=1\big{\\}}\cong\mathrm{SO}_{0}(2,2)/\mathrm{SO}_{0}(2,1).$

The tangent space $T_{x}(\mathbb{H}^{2,1})$ at $x\in\mathbb{H}^{2,1}$ is
isomorphic to the orthogonal complement $(\mathbb{R}x)^{\bot}$ with respect to
$Q$. Then $-Q|_{(\mathbb{R}x)^{\bot}}$ is a quadratic form of signature
$(2,1)$ on $T_{x}(\mathbb{H}^{2,1})\cong(\mathbb{R}x)^{\bot}$ and thus $-Q$
induces a Lorentzian structure on $\mathbb{H}^{2,1}$ with constant sectional
curvature $-1$. The $3$-dimensional anti-de Sitter space

$\displaystyle\mathrm{AdS}^{3}:=\mathbb{H}^{2,1}/\\{\pm
1\\}\cong\mathrm{SO}_{0}(2,2)/(\\{\pm 1\\}\times\mathrm{SO}_{0}(2,1)),$

inherits a Lorentzian structure through the double covering
$\pi\colon\mathbb{H}^{2,1}\rightarrow\mathrm{AdS}^{3}$.

###  2.1 Some coordinates and “pseudo-balls”

In this subsection, we work with coordinates on $\mathbb{H}^{2,1}$ and
consider “pseudo-balls” in $\mathrm{AdS}^{3}$. We identify $\mathbb{H}^{2,1}$
with $\mathrm{SL}(2,\mathbb{R})$ using the isomorphism

$\displaystyle\begin{array}[]{@{}ccc}\mathbb{H}^{2,1}&\xrightarrow{\cong}&\mathrm{SL}(2,\mathbb{R}),\\\
x=(x_{1},x_{2},x_{3},x_{4})&\longmapsto&\begin{pmatrix}x_{1}+x_{4}&-x_{2}+x_{3}\\\
x_{2}+x_{3}&x_{1}-x_{4}\end{pmatrix}\\!.\end{array}$ (2.3)

For $t\geq 0$ and $\theta\in\mathbb{R}$, we use the notations

$\displaystyle k(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\\!,\qquad a(t)=\begin{pmatrix}{\rm
e}^{t}&0\\\ 0&{\rm e}^{-t}\end{pmatrix}\\!.$ (2.4)

We embed $\mathbb{H}^{2,1}$ into $\mathbb{C}^{2}$ by

$\displaystyle
x\mapsto(z_{1},z_{2})=\big{(}x_{1}+\sqrt{-1}x_{2},x_{3}+\sqrt{-1}x_{4}\big{)}.$
(2.5)

We note that $z_{1}\neq 0$ if $x\in\mathbb{H}^{2,1}$. Via the identification
(2.3), we have

$\displaystyle(z_{1},z_{2})=\big{(}(\cosh t){\rm
e}^{\sqrt{-1}(\theta_{1}+\theta_{2})},(\sinh t){\rm
e}^{\sqrt{-1}(\theta_{1}-\theta_{2})}\big{)},$ (2.6)

if $x=k(\theta_{1})a(t)k(\theta_{2})\in\mathrm{SL}(2,\mathbb{R})$ (a “polar
coordinate”). In particular, we have

$\displaystyle\cosh 2t=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}.$

Next, we consider pseudo-balls on $\mathrm{AdS}^{3}$, as a special case of
Kassel–Kobayashi [10] for reductive symmetric spaces.

######  Definition 2.1.

For $x=(x_{1},x_{2},x_{3},x_{4})\in\mathbb{H}^{2,1}$,
$\|x\|\in\mathbb{R}_{\geq 0}$ is defined by

$\displaystyle\cosh\|x\|:=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}\qquad(=\cosh(2t)).$

This function is invariant under $x\mapsto-x$, hence defines a function on
$\mathrm{AdS}^{3}$, to be also denoted by $\|\cdot\|$ (a “pseudo-distance”
from the origin). The compact set

$\displaystyle B(R):=\big{\\{}y\in\mathrm{AdS}^{3}\mid\|y\|\leq R\big{\\}}$

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
$\square_{\mathrm{AdS}^{3}}$ with eigenvalues $\lambda_{m}=4m(m-1)$. We recall
from [10, Section 9] the following decomposition of the open subset
$\\{Q>0\\}$ of the flat pseudo-Riemannian manifold
$\mathbb{R}^{2,2}=\big{(}\mathbb{R}^{4},Q({\rm d}x)\big{)}$:

$\displaystyle\begin{array}[]{@{}ccc}\\{Q>0\\}&\xrightarrow{\cong}&\mathbb{R}_{>0}\times\mathbb{H}^{2,1},\\\\[4.30554pt]
x&\longmapsto&\big{(}\sqrt{Q(x)},x/\sqrt{Q(x)}\big{)}.\end{array}$

Let $r=\sqrt{Q(x)}$. Then one has, see [10, p. 215],

$\displaystyle-r^{2}\square_{\mathbb{R}^{2,2}}=-\left(r\frac{\partial}{\partial
r}\right)^{2}-2r\frac{\partial}{\partial r}+\square_{\mathbb{H}^{2,1}}.$ (2.7)

Let $m$ be a positive integer and $k$ be a non-negative integer. In the
coordinates (2.5), the homogeneous function $z_{1}^{-(k+2m)}z_{2}^{k}$ of
degree $-2m$ is harmonic with respect to $\square_{\mathbb{R}^{2,2}}$, hence
its restriction to the submanifold $\mathbb{H}^{2,1}$ is an eigenfuction of
$\square_{\mathbb{H}^{2,1}}$ with eigenvalue $\lambda_{m}=4m(m-1)$ by the
formula (2.7). Moreover, it is square-integrable with respect to the measure
$\sinh(2t){\rm d}\theta_{1}{\rm d}t{\rm d}\theta_{2}$ in the polar coordinate
(2.6) induced from the Lorentzian metric on $\mathbb{H}^{2,1}$, as in the
$k=0$ case [10, Section 9]. This $L^{2}$-eigenfunction is invariant under
$(z_{1},z_{2})\mapsto(-z_{1},-z_{2})$, hence defines a real analytic
$L^{2}$-eigenfunction on $\mathrm{AdS}^{3}$ with eigenvalue $\lambda_{m}$, to
be denoted by $\psi_{m,k}$. The discrete spectrum
$\mathrm{Spec}_{d}(\square_{\mathrm{AdS}^{3}})$ coincides with
$\\{\lambda_{m}\mid m\in\mathbb{N}\\}$ and
$L^{2}_{\lambda_{m}}\big{(}\mathrm{AdS}^{3}\big{)}$ is generated by
$\psi_{m,0}$ and its complex conjugate $\overline{\psi_{m,0}}$ as a
representation of $\mathrm{SO}_{0}(2,2)$ (see [10, Claim 9.12]). By (2.6), we
have

$\displaystyle\psi_{m,k}(\pi(x))={\rm
e}^{-2\sqrt{-1}(m\theta_{1}+(m+k)\theta_{2})}\tanh^{k}t\cosh^{-2m}t$ (2.8)

for $x=k(\theta_{1})\,a(t)\,k(\theta_{2})\in\mathbb{H}^{2,1}$. We refer to
$\psi_{m,k}$ as a spherical function of type $(-m,m+k)$ in accordance with the
action of $\mathrm{SO}(2)\times\mathrm{SO}(2)$.

###  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 $\mathrm{AdS}^{3}$ setting.
We use the following notation.

######  Notation 2.2.

  * •

Let $\grave{\ }G=\mathrm{PSL}(2,\mathbb{R})=\mathrm{SL}(2,\mathbb{R})/\\{\pm
1\\}$ and $G=\grave{\ }G\times\grave{\ }G$.

  * •

Let $\grave{\ }K=\mathrm{PSO}(2)=\mathrm{SO}(2)/\\{\pm 1\\}$ and $K=\grave{\
}K\times\grave{\ }K$.

  * •

Let $E$ and $\grave{\ }E$ be respectively the identity elements of $G$ and
$\grave{\ }G$.

######  Remark 2.3.

The double covering $\mathrm{SO}_{0}(2,2)\rightarrow G$ induces an isomorphism
$\mathrm{AdS}^{3}\cong G/\mathrm{diag}\grave{\ }G$ $(\cong\grave{\ }G)$. From
now on, we consider only discontinuous groups $\Gamma$ for $\mathrm{AdS}^{3}$
which are discrete subgroups of $G$. This is enough for our purpose.

In order to study
$\operatorname{Spec}_{d}\big{(}\square_{\Gamma\backslash\mathrm{AdS}^{3}}\big{)}$,
Kassel–Kobayashi [10] considered the convergence and non-vanishing of
generalized Poincaré series

$\displaystyle\varphi^{\Gamma}(\Gamma
x):=\sum_{\gamma\in\Gamma}\varphi\big{(}\gamma^{-1}x\big{)}$ (2.9)

for $K$-finite square-integrable eigenfunctions $\varphi$ of
$\square_{\mathrm{AdS}^{3}}$. For this, they used an analytic estimate of
$\varphi$ and a geometric estimate of the number of $\Gamma$-orbits

$\displaystyle N_{\Gamma}(x,R):=\\#\\{\gamma\in\Gamma\mid\gamma x\in B(R)\\}$
(2.10)

in the pseudo-ball $B(R)$ for $R>0$. Since the $\Gamma$-action is properly
discontinuous and $B(R)$ is compact, we have $N_{\Gamma}(x,R)<\infty$.

The convergence of generalized Poincaré series is proved by [10] as follows.
For $g\in G$ and a function $f$ on $\mathrm{AdS}^{3}$, $\ell_{g}^{*}f$ is
defined by $\ell_{g}^{*}f(x)=f\big{(}g^{-1}x\big{)}$.

######  Fact 2.4 (Kassel–Kobayashi [10]).

Let $\Gamma\subset G$ be a discontinuous group for $\mathrm{AdS}^{3}$
satisfying the exponential growth condition

$\displaystyle\exists A,a>0,\qquad\forall x\in\mathrm{AdS}^{3},\qquad\forall
R>0,\qquad N_{\Gamma}(x,R)<A{\rm e}^{aR}.$ (2.11)

Then, for any $K$-finite eigenfunction $\varphi$ of
$\square_{\mathrm{AdS}^{3}}$ with eigenvalue $\lambda_{m}$ and any $g\in G$,
if $m>a$, then $(\ell_{g}^{*}\varphi)^{\Gamma}$ $($see $\eqref{Poincare-
series})$ is continuous and square-integrable on
$\Gamma\backslash\mathrm{AdS}^{3}$ and an eigenfunction of
$\square_{\Gamma\backslash\mathrm{AdS}^{3}}$ with eigenvalue $\lambda_{m}$.

######  Remark 2.5.

  1. (1)

Fact 2.4 does not assert the non-vanishing of the series
$(\ell_{g}^{*}\varphi)^{\Gamma}$ which is more involved. Kassel–Kobayashi [10]
proved that there exists $g\in G$ such that
$(\ell^{*}_{g}\psi_{m,0})^{\Gamma}\neq 0$ for sufficiently large
$m\in\mathbb{N}$.

  2. (2)

By [10, Lemma 4.6.4], if a discontinuous group $\Gamma$ is sharp in the sense
of [10, Definition 4.2], then $\Gamma$ satisfies the exponential growth
condition (2.11). Moreover, Kassel [8] and Guériataud–Kassel [6] proved that
finitely generated discontinuous groups for $\mathrm{AdS}^{3}$ are always
sharp (see Fact 4.5 below).

  3. (3)

There exist discontinuous groups which do not satisfy the exponential growth
condition (2.11). Indeed, for any increasing function
$f\colon\mathbb{R}\rightarrow\mathbb{R}_{>0}$ and any $x\in\mathrm{AdS}^{3}$,
we constructed a discontinuous group $\Gamma_{f,x}$ for $\mathrm{AdS}^{3}$
satisfying $N_{\Gamma_{f,x}}(x,R)>f(R)$ for sufficiently large $R>0$ in [7].

The conclusion of Fact 2.4 still holds if we drop the assumption that $\Gamma$
acts freely on $X=\mathrm{AdS}^{3}$. In this case, the quotient space
$X_{\Gamma}=\Gamma\backslash X$ is an orbifold. To formulate more precisely in
the orbifold case, we observe that the quotient space $X_{\Gamma}$ 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 $X_{\Gamma}$
is smooth if the pull-back $p_{\Gamma}^{*}g$ is a smooth function on $X$,
where $p_{\Gamma}\colon X\rightarrow X_{\Gamma}$ is the natural quotient map.
We write $C_{c}^{\infty}(X_{\Gamma})$ for the set of smooth functions on
$X_{\Gamma}$ with compact support. For $g\in C_{c}^{\infty}(X_{\Gamma})$, we
define $\square_{X_{\Gamma}}g\in C_{c}^{\infty}(X_{\Gamma})$ by identifying it
with the $\Gamma$-invariant function $\square_{X}(p_{\Gamma}^{*}g)$. For
$\lambda\in\mathbb{C}$, we define

$\displaystyle L^{2}_{\lambda}(X_{\Gamma}):=\big{\\{}f\in
L^{2}(X_{\Gamma})\mid\forall g\in C_{c}^{\infty}(X_{\Gamma}),\langle
f,\square_{X_{\Gamma}}g\rangle_{X_{\Gamma}}=\lambda\langle
f,g\rangle_{X_{\Gamma}}\big{\\}}.$

The discrete spectrum $\mathrm{Spec}_{d}(\square_{X_{\Gamma}})$ and its
multiplicity $\mathcal{N}_{X_{\Gamma}}$ are defined similarly to the case
where $\Gamma$ acts also freely.

###  2.4 “Injectivity radii” of anti-de Sitter 3-manifolds

Let $\Gamma$ be a discontinuous group for $\mathrm{AdS}^{3}$. In this
subsection, we give a uniform estimate of the pseudo-distance between the
origin and the second closest point of each $\Gamma$-orbit.

We recall that $\Gamma(\subset\grave{\ }G\times\grave{\ }G)$ acts
isometrically on $\mathrm{AdS}^{3}(\cong\grave{\ }G)$ by
$(\gamma_{1},\gamma_{2})x=\gamma_{1}x\gamma_{2}^{-1}$ for
$(\gamma_{1},\gamma_{2})\in\Gamma$ and $x\in\grave{\ }G$. We set

$\displaystyle\varepsilon_{\Gamma}:=\inf_{(\gamma_{1},\gamma_{2})\in\Gamma\setminus\\{E\\}}\frac{1}{3}\big{|}\|\gamma_{1}\|-\|\gamma_{2}\|\big{|}.$
(2.12)

By the inequality (see, e.g., [7, Lemma 5.5])

$\displaystyle\|(g_{1},g_{2})x\|\geq\big{|}\|g_{1}\|-\|g_{2}\|\big{|}-\|x\|\qquad\text{for}\quad(g_{1},g_{2})\in
G\quad\text{and}\quad x\in\mathrm{AdS}^{3},$

we get:

######  Lemma 2.6.

If $\varepsilon_{\Gamma}>0$, then $\gamma B(\varepsilon_{\Gamma})\cap
B(\varepsilon_{\Gamma})=\varnothing$ for all
$\gamma\in\Gamma\setminus\\{E\\}$.

######  Proposition 2.7.

Let $\Gamma$ be a discrete subgroup of $G$ acting properly discontinuously on
$\mathrm{AdS}^{3}$. Then there exists $g\in G$ satisfying
$\varepsilon_{g^{-1}\Gamma g}>0$.

######  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
$\Gamma$-action and the following lemma:

######  Lemma 2.9.

For any countable subset $\Gamma$ of $G$, there exists $g\in G$ such that
$\|\gamma_{1}\|\neq\|\gamma_{2}\|$ for all $\gamma=(\gamma_{1},\gamma_{2})\in
g^{-1}\Gamma g\setminus\\{E\\}$.

###### Proof of Lemma 2.9.

For $\gamma\in\Gamma$, the map $f_{\gamma}\colon G\rightarrow G$ defined by
$g\mapsto g^{-1}\gamma g$ is real analytic. For the analytic subset
$F=\\{(g_{1},g_{2})\in G\mid\|g_{1}\|=\|g_{2}\|\\}$ of $G$, we claim that the
set $f_{\gamma}^{-1}(F)$ is a proper subset of $G$ if $\gamma\neq E$. For
this, we may assume $\gamma_{1}\neq\grave{\ }E$ without loss of generality.
Then there exists $g_{1}\in\grave{\ }G$ satisfying
$\|g_{1}^{-1}\gamma_{1}g_{1}\|\neq\|\gamma_{1}\|$ as one can find $g_{1}$
depending on the three cases where $\gamma_{1}$ is hyperbolic, parabolic, or
elliptic. Hence $(g_{1},\grave{\ }E)\notin f_{\gamma}^{-1}(F)$ if
$\|\gamma_{1}\|=\|\gamma_{2}\|$, and $E\notin f_{\gamma}^{-1}(F)$ if not. Thus
$f_{\gamma}^{-1}(F)$ is a proper subset of $G$.

Therefore the analytic set $f_{\gamma}^{-1}(F)$ has no interior point, and
thus so does the countable union
$\bigcup_{\gamma\in\Gamma\setminus\\{E\\}}f_{\gamma}^{-1}(F)$ by the Baire
category theorem (see, e.g., [22, Theorem 2.2]). Hence there exists an element
$g$ of $G\setminus\bigcup_{\gamma\in\Gamma\setminus\\{E\\}}f_{\gamma}^{-1}(F)$
and we have $\|\gamma_{1}\|\neq\|\gamma_{2}\|$ for all
$\gamma=(\gamma_{1},\gamma_{2})\in g^{-1}\Gamma g\setminus\\{E\\}$. ∎

##  3 Proof of Theorem 1.1

In this section, we prove Theorem 1.1.

More generally, without finitely generated assumption of $\Gamma$, we study
linear independence of the generalized Poincaré series of the spherical
functions $\psi_{m,k}$ of type $(-m,m+k)$ defined in Section 2.2. By choosing
$k=3^{j}$ ($j=0,1,2,\ldots$), we prove:

######  Theorem 3.1.

If $\Gamma$ is a discontinuous group for $\mathrm{AdS}^{3}$ satisfying the
exponential growth condition (2.11), then

$\displaystyle\lim_{m\to\infty}\mathcal{N}_{\Gamma\backslash\mathrm{AdS}^{3}}(\lambda_{m})=\infty.$

Theorem 1.1 is a direct consequence of Theorem 3.1 by Remark 2.5(2).

######  Proposition 3.2.

Let $\Gamma$ be a discrete subgroup of $G$ acting properly discontinuously on
$\mathrm{AdS}^{3}$ and satisfying the exponential growth condition (2.11). If
$\varepsilon_{\Gamma}>0$, then there exists a real number $m_{\Gamma}(k)$
$($given explicitly by $\eqref{Poincare_estimate})$ for $k\in\mathbb{N}$ such
that $\\{(\operatorname{Re}(\psi_{m,3^{j}}))^{\Gamma}\\}_{j=0}^{k-1}\subset
L^{2}_{\lambda_{m}}(\Gamma\backslash\mathrm{AdS}^{3})$ are linearly
independent for all integers $m>m_{\Gamma}(k)$.

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 $L^{2}$-eigenvalues,
$\mathcal{N}_{\Gamma\backslash\mathrm{AdS}^{3}}=\mathcal{N}_{(g^{-1}\Gamma
g)\backslash\mathrm{AdS}^{3}}$ for any $g\in G$ through the natural
isomorphism $\Gamma\backslash\mathrm{AdS}^{3}\cong\big{(}g^{-1}\Gamma
g\big{)}$ $\backslash\mathrm{AdS}^{3}$ as Lorentzian manifolds. By replacing
$\Gamma$ with $g^{-1}\Gamma g$ if necessary, we may and do assume
$\varepsilon_{\Gamma}>0$ by Proposition 2.7. Then Proposition 3.2 implies that
$L^{2}_{\lambda_{m}}\big{(}\Gamma\backslash\mathrm{AdS}^{3}\big{)}$ contains
at least $k$ linearly independent elements if $m>m_{\Gamma}(k)$ for any fixed
$k\in\mathbb{N}$, which means
$\dim_{\mathbb{C}}L^{2}_{\lambda_{m}}(\Gamma\backslash\mathrm{AdS}^{3})\geq
k.$ Hence Theorem 3.1 follows. ∎

Kassel–Kobayashi [10] proved the non-vanishing of the generalized Poincaré
series $(\psi_{m,0})^{\Gamma}$ for sufficiently large $m\in\mathbb{N}$ 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 $\psi_{m,0}(\grave{\ }E)=1$. Our strategy for the proof of
Proposition 3.2 is along the same line, however, there are some technical
difficulties since $\psi_{m,k}$ for $k\geq 1$ vanishes at the origin. We then
make use of an observation that $\psi_{m,k}$ decays more slowly at the origin
than at infinity, to be precise, by the following formula, see (2.8):

$\displaystyle|\psi_{m,k}(x)|=\cosh^{-2m}(\|x\|/2)\tanh^{k}(\|x\|/2).$

Actually, we use an analytic lemma (Lemma 3.3) to prove that the first term in
the generalized Poincaré series $(\psi_{m,k})^{\Gamma}$ is larger at points
sufficiently close to the origin than the sum of the remaining terms if $m\gg
0$. Moreover, we use a combinatorial lemma (Lemma 3.4) to find points at which
leading terms of $(\operatorname{Re}(\psi_{m,k}))^{\Gamma}$ do not cancel each
other for any linear combination.

For $C,a,\varepsilon>0$ and $s\in\mathbb{N}$, we set

$\displaystyle m(C,a,\varepsilon,s):=\frac{(\log
2)s+2a\varepsilon+\log\big{(}1+2^{s}C{\rm
e}^{6a\varepsilon}\big{)}}{\log\cosh\varepsilon}$

and

$\displaystyle\tilde{m}(C,a,\delta,s):=\inf_{0<\varepsilon<\delta}m(C,a,\varepsilon,s).$

Note that $\tilde{m}(C,a,\delta,s)=O\big{(}\delta^{-2}\big{)}$ as $\delta\to
0$ and $=O(1)$ as $\delta\to\infty$.

######  Lemma 3.3.

For any integer $m>m(C,a,\varepsilon,s)$ and any one-variable polynomial $f$
of degree $\leq s$ with non-negative coefficients,

$\displaystyle C\sum_{n=1}^{\infty}{\rm e}^{4a(n+1)\varepsilon}(\cosh
2n\varepsilon)^{-m}f(\tanh
2(n+1)\varepsilon)<(\cosh\varepsilon)^{-m}f(\tanh\varepsilon).$

###### Proof.

We may assume that $f(x)=x^{j}$ for $j=0,1,\ldots,s$. Since

$\displaystyle 1\leq\frac{\tanh nx}{\tanh x}\leq n,\ (\cosh x)^{n}\leq\cosh
nx$

for $x\in\mathbb{R}$, we have

(LHS)/(RHS) $\displaystyle=C\sum_{n=1}^{\infty}{\rm
e}^{4a(n+1)\varepsilon}\bigg{(}\frac{\cosh
2n\varepsilon}{\cosh\varepsilon}\bigg{)}^{-m}\bigg{(}\frac{\tanh
2(n+1)\varepsilon}{\tanh\varepsilon}\bigg{)}^{j}$ $\displaystyle\leq C{\rm
e}^{6a\varepsilon}\sum_{n=1}^{\infty}\big{(}{\rm
e}^{2a\varepsilon}(\cosh\varepsilon)^{-m}\big{)}^{2n-1}(2(n+1))^{s}.$

We set $d:={\rm e}^{2a\varepsilon}(\cosh\varepsilon)^{-m}$. Then $d<1$ by
$m>m(C,a,\varepsilon,s)$. Since $n+1\leq 2^{n}$ for all $n\in\mathbb{N}$, we
have

$\displaystyle\text{(LHS)/(RHS)}\leq 2^{s}C{\rm
e}^{6a\varepsilon}\sum_{n=1}^{\infty}(2^{s}d)^{n}=2^{s}C{\rm
e}^{6a\varepsilon}\frac{2^{s}d}{1-2^{s}d}.$

Again by $m>m(C,a,\varepsilon,s)$, we have $2^{s}d<\big{(}1+2^{s}C{\rm
e}^{6a\varepsilon}\big{)}^{-1}$. Therefore we obtain

$\displaystyle\text{(LHS)/(RHS)}<1.$ ∎

Let $\chi\colon\\{\pm 1\\}\rightarrow\\{0,1\\}$ be the map defined by
$\chi(1)=0$ and $\chi(-1)=1$. For $a=(a_{j})_{j=0}^{k-1}\in\\{\pm 1\\}^{k}$
and an odd integer $N\geq 3$, we set

$\displaystyle\theta_{a,N}:=\pi\sum_{i=0}^{k-1}(\chi(a_{i})-\chi(a_{i-1}))N^{-i}.$

Here we use the convention $a_{-1}=1$.

######  Lemma 3.4.

For any $a=(a_{0},\ldots,a_{k-1})\in\\{\pm 1\\}^{k}$ and any odd integer $N$,
we have

$\displaystyle a_{j}\cos\big{(}N^{j}\theta_{a,N}\big{)}>0\qquad\text{for}\quad
j=0,1,\ldots,k-1.$

###### Proof.

Since $N^{k-1}\theta_{a,N}\equiv\pi\chi(a_{k-1})\ (\mathrm{mod}\ 2\pi)$, we
have $\cos\big{(}N^{k-1}\theta_{a,N}\big{)}=a_{k-1}$. It is easy to check that
$|N^{j}\theta_{a,N}-N^{j}\theta_{(a_{0},\cdots,a_{j}),N}|<\pi/2$ for
$j=0,1,\ldots,k-1$, hence the signature of $\cos(N^{j}\theta_{a,N})$ is equal
to that of $\cos(N^{j}\theta_{(a_{0},\cdots,a_{j}),N})=a_{j}$. ∎

######  Remark 3.5.

We have used the geometric progression $(N^{j})_{j=0}^{k-1}$ 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 $\theta\in\mathbb{R}$
satisfying $a_{j}\cos m_{j}\theta>0$ for all $j=0,1,2,3,4$ if we choose
$(a_{j})_{j=0}^{4}=(1,1,1,-1,1)$ and an arithmetic progression
$(m_{j})_{j=0}^{4}$.

For a discontinuous group $\Gamma$ and $k\in\mathbb{N}$, one can take
$m_{\Gamma}(k)$ in Proposition 3.1 by

$\displaystyle
m_{\Gamma}(k)=\inf_{(A,a)\in\mathrm{C_{exp}}(\Gamma)}\max\big{\\{}\tilde{m}\big{(}3^{k-1}A,a,\varepsilon_{\Gamma}/4,3^{k-1}\big{)}/2,a\big{\\}},$
(3.1)

where $\mathrm{C_{exp}}(\Gamma):=\big{\\{}(A,a)\in\mathbb{R}^{2}\mid\forall
x\in\mathrm{AdS}^{3},\forall R>0,N_{\Gamma}(x,R)<A{\rm e}^{aR}\big{\\}}$.
Here, we adopt the convention that $\inf_{\varnothing}f=\infty$ for a real-
valued function $f$. In particular, $m_{\Gamma}(k)=\infty$ when
$\mathrm{C_{exp}}(\Gamma)=\varnothing$ or $\varepsilon_{\Gamma}=0$.

###### Proof of Proposition 3.2.

By the exponential growth condition (2.11),
$\mathrm{C_{exp}}(\Gamma)\neq\varnothing$ and thus $m_{\Gamma}(k)<\infty$. We
take an integer $m>m_{\Gamma}(k)$. Then there exist $\varepsilon$ with
$0<\varepsilon<\varepsilon_{\Gamma}/4$ and $(A,a)\in\mathrm{C_{exp}}(\Gamma)$
satisfying the inequality
$m>\max\big{\\{}m\big{(}3^{k-1}A,a,\varepsilon,3^{k-1}\big{)}/2,a\big{\\}}$.

To see $\mathbb{C}$-linear independence of the real-valued functions
$\big{\\{}(\operatorname{Re}(\psi_{m,3^{j}}))^{\Gamma}\big{\\}}_{j=0}^{k-1}$,
it is enough to prove the non-vanishing of the real part
$\operatorname{Re}\big{(}\psi_{m,b}^{\Gamma}\big{)}=(\operatorname{Re}(\psi_{m,b}))^{\Gamma}$
of the generalized Poincaré series of a linear combination

$\displaystyle\psi_{m,b}:=\sum_{j=0}^{k-1}b_{j}\psi_{m,3^{j}}$

for any $b=(b_{0},b_{1},\ldots,b_{k-1})\in\mathbb{R}^{k}\setminus\\{0\\}$. By
Lemma 2.6, for $x\in B(4\varepsilon)$, we have

$\displaystyle\psi_{m,b}^{\Gamma}(\Gamma
x)=\psi_{m,b}(x)+\sum_{\begin{subarray}{c}\gamma\in\Gamma\\\
\|\gamma^{-1}x\|>4\varepsilon\end{subarray}}\psi_{m,b}\big{(}\gamma^{-1}x\big{)}.$
(3.2)

By (2.8), for any $y\in\mathrm{AdS}^{3}$, we get

$\displaystyle|\psi_{m,b}(y)|\leq\bigg{(}\\!\cosh\frac{\|y\|}{2}\bigg{)}^{-2m}\sum_{j=0}^{k-1}|b_{j}|\bigg{(}\\!\tanh\frac{\|y\|}{2}\bigg{)}^{3^{j}}.$

We define $a=(a_{j})_{j=0}^{k-1}$ by $a_{j}=1$ for $b_{j}\geq 0$ and
$a_{j}=-1$ for $b_{j}<0$, and set

$\displaystyle
f_{b}(u):=\sum_{j=0}^{k-1}b_{j}\cos\big{(}3^{j}\theta_{a,3}\big{)}u^{3^{j}}.$

We note that all the coefficients of $f_{b}$ are non-negative by Lemma 3.4.
Moreover, we get $\big{|}\cos\big{(}3^{j}\theta_{a,3}\big{)}\big{|}^{-1}\leq
3^{k-1}$ for all $j=0,1,\ldots,k-1$ by using the inequality $\sin(\pi x/2)\geq
x$ for $0\leq x\leq 1$. Thus

$\displaystyle|\psi_{m,b}(y)|\leq
3^{k-1}\bigg{(}\\!\cosh\frac{\|y\|}{2}\bigg{)}^{-2m}f_{b}\bigg{(}\\!\tanh\frac{\|y\|}{2}\bigg{)}$

and, for any $x\in B(4\varepsilon)$, we have

$\displaystyle\bigg{|}\\!\\!\\!\sum_{\begin{subarray}{c}\gamma\in\Gamma\\\
\|\gamma^{-1}x\|>4\varepsilon\end{subarray}}\operatorname{Re}\big{(}\psi_{m,b}\big{(}\gamma^{-1}x\big{)}\big{)}\bigg{|}$
$\displaystyle\leq\sum_{n=1}^{\infty}\sum_{\begin{subarray}{c}\gamma\in\Gamma\\\
4\varepsilon n<\|\gamma^{-1}x\|\leq
4\varepsilon(n+1)\end{subarray}}|\psi_{m,b}(\gamma^{-1}x)|$ $\displaystyle\leq
3^{k-1}\sum_{n=1}^{\infty}N_{\Gamma}(x,4\varepsilon(n+1))\left(\cosh
2\varepsilon n\right)^{-2m}f_{b}\left(\tanh 2\varepsilon(n+1)\right)$
$\displaystyle\leq 3^{k-1}A\sum_{n=1}^{\infty}{\rm
e}^{4a\varepsilon(n+1)}\left(\cosh 2\varepsilon n\right)^{-2m}f_{b}\left(\tanh
2\varepsilon(n+1)\right)$
$\displaystyle<\left(\cosh\varepsilon\right)^{-2m}f_{b}\left(\tanh\varepsilon\right).$
(3.3)

The third and forth inequalities respectively follow from the exponential
growth condition (2.11) and Lemma 3.3. On the other hand, we set

$\displaystyle
x_{a,\varepsilon}:=k\bigg{(}\frac{\theta_{a,3}}{2}\bigg{)}a(\varepsilon)k\bigg{(}\frac{\theta_{a,3}}{2}\bigg{)}^{-1}\in
B(4\varepsilon).$

Then it follows from (2.8) that

$\displaystyle\operatorname{Re}\psi_{m,b}(x_{a,\varepsilon})=\left(\cosh\varepsilon\right)^{-2m}f_{b}\left(\tanh\varepsilon\right).$
(3.4)

By (3.2), (3.3), and (3.4), we obtain
$(\operatorname{Re}(\psi_{m,b}))^{\Gamma}(\Gamma x_{a,\varepsilon})\neq 0$.
Hence we complete the proof by the continuity of $\psi_{m,b}^{\Gamma}$ (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 $\Delta$ to have torsion.

######  Setting 4.1.

  * •

$\Delta$ is a discrete subgroup of $\grave{\ }G=\mathrm{PSL}(2,\mathbb{R})$.

  * •

$j,\rho\colon\Delta\rightarrow\grave{\ }G$ are two group homomorphisms with
$j$ injective and discrete.

  * •

$\Delta^{j,\rho}$ is a discrete subgroup of $G=\grave{\ }G\times\grave{\ }G$
given by $\\{(j(\gamma),\rho(\gamma))\mid\gamma\in\Delta\\}$.

We use the following structural results of discontinuous groups for the proof
of Theorem 1.4.

######  Fact 4.2 ([10, Lemma 9.2]).

Let $\Gamma$ be a finitely generated discrete subgroup of $G$ acting properly
discontinuously on $\mathrm{AdS}^{3}$. Then $\Gamma$ is of either type $(i)$
or $(ii)$ as follows:

  1. type$i$ $(i)$

$\Gamma$ is of the form $\Delta^{j,\rho}$ up to switching the two factors,

  2. type $(ii)$

$\Gamma$ is contained in a conjugate of $\grave{\ }G\times\grave{\ }K$ or
$\grave{\ }K\times\grave{\ }G$.

A non-elementary discrete subgroup $\Gamma$ of a connected linear real
reductive Lie group $L$ of real rank $1$ is called convex cocompact if
$\Gamma$ 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 $\mathrm{AdS}^{3}$:

######  Definition 4.3 ([10, Definition 9.1]).

A discontinuous group $\Gamma$ for $\mathrm{AdS}^{3}$ is called convex
cocompact if $\Gamma$ is of the form $\Delta^{j,\rho}$ up to finite index and
switching the two factors, where $\Delta$ is torsion-free and $j(\Delta)$ is
convex cocompact in $\grave{\ }G$.

We note that a discontinuous group $\Delta^{j,\rho}$ acts cocompactly on
$\mathrm{AdS}^{3}$ if and only if $j(\Delta)$ is cocompact in $\grave{\ }G$
because $\Delta^{j,\rho}$ is isomorphic to $j(\Delta)$ as abstract groups. By
Fact 4.2, discontinuous groups acting cocompactly on $\mathrm{AdS}^{3}$ are
convex cocompact.

###  4.1 Proof of Theorem 1.4 for $\boldsymbol{\Gamma}$ of type
($\boldsymbol{i}$)

In this subsection, we prove Theorem 1.4 for $\Gamma$ of type $(i)$. For this,
we use the constant $C_{\rm Lip}(j,\rho)$ introduced by Kassel [8] and
Guéritaud–Kassel [6], which quantifies the properness of the action of
$\Delta^{j,\rho}$ on $\mathrm{AdS}^{3}$.

######  Definition 4.4.

Let $d_{\mathbb{H}^{2}}$ be the hyperbolic distance of the $2$-dimensional
hyperbolic space $\mathbb{H}^{2}(\cong\grave{\ }G/\grave{\ }K)$. In Setting
4.1, we denote by $C_{\rm Lip}(j,\rho)$ the infimum of Lipschitz constants

$\displaystyle\operatorname{Lip}(f)=\sup_{y\neq
y^{\prime}}\frac{d_{\mathbb{H}^{2}}(f(y),f(y^{\prime}))}{d_{\mathbb{H}^{2}}(y,y^{\prime})}$

of maps $f\colon\mathbb{H}^{2}\rightarrow\mathbb{H}^{2}$ that are
$(j,\rho)$-equivariant.

The map $(j,\rho)\mapsto C_{\rm Lip}(j,\rho)$ is continuous over the set of
$(j,\rho)\in\operatorname{Hom}(\Delta,\grave{\ }G)^{2}$ such that $j$ is
injective and $j(\Delta)$ is convex cocompact in $\grave{\ }G$ [6, Proposition
1.5].

######  Fact 4.5 ([6, 8]).

Assume that $\Delta$ is finitely generated. Then the action of
$\Delta^{j,\rho}$ on $\mathrm{AdS}^{3}$ is properly discontinuous if and only
if $\operatorname{min}\big{\\{}C_{\rm Lip}(j,\rho),C_{\rm
Lip}(\rho,j)\big{\\}}<1$.

######  Remark 4.6.

In the setting of Fact 4.5, if $C_{\rm Lip}(\rho,j)<1$, then $\rho$ is
injective and discrete. Moreover, if $j(\Delta)$ is convex cocompact, then so
is $\rho(\Delta)$.

Therefore, Theorem 1.4 for $\Gamma$ of type $(i)$ reduces to the following:

######  Theorem 4.7.

In Setting 4.1, we assume that $\Delta$ is finitely generated and that $C_{\rm
Lip}(j,\rho)<1$. Then there exists a constant $\mu_{1}>0$ independent of
$j,\rho$ and $\Delta$ such that for any $m,k\in\mathbb{N}$ with
$m>3^{k}\mu_{1}(1-C_{\rm Lip}(j,\rho))^{-2}$,

$\displaystyle\mathcal{N}_{\Delta^{j,\rho}\backslash\mathrm{AdS}^{3}}(\lambda_{m})\geq
k.$

For the proof of Theorem 4.7, we need two results from Kassel–Kobayashi [10]
applied to our setting $G=\grave{\ }G\times\grave{\ }G$. If a discontinuous
group $\Gamma$ satisfies the assumption of Fact 4.8 below, then it is
$((1-\alpha)/2,0)$-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 $\Gamma\subset G$ be a discontinuous group for $\mathrm{AdS}^{3}$. We
assume that there exists $0\leq\alpha<1$ such that
$\|\gamma_{2}\|\leq\alpha\|\gamma_{1}\|$ or
$\|\gamma_{1}\|\leq\alpha\|\gamma_{2}\|$ for any
$(\gamma_{1},\gamma_{2})\in\Gamma$. Then there exists $c>0$ independent of
$\alpha$ and $\Gamma$ such that for any $x\in\mathrm{AdS}^{3}$ and any $R>0$,

$\displaystyle N_{\Gamma}(x,R)\leq\\#(\Gamma\cap K)c{\rm
e}^{8R(1-\alpha)^{-1}}.$

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 $\grave{\ }\Gamma$ of $\grave{\ }G$, there exists $\grave{\
}g\in\grave{\ }G$ such that $\|\grave{\ }\gamma\|\geq r$ for all $\grave{\
}\gamma\in\grave{\ }g^{-1}\grave{\ }\Gamma\grave{\ }g\setminus\\{\grave{\
}E\\}$.

In the following, we use the upper half plane model
$\big{\\{}z=x+\sqrt{-1}y\in\mathbb{C}\mid\operatorname{Im}z>0\big{\\}}$
equipped with the metric tensor ${\rm d}s^{2}=\big{(}{\rm d}x^{2}+{\rm
d}y^{2}\big{)}/y^{2}$ for the hyperbolic space $\mathbb{H}^{2}$. Then
$\|\grave{\ }g\|$ is equal to the hyperbolic distance
$d_{\mathbb{H}^{2}}\big{(}\grave{\ }g\sqrt{-1},\sqrt{-1}\big{)}$ for $\grave{\
}g\in\mathrm{AdS}^{3}\cong\grave{\ }G$ (see, e.g., [6, equation (A.1)]).

###### 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 $\grave{\ }G$, we may assume $\|j(\gamma)\|\geq r$ for any
$\gamma\in\Delta\setminus\\{\grave{\ }E\\}$. In particular, $\Gamma\cap
K=\\{E\\}$ for such $j$ and for any $\rho$. We fix $\delta>0$ such that

$\displaystyle\alpha:=C_{\rm Lip}(j,\rho)+\delta<1.$

Then, replacing $\rho$ by some conjugate under $\grave{\ }G$, we may assume

$\displaystyle\|\rho(\gamma)\|\leq\alpha\|j(\gamma)\|\qquad\text{for
any}\quad\gamma\in\Delta.$ (4.1)

Indeed, by Definition 4.4, there exists a $(j,\rho)$-equivariant map
$f_{\delta}\colon\mathbb{H}^{2}\rightarrow\mathbb{H}^{2}$ satisfying
$\operatorname{Lip}(f_{\delta})$ $<\alpha$. We take $g_{\delta}\in\grave{\ }G$
such that $g_{\delta}\sqrt{-1}=f_{\delta}\big{(}\sqrt{-1}\big{)}$. Then, for
any $\gamma\in\Delta$, we have

$\displaystyle\big{\|}g_{\delta}^{-1}\rho(\gamma)g_{\delta}\big{\|}=d_{\mathbb{H}^{2}}\big{(}f_{\delta}(\sqrt{-1}),\rho(\gamma)f_{\delta}\big{(}\sqrt{-1}\big{)}\big{)}<\alpha
d_{\mathbb{H}^{2}}\big{(}\sqrt{-1},j(\gamma)\sqrt{-1}\big{)}=\alpha\|j(\gamma)\|.$

Hence (4.1) holds by replacing $\rho$ with
$g_{\delta}^{-1}\rho(\cdot)g_{\delta}$, and therefore we get

$\displaystyle N_{\Gamma}(x,R)\leq c{\rm e}^{8R(1-(C_{\rm
Lip}(j,\rho)+\delta))^{-1}}$

by Fact 4.8. Then the constant $\varepsilon_{\Gamma}$ in (2.12) has the
following lower bound:

$\displaystyle 3\varepsilon_{\Gamma}=\inf_{\gamma\in\Delta\setminus\\{\grave{\
}E\\}}\left|\|j(\gamma)\|-\|\rho(\gamma)\|\right|\geq\inf_{\gamma\in\Delta\setminus\\{\grave{\
}E\\}}(1-\alpha)\|j(\gamma)\|\geq r(1-\alpha).$

Note that $\log\cosh t=O\big{(}t^{2}\big{)}$ as $t\to 0$. By the explicit
description (3.1) of $m_{\Gamma}(k)$, Theorem 4.7 follows from Proposition
3.2. ∎

###  4.2 Proof of Theorem 1.4 for $\boldsymbol{\Gamma}$ of type
($\boldsymbol{ii}$)

In this subsection, we prove Theorem 1.4 for the case where $\Gamma$ is
standard. For this, we use the following fact by Kobayashi [17] and Kassel [9]
applied to our $\mathrm{AdS}^{3}$ setting, which gives the stability for
properness under any small deformation of standard convex cocompact
discontinuous groups.

######  Fact 4.10 ([9, Theorem 1.4]).

Let $\Gamma$ be a convex cocompact discrete subgroup of $\grave{\
}G\times\grave{\ }K$. Then for any $\alpha,\beta>0$, there exists a
neighborhood $W\subset\operatorname{Hom}(\Gamma,G)$ of the natural inclusion
$\Gamma\subset G$ such that for any $\varphi\in W$,

$\displaystyle\left|\mu(\varphi(\gamma))-\mu(\gamma)\right|\leq\begin{cases}\alpha\left|\mu(\gamma)\right|&\text{if}\quad\gamma\in\Gamma\setminus
K,\\\ \beta&\text{if}\quad\gamma\in\Gamma\cap K,\end{cases}$

where $\mu(g_{1},g_{2}):=(\|g_{1}\|,\|g_{2}\|)\in\mathbb{R}^{2}$ for
$(g_{1},g_{2})\in G$, $\|\cdot\|$ is given in Definition 2.1, and
$\left|(x_{1},x_{2})\right|:=\sqrt{x_{1}^{2}+x_{2}^{2}}$ for
$(x_{1},x_{2})\in\mathbb{R}^{2}$.

We introduce the following terminology for the estimate of the discrete
spectrum since a discontinuous group $\Gamma$ is not necessarily torsion-free.
Let $\operatorname{pr}_{j}\colon G=\grave{\ }G\times\grave{\
}G\rightarrow\grave{\ }G$ be the $j$-th projection ($j=1,2$). In the following
definition, we assume that $\operatorname{pr}_{2}(\Gamma)$ is bounded. Then
the group $\Gamma_{1}:=\mathrm{ker}(\operatorname{pr}_{1}|_{\Gamma})$ is
cyclic since $\Gamma_{1}$ is a discrete subgroup of a conjugate of the product
group $\\{\grave{\ }E\\}\times\grave{\ }K$ $(\cong\mathbb{R}/\mathbb{Z})$.

######  Definition 4.11.

A discrete subgroup $\Gamma$ of $G$ is said to be standard of class $n$ if
$\operatorname{pr}_{2}(\Gamma)$ is bounded and the cyclic group
$\Gamma_{1}=\mathrm{ker}(\operatorname{pr}_{1}|_{\Gamma})$ is of order $n$.

######  Remark 4.12.

  1. (1)

If $\Gamma$ is torsion-free, then it is of class $1$.

  2. (2)

If $\operatorname{pr}_{2}(\Gamma)$ is bounded for a discrete subgroup $\Gamma$
of $G$, then the group $\operatorname{pr}_{1}(\Gamma)$ is discrete in
$\grave{\ }G$. Moreover, if $\Gamma$ is of class $1$, then it is of the form
$\Delta^{j,\rho}$ such that $\Delta=\operatorname{pr}_{1}(\Gamma)$ and $C_{\rm
Lip}(j,\rho)=0$.

Let $r>0$ be the constant in Fact 4.9. For an integer $n\geq 2$, we define a
positive number $\eta_{n}$ by

$\displaystyle\cosh\eta_{n}:=1+2\bigg{(}\sinh\frac{r}{4}\sin\frac{\pi}{n}\bigg{)}^{2}.$

We get the following by easy computations:

######  Lemma 4.13.

By an abuse of notation, we regard $k(\theta)$, $a(t)$ in (2.4) as elements of
$\grave{\ }G=\mathrm{PSL}(2,\mathbb{R})$. Then

$\displaystyle\bigg{\|}a\bigg{(}\frac{r}{8}\bigg{)}^{-1}k\bigg{(}\frac{j\pi}{n}\bigg{)}a\bigg{(}\frac{r}{8}\bigg{)}\bigg{\|}\geq\eta_{n}\qquad\text{for}\quad
j=1,\ldots,n-1.$

We give a uniform estimate of $\varepsilon_{\Gamma}$ in (2.12) and
$N_{\Gamma}(x,R)$ in (2.10) for standard discrete subgroups $\Gamma$ of class
$n$ after taking a conjugation of $\Gamma$.

######  Lemma 4.14.

Let $\Gamma$ be a standard discrete subgroup of class $n\geq 2$. There exists
$g\in G$ such that $\varepsilon_{g^{-1}\Gamma
g}\geq\operatorname{min}\\{\eta_{n}/3,r/6\\}$ and $N_{g^{-1}\Gamma
g}(x,R)<c{\rm e}^{16R}$ for any $x\in\mathrm{AdS}^{3}$ and any $R>0$.

###### Proof.

Let $\Gamma_{1}=\mathrm{ker}(\operatorname{pr}_{1}|_{\Gamma})$ as in
Definition 4.11. Since $\Gamma$ is of class $n$, the group
$\operatorname{pr}_{2}(\Gamma_{1})$ is generated by $k(\pi/n)\in\grave{\
}G=\mathrm{PSL}(2,\mathbb{R})$. We take $\grave{\ }g\in\grave{\ }G$ in Fact
4.9 applied to $\grave{\ }\Gamma=\operatorname{pr}_{1}(\Gamma)$ and set
$g:=(\grave{\ }g,a(r/8))\in G$. Replacing $\Gamma$ by $g^{-1}\Gamma g$, we get
$\|\gamma_{1}\|\geq r$ for
$(\gamma_{1},\gamma_{2})\in\Gamma\setminus\Gamma_{1}$ by Fact 4.9 and
$\|\gamma_{2}\|\geq\eta_{n}$ for
$(\gamma_{1},\gamma_{2})\in\Gamma_{1}\setminus\\{E\\}$ by Lemma 4.13.
Moreover, if $(\gamma_{1},\gamma_{2})\in\Gamma$, then
$\|\gamma_{2}\|=\|a(r/8)^{-1}ka(r/8)\|$ for some $k\in\grave{\ }K$, hence
$\|\gamma_{2}\|\leq r/2$ because $\|g_{1}g_{2}\|\leq\|g_{1}\|+\|g_{2}\|$ for
$g_{1},g_{2}\in\grave{\ }G$ and since $\|a(t)\|=2t$ for $t\geq 0$ and
$\|k\|=0$ for $k\in\grave{\ }K$. To summarize,

$\displaystyle\begin{cases}\|\gamma_{2}\|\leq\frac{r}{2}\leq\frac{\|\gamma_{1}\|}{2}&\text{if}\quad(\gamma_{1},\gamma_{2})\in\Gamma\setminus\Gamma_{1},\\\\[4.30554pt]
\|\gamma_{2}\|\geq\eta_{n}&\text{if}\quad(\gamma_{1},\gamma_{2})\in\Gamma_{1}\setminus\\{E\\}.\end{cases}$

Then $\varepsilon_{\Gamma}\geq\operatorname{min}\\{\eta_{n}/3,r/6\\}$ and
$\Gamma\cap K=\\{E\\}$. Moreover, $\|\gamma_{1}\|\leq\|\gamma_{2}\|/2$ or
$\|\gamma_{2}\|\leq\|\gamma_{1}\|/2$ for any
$(\gamma_{1},\gamma_{2})\in\Gamma$ and thus $N_{\Gamma}(x,R)<c{\rm e}^{16R}$
for any $x\in\mathrm{AdS}^{3}$ and any $R>0$ by Fact 4.8. ∎

######  Theorem 4.15.

There exists a constant $\mu_{n}>0$ depending only on $n$ such that for any
convex cocompact standard discrete subgroup $\Gamma$ of class $n$ and any
$m,k\in\mathbb{N}$ with $m>3^{k}\mu_{n}$,

$\displaystyle\widetilde{\mathcal{N}}_{\Gamma\backslash\mathrm{AdS}^{3}}(\lambda_{m})\geq
k.$

###### Proof.

If $n=1$, then this follows from Theorem 4.7 since convex cocompact
discontinuous groups are finitely generated, hence we assume that $n\geq 2$.
In this case, we shall prove that $\Gamma$ and its small deformation are
standard of class $n$. When $n\geq 2$, the group
$\Gamma_{1}=\mathrm{ker}(\operatorname{pr}_{1}|_{\Gamma})$ is a cyclic group
of order $n$. By Fact 4.9, replacing $\Gamma$ by some conjugate under
$\grave{\ }G\times\\{\grave{\ }E\\}$, we may and do assume $\|\gamma_{1}\|\geq
r$ for any $(\gamma_{1},\gamma_{2})\in\Gamma\setminus\Gamma_{1}$. By Fact
4.10, there exists a neighborhood $W$ of the natural inclusion $\Gamma\subset
G$ such that for any $\varphi\in W$, the restriction of $\varphi$ to the
finite subgroup $\Gamma_{1}$ is injective and the inequalities

$\displaystyle\begin{cases}\|\varphi_{1}(\gamma)\|\geq\frac{1}{2}r,\quad\|\varphi_{2}(\gamma)\|\leq\frac{1}{2}\|\varphi_{1}(\gamma)\|&\text{if}\quad\gamma\in\Gamma\setminus\Gamma_{1},\\\\[4.30554pt]
\left|\mu(\varphi(\gamma))\right|<\frac{1}{2}r&\text{if}\quad\gamma\in\Gamma_{1}\end{cases}$
(4.2)

hold where $\varphi_{i}=\operatorname{pr}_{i}\circ\varphi$ for $i=1,2$. Then
$\varphi$ is injective and discrete.

We claim $\varphi_{1}(\Gamma_{1})$ is trivial. Indeed, if there exists
$\gamma\in\Gamma_{1}\setminus\\{E\\}$ such that
$\varphi_{1}(\gamma)\neq\grave{\ }E$, then the normalizer of
$\varphi(\Gamma_{1})$ in $G$ is contained in $\grave{\ }K_{1}\times\grave{\
}G$, where $\grave{\ }K_{1}$ is the maximal compact subgroup of $\grave{\ }G$
containing $\varphi_{1}(\Gamma_{1})$. Hence $\varphi(\Gamma)\subset\grave{\
}K_{1}\times\grave{\ }G$. By the inequalities (4.2), $\varphi(\Gamma)$ is
finite, hence $\Gamma$ is also finite. This contradicts the assumption that
$\Gamma$ is non-elementary. Thus $\varphi_{1}(\Gamma_{1})$ is trivial and
$\varphi_{2}(\Gamma_{1})$ is non-trivial. Hence the normalizer of
$\varphi(\Gamma_{1})$ in $G$ is contained in $\grave{\ }G\times\grave{\
}K_{2}$, where $\grave{\ }K_{2}$ is the maximal compact subgroup of $\grave{\
}G$ containing $\varphi_{2}(\Gamma_{1})$. Therefore
$\operatorname{pr}_{2}(\varphi(\Gamma))$ is bounded. Moreover
$\varphi(\Gamma)_{1}=\varphi(\Gamma_{1})$ by the inequalities (4.2), hence the
discrete subgroup $\varphi(\Gamma)$ is standard of class $n$. By the explicit
description (3.1) of $m_{\Gamma}(k)$ 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 $\Gamma$ of class $n\geq 2$ and its small deformation are standard of
class $n$. Therefore we obtain a stronger result that

$\displaystyle\widetilde{\mathcal{N}}_{\Gamma\backslash\mathrm{AdS}^{3}}(\lambda_{m})=\infty$

for any convex cocompact standard discrete subgroup $\Gamma$ of class $n\geq
2$ and any integer $m>3\mu_{n}$ if the following statement holds:
$\mathcal{N}_{\Gamma\backslash\mathrm{AdS}^{3}}(\lambda_{m})=\infty$ for any
standard discrete subgroup $\Gamma$ and any $m\in\mathbb{N}$ such that
$\mathcal{N}_{\Gamma\backslash\mathrm{AdS}^{3}}(\lambda_{m})\geq 1$. 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
$L^{2}$-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 $m(C,a,\varepsilon,s)$ 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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