Counting orbits of certain infinitely generated non-sharp discontinuous groups for the anti-de Sitter space

TL;DR

This paper constructs new examples of infinitely generated discontinuous groups acting on 3D anti-de Sitter space, analyzes their orbit counting functions, and demonstrates the existence of groups with arbitrarily large orbit counts and non-sharp properties.

Contribution

It introduces a family of non-sharp discontinuous groups for AdS3, provides bounds on orbit counting functions, and shows existence of groups with infinitely many Laplacian eigenvalues and arbitrarily large orbit counts.

Findings

Constructed a family of non-sharp discontinuous groups for AdS3.

Derived bounds for orbit counting functions as radius grows.

Proved existence of groups with infinitely many Laplacian eigenvalues.

Abstract

Inspired by an example of Gueritaud-Kassel [Geom. Topol. 2017], we construct a family of infinitely generated discontinuous groups $\Gamma$ for the 3-dimensional anti-de Sitter space $\mathrm{AdS}^{3}$. These groups are not necessarily sharp (a kind of "strong" properly discontinuous condition introduced by Kassel and Kobayashi [Adv. Math. 2016]), and we give its criterion. Moreover, we find upper and lower bounds of the counting $N_{\Gamma}(R)$ of a $\Gamma$-orbit contained in a pseudo-ball $B(R)$ as the radius $R$ tends to infinity. We then find a non-sharp discontinuous group $\Gamma$ for which there exist infinitely many $L^2$-eigenvalues of the Laplacian on the noncompact anti-de Sitter manifold $\Gamma\backslash\mathrm{AdS}^{3}$, by applying the method established by Kassel-Kobayashi. We also prove that for any increasing function $f$, there exists a discontinuous group $\Gamma$ for $\mathrm{AdS}^{3}$ such that the counting $N_{\Gamma}(R)$ of a $\Gamma$-orbit is larger than $f(R)$ for sufficiently large $R$.

Tables

Table 1. Table 1.1: The asymptotic behaviors of N Γ ​ ( x , R ) subscript 𝑁 Γ 𝑥 𝑅 N_{\Gamma}(x,R)

	$\Gamma$	$x\in {\mathrm{AdS}}^3$	$N_{\Gamma }(x,R)$
Eskin-McMullen [2]	$\forall$ lattice in $G$	special $x$	$\sim A{e}^R$
Kassel-Kobayashi [6]	$\forall$ sharp discont. gp.	general $x$	$\le A{e}^{aR}$
Theorem 1.1	$\exists$ non-sharp discont. gp.	general $x$	$\le A{e}^{aR}$
Theorem 1.2	$\exists$ non-sharp discont. gp.	general $x$	$\gg \exp \left(e^R\right)$

Table 2. Table 3.1: Examples of ( a − , a + , r , R ) subscript 𝑎 subscript 𝑎 𝑟 𝑅 (a_{-},a_{+},r,R) satisfying Assumptions 1 and 2

	$a_{-}\left(k\right)$	$a_{+}\left(k\right)$	$r\left(k\right)$	$R\left(k\right)$	$N_{\Gamma }(x,R)$
(1)	$\exp \left(e^k\right)$	$\exp \left(e^{k+\frac{1}{2}}\right)$	$1$	$e^k$	$\le 2^{2R}$
(2)	$k^2$	$k^2+k$	$1$	$\log k$
(3)	$\log k$	$\log \left(k+\frac{1}{2}\right)$	${\left(k^2\log k\right)}^{-1}$	$k^{-2}$	$\ge \exp \left(e^{\frac{R}{4}}\right)-\nu$

Equations

$$N_{\Gamma }(x,R):=\#(\Gamma x\cap B(R))\text{ for }x\in X\text{ and }R>0$$

$$\exists A>0,\exists a>0,\forall x\in X,\forall R>0,N_{\Gamma }(x,R)\le A{e}^{aR}.$$

$$\exists a>0,\forall x\in X,\underset{R\to \infty }{\mathrm{limsup}}\frac{N_{\Gamma }(x,R)}{e^{aR}}<\infty .$$

$$\underset{R\to \infty }{\lim }\frac{N_{\Gamma }(x,R)}{f(R)}=\infty .$$

$$\underset{R\to \infty }{\lim }\frac{N_{\Gamma }(x,R)}{\mathrm{vol}(B(R+c))}=\infty$$

$$\forall x\in X,\exists c>0,\underset{R\to \infty }{\mathrm{limsup}}\frac{\#(\Gamma x\cap B(R))}{\mathrm{vol}(B(R+c))}<\infty .$$

$$\underset{R\to \infty }{\lim }\frac{N_{\Gamma }(o,R)}{\mathrm{vol}(B(R))}=\frac{\mathrm{vol}(\Gamma \backslash G)}{\mathrm{vol}((\Gamma \cap H)\backslash H)}.$$

$${\mathrm{Spec}}_d({\square }_{X_{\Gamma }}):=\{\lambda \in \mathbb{C}\mid \exists f\in L^2(X_{\Gamma })\setminus \{0\},{\square }_{X_{\Gamma }}f=\lambda f\text{ in the weak sense}\}.$$

$${\mathrm{Spec}}_d({\square }_{X_{\Gamma }})\supset \{4m(m-1)\mid m\in \mathbb{Z}\text{ and }m>m_0\}.$$

$${\mathrm{AdS}}^3\cong (\mathrm{SL}(2,\mathbb{R})\times \mathrm{SL}(2,\mathbb{R}))/\mathrm{diag}(\mathrm{SL}(2,\mathbb{R})).$$

$$2\cosh \Vert g\Vert =\mathrm{Tr}{(}^{t}gg).$$

$$\Vert g\Vert =2t.$$

$$B(R):=\{g\in \mathrm{SL}(2,\mathbb{R})\mid \Vert g\Vert \le R\}$$

$$\Vert g\Vert =d_{{\mathbb{H}}^2}(g\sqrt{-1},\sqrt{-1}).$$

$$2\cosh d_{{\mathbb{H}}^2}(x+\sqrt{-1}y,\sqrt{-1})=\frac{x^2+y^2+1}{y}.$$

$${\int }_{\mathrm{SL}(2,\mathbb{R})}f(g)dg={\int }_0^{2\pi }{\int }_0^{\infty }{\int }_0^{2\pi }f(k({\theta }_1)a(t)k({\theta }_2))\sinh (2t)d{\theta }_{1}dtd{\theta }_2.$$

$$\tau =\tau (x_{-},x_{+},u):=\frac{1}{u}\left(\begin{array}{cc}{x}_{+}& -(x_{-}{x}_{+}+u^2)\\ 1& -x_{-}\end{array}\right)\in \mathrm{SL}(2,\mathbb{R}).$$

$${\alpha }_k:=\tau (a_{-}(k),a_{+}(k),r(k)),{\beta }_k:=\tau (a_{-}(k),a_{+}(k),R(k))\in \mathrm{SL}(2,\mathbb{R}).$$

$${\Gamma }_{\nu }(a_{-},a_{+},r,R)=\{(j(\gamma ),\rho (\gamma ))\mid \gamma \in F_{\nu }^{\infty }\}.$$

$$r(k)$$

$$a_{-}(k)+R(k)$$

$$a_{+}(k)+R(k)$$

$$A_k^{ϵ}:=\{z\in {\mathbb{H}}^2\mid |z-a_{ϵ}(k)|\le r(k)\},B_k^{ϵ}:=\{z\in {\mathbb{H}}^2\mid |z-a_{ϵ}(k)|\le R(k)\}.$$

$$|z-x_{-}|>u\text{ if and only if }|\tau (z)-x_{+}|<u\text{ for }z\in {\mathbb{H}}^2,$$

$$\tau (z)-x_{+}=-u^2(z-x_{-}{)}^{-1}.$$

$$\underset{k\to \infty }{\lim }(|z-a_{ϵ}(k)|-r(k))=\infty$$

$$|z-a_{ϵ}(k)|-r(k)\ge a_{ϵ}(k)-|z|-r(k)\ge a_{+}(k-1)-|z|.$$

$$\eta (\nu ):=\sup \left\{\left|\frac{R(k)}{a_{\delta }(k)-a_{ϵ}(\ell )}\right||\delta ,ϵ\in \{+,-\}\text{ and }k,\ell \ge \nu \text{ s.t. }(k,\delta )\ne (\ell ,ϵ)\right\}.$$

$$\epsilon (\nu ):=\max \left\{\underset{k\ge \nu }{\max }\left\{\frac{24R(k)}{a_{-}(k)},\frac{6(R(k{)}^2+1)}{(a_{-}(k)-R(k){)}^2}\right\},8\eta (\nu )\right\}.$$

$$\underset{\nu \to \infty }{\lim }\epsilon (\nu )=0.$$

Full text

Keywords Laplace-Beltrami operator, discrete spectrum, anti-de Sitter space, properly discontinuous action, non-sharp action, counting problem.

MSC2010 Primary 58J50; Secondary 53C50, 22E40.

1. 1 Introduction

2. 1.1 Construction of $\Gamma_{\nu}(a_{-},a_{+},r,R)$ and the counting

3. 1.2 Spectrum of the Laplacian on $\Gamma_{\nu}\backslash\mathrm{AdS}^{3}$

4. 1.3 Organization of the paper

5. 2 Preliminary results about $\mathrm{AdS}^{3}$

6. 2.1 Pseudo-balls in $\mathrm{AdS}^{3}$

7. 2.2 Discontinuous groups for $\mathrm{AdS}^{3}$

8. 3 Discontinuous groups $\Gamma_{\nu}(a_{-},a_{+},r,R)$ for $\mathrm{AdS}^{3}$

9. 3.1 Construction of discrete subgroups $\Gamma_{\nu}(a_{-},a_{+},r,R)$

10. 3.2 A key proposition

11. 3.3 Proper discontinuity of the action of $\Gamma_{\nu}(a_{-},a_{+},r,R)$

12. 3.4 Sharpness of the $\Gamma_{\nu}(a_{-},a_{+},r,R)$-action

13. 4 Construction of $\Gamma$ with large counting

14. 4.1 Construction of $\Gamma$ and a lower bound of $N_{\Gamma}(o,R)$

15. 4.2 Proof of Theorem 1.2

16. 5 Application to the spectral analysis

17. A Proof of the formula (1.3)

1 Introduction

1.1 Construction of $\Gamma_{\nu}(a_{-},a_{+},r,R)$

and the counting

In this paper, we construct a family of discrete groups $\Gamma$ of isometries of the $3$-dimensional anti-de Sitter space $\mathrm{AdS}^{3}$ such that

• •

$\Gamma$ act properly discontinuously on $\mathrm{AdS}^{3}$;

• •

the counting has an arbitrary growth rate at infinity,

generalizing an example of Guéritaud-Kassel [3, Sect. 10.1]. By counting, we mean the number of points in a $\Gamma$-orbit contained in a compact set called a pseudo-ball $B(R)$ of radius $R>0$.

In contrast to the Riemannian case, a discrete group of isometries of a pseudo-Riemannian manifold such as $\mathrm{AdS}^{3}$ may act with non-closed orbits. We recall some basic notions and facts. A pseudo-Riemannian manifold is a smooth manifold $X$ equipped with a smooth non-degenerate symmetric bilinear form of signature $(p,q)$. It is Riemannian if $q=0$ and Lorentzian if $q=1$. A discrete group $\Gamma$ of isometries of a pseudo-Riemannian manifold $X$ is called a discontinuous group for $X$ if $\Gamma$ acts on $X$ properly discontinuously and freely (we include freeness in the definition as in Kobayashi [11, Def. 1.3]). Then there are at most finite elements in any orbit of a discontinuous group $\Gamma$ contained in any compact subset of $X$, hence we can count them. A semisimple symmetric space $X=G/H$ is a typical example of a pseudo-Riemannian manifold, of which the isometry group is “large”. Kassel and Kobayashi proved in [6] for a discontinuous group $\Gamma(\subset G)$ for an arbitrary semisimple symmetric space $G/H$ that the counting is at most of exponential growth if $\Gamma$ is sharp (a notion for “strong” proper discontinuity, see [6, Def. 4.2]).

The $3$-dimensional anti-de Sitter space $\mathrm{AdS}^{3}$ is the simplest example of a Lorentzian semisimple symmetric space that admits infinite discontinuous groups. Let us recall the counting result of Kassel-Kobayashi [6] in this specific setting where $X=\mathrm{AdS}^{3}$ and $G=\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$. They considered a compact subset $B(R)$ of $X$ called a pseudo-ball of radius $R>0$, of which the volume is of exponential growth as $R\to\infty$, see Section 2.1. They proved that if a discontinuous group $\Gamma\subset G$ is sharp, then the counting

[TABLE]

has at most an exponential growth uniformly on $x\in X$ ([6, Lem. 4.6 (4)]):

[TABLE]

In particular, one has

[TABLE]

Any finitely generated discontinuous group for $\mathrm{AdS}^{3}$ is sharp by the results of Kassel [5, Thm. 0.2.13] and Guéritaud-Kassel [3, Thm. 1.8], hence its counting always satisfies the exponential growth condition (1.1).

On the other hand, the counting for a non-sharp discontinuous group has not been well-understood. In this paper, we investigate what can happen about the asymptotic behavior for the counting $N_{\Gamma}(x,R)$ when $\Gamma$ is non-sharp. For this, we construct a family of subgroups $\Gamma_{\nu}\equiv\Gamma_{\nu}(a_{-},a_{+},r,R)$ of $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ for sufficiently large $\nu\in\mathbb{N}$ associated to quadruples $(a_{-},a_{+},r,R)$ of real-valued sequences in Section 3, and study how the properties of $\Gamma_{\nu}$ depend on the data $(a_{-},a_{+},r,R)$. For instance, we find a necessary and sufficient condition for the quadruple $(a_{-},a_{+},r,R)$ that $\Gamma_{\nu}$ is a discontinuous group for $\mathrm{AdS}^{3}$ in Proposition 3.13. Moreover we determine when the $\Gamma_{\nu}$-action on $\mathrm{AdS}^{3}$ is sharp in Proposition 3.16. With these criteria, we present various non-sharp discontinuous groups for which different phenomena happen about the counting by choosing appropriate data $(a_{-},a_{+},r,R)$:

Theorem 1.1**.**

There exists a non-sharp discontinuous group $\Gamma$ for $\mathrm{AdS}^{3}$ such that for any $x\in\mathrm{AdS}^{3}$ and any $R>0$, the counting $N_{\Gamma}(x,R)$ has at most an exponential growth.

Theorem 1.2**.**

Let $x\in\mathrm{AdS}^{3}$. For any increasing function $f\colon\mathbb{R}\to\mathbb{R}_{>0}$, there exists a discontinuous group $\Gamma\equiv\Gamma_{f,x}$ for $\mathrm{AdS}^{3}$ satisfying

[TABLE]

Remark 1.3**.**

Theorem 1.2 applied to the function $f(R)=\exp(\exp(R))$ shows the existence of a discontinuous group $\Gamma$ for $\mathrm{AdS}^{3}$ such that

[TABLE]

for any $c>0$ since the volume $\mathrm{vol}(B(R))$ is of exponential growth as $R\to\infty$. Thus an analogue of the Riemannian case (1.3) below does not hold.

The above theorems deal with the setting where the metric tensor of $X$ is indefinite and $\Gamma$ is a discontinuous group for $X$. Let us compare them with some known results in the following different settings:

• •

$\Gamma$ is a discontinuous group for $X$, but $X$ is Riemannian (the metric tensor of $X$ is positive definite);

• •

the metric tensor of $X$ is indefinite, but $\Gamma$ is not a discontinuous group for $X$ (e.g. $\Gamma$ is a lattice of the isometry group of $X$).

Suppose that $X$ is a complete Riemannian manifold, and that $\Gamma$ is a discrete group of isometries of $X$. We write $B(R)$ for the ball of radius $R$ centered at a fixed point in $X$. Then we have (cf. Milnor [12, Thm. 1])

[TABLE]

The estimate (1.3) does not require that $\Gamma$ is finitely generated, but the Riemannian assumption is crucial as shown in Remark 1.3. The inequality (1.3) is probably well known, but for the reader’s convenience, we will give a proof in the appendix.

A semisimple symmetric space $X=G/H$ admits a $G$-invariant pseudo-Riemannian structure. Eskin-McMullen [2] studied the counting of an orbit of a lattice $\Gamma$ of $G$ in $X=G/H$. Let us apply their result [2, Thm. 1.4] to the specific case where $X=\mathrm{AdS}^{3}$ and $H=\mathrm{diag}(\mathrm{SL}(2,\mathbb{R}))\subset G=\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$. Then it tells us that if $\Gamma\cap H$ is a lattice in $H$, then at the base point $o=eH\in X$

[TABLE]

In the right-hand side, the Haar measures of $G$ and $H$ (and therefore, the induced measures of $\Gamma\backslash G$ and $(\Gamma\cap H)\backslash H$) are normalized such that the Fubini theorem for the fibration $H\rightarrow G\rightarrow X=G/H$ is given by $dg=dxdh$, where $dx$ is the volume element of the anti-de Sitter space $X=\mathrm{AdS}^{3}$ (see Section 2). We note that their setting is different from ours: $\Gamma$ in [2] is a lattice of $G$, hence does not act properly discontinuously on $X$.

We summarize these results about the asymptotic behaviors of $N_{\Gamma}(x,R)$ in each setting in Table 1.1 below:

Remark 1.4**.**

Kassel-Kobayashi [6] gave a uniform estimate of $N_{\Gamma}(x,R)$ with respect to $x\in\mathrm{AdS}^{3}$. We prove such a uniform estimate for Theorem 1.1, but not for Theorem 1.2.

1.2 Spectrum of the Laplacian on

$\Gamma_{\nu}\backslash\mathrm{AdS}^{3}$

Let $\Gamma$ be a discontinuous group for the anti-de Sitter space $X:=\mathrm{AdS}^{3}$. Then the quotient space $X_{\Gamma}:=\Gamma\backslash X$ is a $C^{\infty}$-manifold and the quotient map $X\rightarrow X_{\Gamma}$ is a smooth covering. Thus $X_{\Gamma}$ inherits an anti-de Sitter structure from $X$, and in particular, is a Lorentzian manifold. As in the Riemannian case, one defines the Laplacian $\square_{X_{\Gamma}}:=\mathrm{div}\circ\mathrm{grad}$, a second-order differential operator on $X_{\Gamma}$.

Kassel-Kobayashi [6] initiated the study of global analysis on the anti-de Sitter manifold $X_{\Gamma}$ (actually in a much more general setting). They studied the discrete spectrum, namely the set of $L^{2}$-eigenvalues of the Laplacian $\square_{X_{\Gamma}}$ on $X_{\Gamma}$, denoted by

[TABLE]

Here $L^{2}(X_{\Gamma})$ is the Hilbert space of square integrable functions on $X_{\Gamma}$ with respect to the Radon measure induced by the Lorentzian structure. We note that in contrast to the Riemannian case where the Laplacian is an elliptic differential operator, the Laplacian for the Lorentzian manifold $X_{\Gamma}$ is a hyperbolic operator, and thus eigenfunctions may and may not be smooth functions by the failure of the elliptic regularity theorem (see [7, Sect. 3.1] for example). Kassel-Kobayashi [6] proved the following: if $\Gamma$ is sharp, then there exists $m_{0}=m_{0}(\Gamma)>0$ such that

[TABLE]

In particular, they proved that the discrete spectrum $\mathrm{Spec}_{d}(\square_{X_{\Gamma}})$ is infinite in the setting where $\Gamma$ is sharp.

A natural question would be whether the Laplacian $\square_{X_{\Gamma}}$ still has an $L^{2}$-eigenvalue if the discontinuous group $\Gamma$ is non-sharp. As an application of the sharpness criterion (Proposition 3.16) and an upper estimate of the counting as in Theorem 1.1, we see that the machinery developed in [6] also can be applied to some non-sharp discontinuous groups, and prove:

Theorem 1.5** (see Theorem 5.1 and Example 5.2).**

There exist a non-sharp discontinuous group $\Gamma$ for $\mathrm{AdS}^{3}$ and $m_{0}=m_{0}(\Gamma)>0$ such that

[TABLE]

1.3 Organization of the paper

In Section 2, we give preliminary results, including a pseudo-ball $B(R)$ and the Kobayashi-Benoist properness criterion applied to our $\mathrm{AdS}^{3}$ setting. In Section 3, we construct a family of infinitely generated Schottky-like discontinuous groups $\Gamma_{\nu}\equiv\Gamma_{\nu}(a_{-},a_{+},r,R)$ for $\mathrm{AdS}^{3}$ associated to quadruples $(a_{-},a_{+},r,R)$ of real-valued sequences satisfying some conditions for a sufficiently large $\nu\in\mathbb{N}$. Moreover, we recall the notion of sharpness for discontinuous groups, and find a necessary and sufficient condition on the quadruple $(a_{-},a_{+},r,R)$ such that $\Gamma_{\nu}$ is sharp. In Section 4, we find a lower bound for the counting $N_{\Gamma_{\nu}}(x,R)$, and prove Theorem 1.2. In Section 5, we find a sufficient condition on the quadruple $(a_{-},a_{+},r,R)$ such that the counting $N_{\Gamma_{\nu}}(x,R)$ is at most of exponential growth, and complete the proof of Theorem 1.1 with the sharpness criterion given in Section 3. The proof of Theorem 1.5 is then given by applying the method established by Kassel-Kobayashi [6].

Notation. $\mathbb{N}=\{0,1,2,\ldots\}$ and $\mathbb{N}_{+}=\{1,2,3,\ldots\}$.

2 Preliminary results about $\mathrm{AdS}^{3}$

In this section, we collect some preliminary results about $\mathrm{AdS}^{3}$ that will be needed for later sections.

Let $V$ be a four-dimensional real vector space equipped with a quadratic form $Q$ of signature $(2,2)$ on $V$, and $X$ the hypersurface given by $X=\{v\in V\mid Q(v)=1\}$. The tangent space $T_{v}X$ at $v\in X$ is identified with the orthogonal complement $(\mathbb{R}v)^{\bot}$ in $V$ with respect to $Q$. The restriction of $-Q$ to the hyperplane $(\mathbb{R}v)^{\bot}$ is a quadratic form of signature $(2,1)$, which induces a Lorentzian structure on $X$ with constant sectional curvature $-1$. The resulting Lorentzian manifold is called the $3$-dimensional anti-de Sitter space $\mathrm{AdS}^{3}$.

2.1 Pseudo-balls in $\mathrm{AdS}^{3}$

In this subsection, we consider pseudo-balls $B(R)$ on the Lorentzian manifold $\mathrm{AdS}^{3}$. We work with coordinates on $\mathrm{AdS}^{3}$ by choosing $V=\mathrm{M}(2,\mathbb{R})$ and $Q=\mathrm{det}$. Then $\mathrm{AdS}^{3}$ is identified with $\mathrm{SL}(2,\mathbb{R})$. The direct product group $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ acts on $V=\mathrm{M}(2,\mathbb{R})$ by left and right multiplication, which induces an isometric and transitive action on $\mathrm{AdS}^{3}$. Thus

[TABLE]

Let $o$ be the base point in $\mathrm{AdS}^{3}$ corresponding to the identity matrix in $\mathrm{SL}(2,\mathbb{R})$. The pseudo-distance $\|g\|(\geq 0)$ of $g\in\mathrm{SL}(2,\mathbb{R})$ from the base point $o$ is defined by the formula

[TABLE]

We give two alternative definitions of the pseudo-distance $\|g\|$ as below.

First, for $\theta\in[0,2\pi]$ and $t\geq 0$, we set $k(\theta):=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}$ and $a(t):=\begin{pmatrix}e^{t}&0\\ 0&e^{-t}\end{pmatrix}$. Any element $g\in\mathrm{SL}(2,\mathbb{R})$ can be expressed by the Cartan decomposition $g=k(\theta_{1})a(t)k(\theta_{2})$ with $\theta_{1},\theta_{2}\in[0,2\pi]$ and unique $t\geq 0$. Then (2.1) implies

[TABLE]

This interpretation shows readily that the map $\|\cdot\|\colon\mathrm{SL}(2,\mathbb{R})\rightarrow[0,\infty)$ is proper and that for any $R>0$,

[TABLE]

is a compact subset of $\mathrm{SL}(2,\mathbb{R})$, to which we refer as the pseudo-ball of radius $R$. The family $\{B(R)\}_{R>0}$ is well-rounded (Eskin-McMullen [2, Thm. 6.1]).

Second, we realize the hyperbolic space $\mathbb{H}^{2}$ as the upper-half plane $\{x+\sqrt{-1}y\in\mathbb{C}\mid y>0\}$ endowed with the metric tensor $ds^{2}=y^{-2}(dx^{2}+dy^{2})$. We write $d_{\mathbb{H}^{2}}$ for the hyperbolic distance of $\mathbb{H}^{2}$. The group $\mathrm{SL}(2,\mathbb{R})$ acts isometrically on $\mathbb{H}^{2}$ by linear fractional transformations. In this model, the pseudo-distance $\|g\|$ is computed by (2.2) as follows:

Lemma 2.1** (see e.g. [3, (A.1) and (A.2)]).**

For any $g\in\mathrm{SL}(2,\mathbb{R})$,

[TABLE]

In particular, for any point $x+\sqrt{-1}y\in\mathbb{H}^{2}$,

[TABLE]

The following properties of the pseudo-distance follow from Lemma 2.1:

Lemma 2.2**.**

For $g,g^{\prime}\in\mathrm{SL}(2,\mathbb{R})$,

1. $(1)$

$\|g^{-1}\|=\|g\|$. 2. $(2)$

$|\|g\|-\|g^{\prime}\||\leq\|gg^{\prime}\|\leq\|g\|+\|g^{\prime}\|$.

The Jacobian of the Cartan decomposition $(0,2\pi)\times(0,\infty)\times(0,2\pi)\rightarrow\mathrm{SL}(2,\mathbb{R})$ defined by $(\theta_{1},t,\theta_{2})\mapsto k(\theta_{1})a(t)k(\theta_{2})$ equals $\sinh(2t)$ with respect to this Lorentzian structure on $\mathrm{SL}(2,\mathbb{R})\cong\mathrm{AdS}^{3}$ and the standard metrics of the intervals $(0,2\pi)$ and $(0,\infty)$. Hence the following integral formula holds:

[TABLE]

Therefore, the volume $\mathrm{vol}(B(R))$ equals $2\pi^{2}(\cosh(R)-1)$ since $k(\theta_{1})a(t)k(\theta_{2})\in B(R)$ if and only if $2t\leq R$.

2.2 Discontinuous groups for $\mathrm{AdS}^{3}$

Let $G$ be a Lie group, $H$ a closed subgroup of $G$, and $\Gamma$ a discrete subgroup of $G$, which acts naturally on $X:=G/H$ from the left. In this subsection, we explain the Kobayashi-Benoist criterion for the proper discontinuity of the $\Gamma$-action on $X$ applied to our specific setting where $G=\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$, $H=\mathrm{diag}(\mathrm{SL}(2,\mathbb{R}))$, and $X=\mathrm{AdS}^{3}$.

Throughout this paper, we mean by a discontinuous group for $X$ a discrete subgroup $\Gamma$ of $G$ acting properly discontinuously and freely on $X$ (Kobayashi [11, Def. 1.3]). A torsion-free discrete subgroup $\Gamma$ of $G$ is a discontinuous group for $X$ if and only if $\Gamma$ acts properly discontinuously on $X$. Proper discontinuity is a serious constraint when the isotropy subgroup of $G$ on $X$ is noncompact. Geometrically, one should note that not every discrete subgroup of isometries can act properly discontinuously on a pseudo-Riemannian manifold $X$. Kobayashi [9] and Benoist [1] established a properness criterion for reductive $G$ generalizing the original properness criterion of Kobayashi [8].

Applying the Kobayashi-Benoist properness criterion to our specific setting, we can determine whether the $\Gamma$-action on $\mathrm{AdS}^{3}$ is properly discontinuous in terms of the pseudo-distance defined in Section 2.1 as follows:

Fact 2.3** (Kobayashi [9, Thm. 3.4] and Benoist [1, Thm. 5.2]).**

Let $\Gamma$ be a discrete subgroup of $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$. The following are equivalent:

1. $(\mathrm{i})$

The action of $\Gamma$ on $\mathrm{AdS}^{3}$ is properly discontinuous. 2. $(\mathrm{ii})$

For any $C>0$, the set $\{(\alpha,\beta)\in\Gamma\mid\left|\|\alpha\|-\|\beta\|\right|<C\}$ is finite.

3 Discontinuous groups

$\Gamma_{\nu}(a_{-},a_{+},r,R)$ for $\mathrm{AdS}^{3}$

In this section, we introduce a family of Schottky-like subgroups $\Gamma_{\nu}(a_{-},a_{+},r,R)$ of $G=\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ in Definition 3.1 associated to the following data:

• •

$\nu\in\mathbb{N}$;

• •

$a_{-},a_{+},r,R\colon\mathbb{N}\rightarrow\mathbb{R}_{>0}$ satisfying Assumptions 1–3 below.

We give a properness criterion and a sharpness criterion for the action of $\Gamma_{\nu}(a_{-},a_{+},r,R)$ on $\mathrm{AdS}^{3}$ for any sufficiently large $\nu\in\mathbb{N}$ in terms of the quadruple $(a_{-},a_{+},r,R)$. Furthermore, we also give sufficient conditions on $\nu$ for discreteness, properness, and sharpness of $\Gamma_{\nu}(a_{-},a_{+},r,R)$. In Section 3.1, we construct an infinitely generated, free discrete subgroup $\Gamma_{\nu}(a_{-},a_{+},r,R)$ of $G$ (Proposition 3.4). In Section 3.2, we introduce a constant $\varepsilon(\nu)$ and prove a key proposition (Proposition 3.8) which will be repeated in later sections. In Section 3.3, based on the Kobayashi-Benoist properness criterion (Fact 2.3), we give a criterion for the $\Gamma_{\nu}(a_{-},a_{+},r,R)$-action on $\mathrm{AdS}^{3}$ to be proper (Proposition 3.12). In Section 3.4, we give a criterion for the $\Gamma_{\nu}(a_{-},a_{+},r,R)$-action on $\mathrm{AdS}^{3}$ to be $(c,0)$-sharp in the sense of Kassel-Kobayashi [6] (Proposition 3.16).

3.1 Construction of discrete subgroups

$\Gamma_{\nu}(a_{-},a_{+},r,R)$

We introduce a coordinate map $\tau\colon\mathbb{R}\times\mathbb{R}\times\mathbb{R}_{>0}\rightarrow\mathrm{SL}(2,\mathbb{R})$ by

[TABLE]

Definition 3.1**.**

Let $(a_{-},a_{+},r,R)$ be a quadruple of positive real valued sequences. Then we define a sequence of elements $(\alpha_{k},\beta_{k})\in G$ by

[TABLE]

For $\nu\in\mathbb{N}$, we define $\Gamma_{\nu}(a_{-},a_{+},r,R)$ as the subgroup of $G$ generated by $\{(\alpha_{k},\beta_{k})\mid k=\nu,\nu+1,\ldots\}$.

Notation 3.2**.**

Let $F^{\infty}$ denote the free group generated by countably many elements $\{\gamma_{k}\}_{k\in\mathbb{N}}$. Let $(\alpha_{k},\beta_{k})\in G$ be a sequence of elements associated to a quadruple $(a_{-},a_{+},r,R)$ by (3.2). Then we write $j\colon F^{\infty}\rightarrow\mathrm{SL}(2,\mathbb{R})$ and $\rho\colon F^{\infty}\rightarrow\mathrm{SL}(2,\mathbb{R})$ for the group homomorphisms such that $j(\gamma_{k})=\alpha_{k}$ and $\rho(\gamma_{k})=\beta_{k}$ for all $k\in\mathbb{N}$. For $\nu\in\mathbb{N}$, let $F_{\nu}^{\infty}$ be the subgroup of $F^{\infty}$ generated by $\{\gamma_{k}\}_{k=\nu}^{\infty}$.

Then, by Definition 3.1,

[TABLE]

Example 3.3**.**

The subgroup $\Gamma_{\nu}(a_{-},a_{+},r,R)$ for $(a_{-}(k),a_{+}(k),r(k),R(k))=(k^{2},k^{2}+k,1,\log k)$ coincides with $\Gamma_{\nu}^{j,\rho}$ in Guéritaud-Kassel [3, Sect. 10.1].

Proposition 3.4**.**

Suppose that a quadruple of positive real valued sequences $(a_{-},a_{+},r,R)$ satisfies the following assumptions:

Assumption 1.

For any sufficiently large integer $k$, we have

[TABLE]

Assumption 2.

$\displaystyle\lim_{k\to\infty}a_{+}(k)=\lim_{k\to\infty}a_{-}(k)=\infty.$**

Let $\nu\in\mathbb{N}$. If (3.3) holds for any integer $k\geq\nu$, then the subgroup $\Gamma_{\nu}(a_{-},a_{+},r,R)$ of $G$ is discrete and free.

The proof is based on the ping-pong lemma. For this, we need some setups. Let $|\cdot|$ denote the Euclidean norm in the upper-half plane $\mathbb{H}^{2}\subset\mathbb{C}$. Associated to the quadruple $(a_{-},a_{+},r,R)$, we set

[TABLE]

for $k\in\mathbb{N}$ and $\epsilon\in\{+,-\}$, see Figure 3.1. Then we claim:

• •

$A_{\nu}^{-},A_{\nu}^{+},A_{\nu+1}^{-},A_{\nu+1}^{+},\ldots$ are disjoint;

• •

$\alpha_{k}(\mathbb{H}^{2}\smallsetminus A_{k}^{-})\subset A_{k}^{+}$ for $k\geq\nu$;

• •

$\bigcup_{k\geq\nu}(A_{k}^{-}\cup A_{k}^{+})$ is a proper closed subset of $\mathbb{H}^{2}$.

The first claim is immediate from the fact that the inequalities (3.3) in Assumption 1 hold for any integer $k\geq\nu$.

The second claim is implied by the following key property of the map $\tau=\tau(x_{-},x_{+},u)$ in (3.1):

[TABLE]

which is readily seen from the identity

[TABLE]

To prove the third claim, it suffices to show

[TABLE]

for any $z\in\mathbb{H}^{2}$ and any $\epsilon\in\{+,-\}$. Recall $a_{\epsilon}(k)>0$ for any $k\in\mathbb{N}$. Hence, by Assumption 1, we have

[TABLE]

By Assumption 2, we obtain (3.7). This proves the third claim.

We are ready to prove Proposition 3.4.

Proof of Proposition 3.4.

The subgroup of $\mathrm{SL}(2,\mathbb{R})$ generated by $\{\alpha_{k}\mid k=\nu,\nu+1,\ldots\}$ is free and discrete by the standard ping-pong argument, namely, by applying Lemma 3.5 below to $H=\mathrm{SL}(2,\mathbb{R})$, $Y=\mathbb{H}^{2}$, and $Y_{k}^{\pm}=A_{k+\nu}^{\pm}$. Hence $\Gamma_{\nu}(a_{-},a_{+},r,R)$ is also free and discrete. ∎

Lemma 3.5** (The ping-pong lemma).**

Let $H$ be a topological group acting continuously on a topological space $Y$, and $\Gamma$ the subgroup generated by $h_{0},h_{1},\ldots\in H$. Suppose that there exist disjoint closed subsets $Y_{0}^{-},Y_{0}^{+},Y_{1}^{-},Y_{1}^{+},\ldots$ of $Y$ satisfying the following:

1. $(i)$

$h_{k}(Y\smallsetminus Y_{k}^{-})\subset Y_{k}^{+}$* for any $k\in\mathbb{N}$.* 2. $(ii)$

$\bigcup_{k\in\mathbb{N}}(Y_{k}^{-}\cup Y_{k}^{+})$* is a proper closed subset of $Y$.*

Then, $\Gamma$ is a free discrete subgroup of $H$.

Although the proof of Lemma 3.5 is standard, we give a proof for the sake of completeness.

Proof.

The conditions (i) and (ii) may be restated as $h_{k}^{-1}(Y\smallsetminus Y_{k}^{+})\subset Y_{k}^{-}$ for any $k\in\mathbb{N}$ and $U:=Y\smallsetminus\bigcup_{i\in\mathbb{N}}(Y_{i}^{-}\cup Y_{i}^{+})$ is a non-empty open subset of $Y$, respectively. Take any $h=h_{i_{0}}^{s_{0}}\cdots h_{i_{n}}^{s_{n}}\in\Gamma$. Suppose that this is a reduced expression, namely, $s_{0},\ldots,s_{n}\in\{1,-1\}$ and $s_{k}=s_{k+1}$ whenever $i_{k}=i_{k+1}$ for $0\leq k<n$. Then we have $h(U)\subset Y_{i_{0}}^{s_{0}}$ and thus $h(U)\cap U=\emptyset$. Hence $h\neq e$ and $\{h_{0},h_{1},\ldots\}$ is a free generator of $\Gamma$. Take a neighborhood $V$ of $e$ in $H$ and a non-empty open subset $W$ of $U$ such that $V\cdot W\subset U$. Then $\Gamma\cap V=\{e\}$ and thus $\Gamma$ is discrete in $H$. ∎

As a byproduct of the above discussion, we also see the following:

Corollary 3.6**.**

Let a quadruple $(a_{-},a_{+},r,R)$ and $\nu\in\mathbb{N}$ be as in the setting of Proposition 3.4. Let $z\in\mathbb{H}^{2}\smallsetminus\bigcup_{k\geq\nu}(B_{k}^{-}\cup B_{k}^{+})$ and $\gamma\in F^{\infty}_{\nu}\smallsetminus\{e\}$. If the expression $\gamma=\gamma_{i_{0}}^{s_{0}}\ldots\gamma_{i_{n}}^{s_{n}}$ is reduced, then we have $j(\gamma)z\in A_{k_{0}}^{s_{0}}$ and $\rho(\gamma)z\in B_{k_{0}}^{s_{0}}$. Here we have used the convention that $A_{k}^{\pm 1}=A_{k}^{\pm}$ and $B_{k}^{\pm 1}=B_{k}^{\pm}$.

Example 3.7**.**

The quadruples $(a_{-},a_{+},r,R)$ in (1)–(3) of Table 3.1 satisfy Assumptions 1 and 2 in Proposition 3.4. For the reader’s convenience, we list in Table 3.1 also the asymptotic behaviors of the counting $N_{\Gamma}(x,R)$ as $R$ tends to infinity where $\Gamma$ are the discontinuous groups $\Gamma_{\nu}(a_{-},a_{+},r,R)$ associated to the quadruples $(a_{-},a_{+},r,R)$ in (1) and (3). We refer to Example 5.2 and Example 4.3 below for details about the counting.

3.2 A key proposition

In this subsection, we prove a key proposition (Proposition 3.8) which will be applied to:

• •

Determine the properness of the $\Gamma_{\nu}(a_{-},a_{+},r,R)$-action on $\mathrm{AdS}^{3}$ (Proposition 3.12);

• •

Determine the sharpness of the $\Gamma_{\nu}(a_{-},a_{+},r,R)$-action on $\mathrm{AdS}^{3}$ in the sense of Kassel-Kobayashi [6](Proposition 3.16);

• •

Estimate an upper bound of the counting $N_{\Gamma_{\nu}(a_{-},a_{+},r,R)}(x,R)$ (Theorem 5.1 $(1)$).

For this, in addition to Assumptions 1 and 2 in Proposition 3.4, we need to impose another condition on a quadruple $(a_{-},a_{+},r,R)$ of positive real valued sequences. For $\nu\in\mathbb{N}$, we set

[TABLE]

From now on, we always assume the following:

Assumption 3.

$\displaystyle\lim_{\nu\to\infty}\eta(\nu)=0$.

Furthermore, we introduce a constant $\varepsilon(\nu)$. Let $(a_{-},a_{+},r,R)$ be a quadruple of positive real valued sequences satisfying Assumptions 1–3. Let $(\alpha_{k},\beta_{k})\in G=\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ be the sequence of the elements associated to the quadruple $(a_{-},a_{+},r,R)$ by (3.2). For $\nu\in\mathbb{N}$, we put

[TABLE]

The sequence $\varepsilon=\{\varepsilon(\nu)\}_{\nu\in\mathbb{N}}$ is monotone decreasing. We claim

[TABLE]

To see this, we note

[TABLE]

In fact, for sufficiently large $k\in\mathbb{N}$, the positive valued sequences $a_{-}=\{a_{-}(k)\}_{k\geq\nu}$ and $a_{+}=\{a_{+}(k)\}_{k\geq\nu}$ are monotone increasing by Assumption 1. Hence we get (3.10) by Assumption 3. Therefore, $\lim_{\nu\to\infty}\varepsilon(\nu)=0$ follows from (3.10) and Assumptions 2,3.

This subsection is devoted to proving the following proposition:

Proposition 3.8**.**

Given a quadruple $(a_{-},a_{+},r,R)$ satisfying Assumptions 1–3 and $\nu\in\mathbb{N}$, let $j$, $\rho$, and $F_{\nu}^{\infty}$ be as in Notation 3.2. Suppose that the inequalities (3.3) hold for any integer $k\geq\nu-1$ and that $\varepsilon(\nu)<1$. Let $\gamma\neq e$ be an arbitrary element of $F_{\nu}^{\infty}$ and $m:=\ell(\gamma)$ the word length of $\gamma$. We write $\gamma=\gamma_{k_{1}}^{s_{1}}\ldots\gamma_{k_{m}}^{s_{m}}$ for the reduced expression where $s_{1},\ldots,s_{m}\in\{1,-1\}$ and $k_{1},\ldots,k_{m}\geq\nu$. Then the following inequalities hold:

1. (1)

$\displaystyle\left|\|j(\gamma)\|-\|\rho(\gamma)\|-2\sum_{i=1}^{\ell(\gamma)}\log\frac{R(k_{i})}{r(k_{i})}\right|\leq\ell(\gamma)\varepsilon(\nu)$. 2. (2)

$\displaystyle\|j(\gamma)\|\leq\left(\sum_{i=1}^{\ell(\gamma)}2\log\frac{a_{+}(k_{i})a_{-}(k_{i})}{r(k_{i})}\right)+\ell(\gamma)\varepsilon(\nu)$.

Remark 3.9**.**

Guéritaud-Kassel [3, Sect. 10.1] gave an upper bound of

[TABLE]

for the quadruple $(a_{-}(k),a_{+}(k),r(k),R(k))=(k^{2},k^{2}+k,1,\log k)$, see Table 3.1 (2). However, since the explanation given there was not clear to the author, we take an alternative approach to prove (1) in our general setting where a quadruple $(a_{-},a_{+},r,R)$ is arbitrary subject to Assumptions 1–3.

We prepare some results needed for the proof of Proposition 3.8.

Lemma 3.10**.**

Let $(a_{-},a_{+},r,R)$ and $\nu\in\mathbb{N}$ be as in Proposition 3.8 and $B_{k}^{\pm}$ the half-disks defined in (3.4). Then, the following assertions hold:

1. $(1)$

$a_{-}(k)>\max\{1,R(k)\}$* for any integer $k\geq\nu$.* 2. $(2)$

$\displaystyle\sqrt{-1}\notin\bigcup_{k\geq\nu}(B_{k}^{-}\cup B_{k}^{+})$* in the upper-half plane $\mathbb{H}^{2}$.* 3. $(3)$

The following inequalities hold:

1. $(3a)$

$\displaystyle\varepsilon(\nu)\geq\max_{k\geq\nu}\left\{6\log\left(1+e^{-2d_{\mathbb{H}^{2}}(B_{k}^{+},\sqrt{-1})}\right),\ 6\log\left(1+e^{-2d_{\mathbb{H}^{2}}(B_{k}^{-},\sqrt{-1})}\right)\right\}$. 2. $(3b)$

$\displaystyle\varepsilon(\nu)\geq\max_{k\geq\nu}\left\{12\left|\log\left(1+\frac{R(k)}{a_{-}(k)}\right)\right|,\ 12\left|\log\left(1-\frac{R(k)}{a_{-}(k)}\right)\right|\right\}$. 3. $(3c)$

$\displaystyle\varepsilon(\nu)\geq\max_{k\geq\nu}\left\{6\log\left(1+\frac{R(k)^{2}+1}{(a_{-}(k)-R(k))^{2}}\right)\right\}$. 4. $(3d)$

$\displaystyle\varepsilon(\nu)\geq\max\left\{4|\log(1+\eta(\nu))|,\ 4|\log(1-\eta(\nu))|\right\}$.

Proof.

Take any integer $\nu$ such that the inequalities (3.3) hold for any integer $k\geq\nu-1$ and that $\varepsilon(\nu)<1$. Then, by $\varepsilon(\nu)<1$, we have $a_{-}(k)>24R(k)>3R(k)$ and $a_{-}(k)-R(k)>\sqrt{6(R(k)^{2}+1)}>\sqrt{6}>2$. Thus we get

[TABLE]

Using the inequality (3.12), we can immediately deduce the assertion $(1)$, and $(2)$ follows obviously from $(1)$.

To prove the assertion $(3)$, we apply the inequalities $t\geq\log(1+t)$ and $2s\geq|\log(1-s)|$ for $t\geq 0$ and $0\leq s\leq\frac{1}{2}$. These inequalities imply $(3b)$–$(3d)$ since we have $\displaystyle\frac{R(k)}{a_{-}(k)}<\frac{1}{2}$ (from (3.11)) and $\displaystyle\eta(\nu)(<\frac{\varepsilon(\nu)}{8})<\frac{1}{2}$.

Let $\delta\in\{+,-\}$. To prove the inequality $(3a)$, we claim

[TABLE]

If we could show this claim, then the inequality $(3a)$ is proved again by the inequality $t\geq\log(1+t)$ for $t\geq 0$.

We now prove (3.13). Again by the inequalities (3.11) and (3.12), we have

[TABLE]

We write $x_{\delta}(k)+\sqrt{-1}y_{\delta}(k)\in\mathbb{H}^{2}$ for the closest point of $B_{k}^{\delta}$ to $\sqrt{-1}$ with respect to the hyperbolic distance. Obviously, we have

[TABLE]

Thus, by Lemma 2.1,

[TABLE]

where the last inequality follows from Assumption 1. Hence, since $e^{-d_{\mathbb{H}^{2}}(B_{k}^{\delta},\sqrt{-1})}\leq 1$, we have

[TABLE]

where the second inequality follows from (3.14). Therefore,

[TABLE]

This proves (3.13) and thus the lemma holds. ∎

Lemma 3.11**.**

In the setting of Proposition 3.8, the following assertions hold for both $\varphi=j$ or $\varphi=\rho$:

1. (1)

$\varphi(\gamma)\sqrt{-1}\in B_{k_{1}}^{s_{1}}$. 2. (2)

$\|\varphi(\gamma)\|\geq d_{\mathbb{H}^{2}}(B_{k_{1}}^{s_{1}},\sqrt{-1})$. 3. (3)

$\mathrm{Re}(\varphi(\gamma)\sqrt{-1})\geq a_{-}(k_{1})-R(k_{1})$. 4. (4)

$\mathrm{Im}(\varphi(\gamma)\sqrt{-1})\leq R(k_{1})$. 5. (5)

$\displaystyle\left|\|\varphi(\gamma)\|-\log(2\cosh\|\varphi(\gamma)\|)\right|\leq\frac{1}{6}\varepsilon(\nu)$. 6. (6)

$\displaystyle\left|2\log\frac{\mathrm{Re}(j(\gamma)\sqrt{-1})}{\mathrm{Re}(\rho(\gamma)\sqrt{-1})}\right|\leq\frac{1}{3}\varepsilon(\nu)$. 7. (7)

$\displaystyle\left|-\log\frac{\mathrm{Im}(j(\gamma)\sqrt{-1})}{\mathrm{Im}(\rho(\gamma)\sqrt{-1})}-2\sum_{i=1}^{m}\log\frac{R(k_{i})}{r(k_{i})}\right|\leq(\ell(\gamma)-1)\varepsilon(\nu)$.

Here $\ell(\gamma)$ is the word length of $\gamma$, and we have used the convention that $B_{k}^{\pm 1}=B_{k}^{\pm}$ and $a_{\pm 1}=a_{\pm}$.

Proof.

By Lemma 3.10 $(1)$, we note that $\sqrt{-1}\notin\bigcup_{k\geq\nu}(B_{k}^{-}\cup B_{k}^{+})$. Hence the assertion (1) follows from Corollary 3.6, and thus we get (3) and (4). By Lemma 2.1, (2) follows from (1).

We have

[TABLE]

where the second and third inequalities follow from Lemmas 3.11 (2) and 3.10, respectively. This proves (5).

By (1), we get $\left|\mathrm{Re}(\varphi(\gamma)\sqrt{-1})-a_{s_{1}}(k_{1})\right|\leq R(k_{1})$. Thus noting $a_{s_{1}}(k_{1})>R(k_{1})$ by $\varepsilon(\nu)<1$, we have

[TABLE]

by Lemma 3.10. Hence (6) follows.

In the following, we set for $k\in\mathbb{N}$

[TABLE]

Then, by (3.6),

[TABLE]

Let us prove (7). Define $\sigma_{0},\ldots,\sigma_{m}\in F_{\nu}^{\infty}$ by $\sigma_{i}=\gamma_{k_{i+1}}^{s_{i+1}}\ldots\gamma_{k_{m}}^{s_{m}}$ for $0\leq i<m$ and $\sigma_{m}=1$. We note $\sigma_{0}=\gamma$. For $0\leq i\leq m$, we set

[TABLE]

We claim:

[TABLE]

[TABLE]

This claims imply (7). Indeed, we get (7) by summing up $(\mathrm{Q}_{i})$ and $(\mathrm{D}_{i})$ for all $i$ because $\log Q(\sigma_{m})=0$.

It remains to verify $(\mathrm{Q}_{i})$ and $(\mathrm{D}_{i})$. Because $\sigma_{i}=\gamma_{k_{i+1}}^{s_{i+1}}\sigma_{i+1}$, we have

[TABLE]

for $0\leq i<m$ by (3.17). Thus ($\mathrm{Q}_{i}$) follows from the formula

[TABLE]

We observe that $(\mathrm{D}_{i})$ for $i=m$ is obvious because $D(j(\sigma_{m}))=D(\rho(\sigma_{m}))=|\sqrt{-1}-a_{-s_{m}}(k_{m})|$. For $0<i<m$, by the triangle inequality, we have

[TABLE]

since $\varphi(\sigma_{i})\sqrt{-1}\in B_{k_{i+1}}^{s_{i+1}}$ by (1). Hence noting

[TABLE]

by $\varepsilon(\nu)<1$, we obtain

[TABLE]

The second and third inequalities follow from the definition (3.8) of $\eta(\nu)$ and Lemma 3.10, respectively. Hence $\left|\log\frac{D(j(\sigma_{i}))}{D(\rho(\sigma_{i}))}\right|\leq\frac{\varepsilon(\nu)}{2}$. Thus ($\mathrm{D}_{i}$) holds for all $i$ and the proof of (7) is completed. ∎

We are ready to prove Proposition 3.8.

Proof of Proposition 3.8.

1. $(1)$

Let $\varphi\colon F^{\infty}_{\nu}\rightarrow\mathrm{SL}(2,\mathbb{R})$ be either $j$ or $\rho$.

By Lemma 3.11 (5), we get

[TABLE]

By Lemma 2.1,

[TABLE]

Therefore, we have

[TABLE]

The second and third inequalities follow from Lemma 3.11 (3), (4) and Lemma 3.10, respectively. Hence

[TABLE]

Summing up Lemma 3.11 (6), (7), (3.18), and (3.20), we obtain (1). 2. $(2)$

We have

[TABLE]

by summing up (5) of Lemma 3.11 and (3.19) for $\varphi=j$. By (3.16), we have

[TABLE]

We claim:

[TABLE]

If we could show this claim, then (2) is proved by summing up (3.21), (3.22), and (3.23) since $m=\ell(\gamma)\geq 1$.

It remains to verify (3.23). As in the proof of Lemma 3.11, $\gamma=\gamma_{k_{1}}^{s_{1}}\cdots\gamma_{k_{m}}^{s_{m}}\in F_{\nu}^{\infty}$ defines a sequence of elements $\sigma_{0},\ldots,\sigma_{m}\in F_{\nu}^{\infty}$ by $\sigma_{i}=\gamma_{k_{i+1}}^{s_{i+1}}\ldots\gamma_{k_{m}}^{s_{m}}$ for $0\leq i<m$ with $\sigma_{0}=\gamma$ and $\sigma_{m}=1$. We set $D(j(\sigma_{i})):=\left|j(\sigma_{i})\sqrt{-1}-a_{-s_{i}}(k_{i})\right|$. Multiplying all ($\mathrm{I}_{i}$) for $0\leq i<m$, with $\varphi=j$, we get $\mathrm{Im}(j(\gamma)\sqrt{-1})=\prod_{i=1}^{m}(r(k_{i})/D(j(\sigma_{i})))^{2}$. Hence

[TABLE]

Since $\sigma_{m}=1$, we have

[TABLE]

where the third and fourth inequalities follow from Assumption 1 and the definition (3.9) of $\varepsilon(\nu)$, respectively.

Note $|s-t|\leq st$ for $s,t>1$ and $a_{+}(k)>a_{-}(k)>1$ for all $k\geq\nu$ by (3.3) and Lemma 3.10 $(1)$. Thus, by ($\mathrm{D}^{\prime}_{i}$) for $1\leq i<m$, we have

[TABLE]

Now the inequality (3.23) follows from (3.24), (3.25), and (3.26). Thus the proof of (2) is completed.

∎

3.3 Proper discontinuity of the action of

$\Gamma_{\nu}(a_{-},a_{+},r,R)$

In this section we give a necessary and sufficient condition for $\Gamma_{\nu}(a_{-},a_{+},r,R)$ to act properly discontinuously on $\mathrm{AdS}^{3}$.

The action of the Schottky-like discrete group $\Gamma_{\nu}(a_{-},a_{+},r,R)$ on $\mathrm{AdS}^{3}$ is not always properly discontinuous. We give a necessary and sufficient condition for this action to be properly discontinuous:

Proposition 3.12**.**

Let $(a_{-},a_{+},r,R)$ be a quadruple of sequences as in Proposition 3.8. The action of $\Gamma_{\nu}(a_{-},a_{+},r,R)$ on $\mathrm{AdS}^{3}$ is properly discontinuous for sufficiently large $\nu\in\mathbb{N}$ if and only if

[TABLE]

In this case, take $\nu\in\mathbb{N}$ as in Proposition 3.8 and assume that $\log(R(k)/r(k))>1$ for any integer $k\geq\nu$. Then the action of $\Gamma_{\nu}(a_{-},a_{+},r,R)$ is properly discontinuous.

Postponing the proof of Proposition 3.12, we state its immediate consequences in Proposition 3.13 and Lemma 4.1 as below. First, since the group $\Gamma_{\nu}(a_{-},a_{+},r,R)$ is torsion-free, any properly discontinuous action is free, hence we obtain:

Proposition 3.13**.**

Let a quadruple $(a_{-},a_{+},r,R)$ and $\nu\in\mathbb{N}$ be as in Proposition 3.12. Assume the condition (3.27). Then $\Gamma_{\nu}(a_{-},a_{+},r,R)$ is a discontinuous group for $\mathrm{AdS}^{3}$.

Example 3.14**.**

All $(a_{-},a_{+},r,R)$ in Table 3.1 apply to Proposition 3.13.

Let us prove Proposition 3.12.

Proof of Proposition 3.12.

Recall $\Gamma_{\nu}(a_{-},a_{+},r,R)=\{(j(\gamma),\rho(\gamma))\mid\gamma\in F^{\infty}_{\nu}\}$. The Kobayashi-Benoist properness criterion (Fact 2.3) tells us that $\Gamma_{\nu}(a_{-},a_{+},r,R)$ acts properly discontinuously on $\mathrm{AdS}^{3}$ if and only if

[TABLE]

We suppose that the action of $\Gamma_{\nu}(a_{-},a_{+},r,R)$ on $\mathrm{AdS}^{3}$ is properly discontinuous for any sufficiently large $\nu\in\mathbb{N}$. Then $\lim_{k\to\infty}\left|\|j(\gamma_{k})\|-\|\rho(\gamma_{k})\|\right|=\infty$ by Fact 2.3. By Proposition 3.8 (1),

[TABLE]

for any $k\geq\nu$ and thus $\lim_{k\to\infty}R(k)/r(k)=\infty$.

Conversely, we suppose $\lim_{k\to\infty}R(k)/r(k)=\infty$. Take $\nu\in\mathbb{N}$ as in Proposition 3.8 and assume that $\log(R(k)/r(k))>1$ for any integer $k\geq\nu$. Since $\varepsilon(\nu)<1$, we note $\|j(\gamma)\|-\|\rho(\gamma)\|\geq 0$ by Proposition 3.8 (1) for any $\gamma\in F^{\infty}_{\nu}$. Assume that $\gamma\in F^{\infty}_{\nu}\smallsetminus\{e\}$ satisfies $|\|j(\gamma)\|-\|\rho(\gamma)\||<C$. Let $m:=\ell(\gamma)$ be the word length of $\gamma$ and we write $\gamma=\gamma_{k_{1}}^{s_{1}}\cdots\gamma_{k_{m}}^{s_{m}}$ for the reduced expression where $s_{1},\ldots,s_{m}\in\{1,-1\}$ and $k_{1},\ldots,k_{m}\geq\nu$. By Proposition 3.8 (1),

[TABLE]

Hence $\ell(\gamma)<C$. Again by Proposition 3.8 (1), we get

[TABLE]

and there are only finitely many $\gamma$ satisfying this inequality. By Fact 2.3, the action of $\Gamma_{\nu}(a_{-},a_{+},r,R)$ on $\mathrm{AdS}^{3}$ is properly discontinuous. ∎

3.4 Sharpness of the $\Gamma_{\nu}(a_{-},a_{+},r,R)$-action

The notion of sharpness was introduced in Kassel-Kobayashi [6], although the idea was already implicit in [10]. It is defined for a general homogeneous space of reductive type. However, in this section, we explain it only for $\mathrm{AdS}^{3}$. Moreover, we find a necessary and sufficient condition that the discontinuous group $\Gamma_{\nu}(a_{-},a_{+},r,R)$ for $\mathrm{AdS}^{3}$ is sharp.

The Cartan projection $\mu$ for the direct product group $G=\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ is given by $\mu(g)=(\|\alpha\|,\|\beta\|)$ for $g=(\alpha,\beta)\in G$, where we recall that $\|\cdot\|$ is the pseudo-distance in $\mathrm{SL}(2,\mathbb{R})$. By Fact 2.3, a discrete subgroup $\Gamma$ of $G$ acts properly discontinuously on $\mathrm{AdS}^{3}$ if and only if $\mu(\Gamma)$ “goes away from the line $x=y$ at infinity”. The condition of sharpness is stronger than the condition of proper discontinuity as Definition 3.15 below, and $\Gamma$ is sharp for $\mathrm{AdS}^{3}$ if $\mu(\Gamma)$ “goes away from the line $x=y$ at infinity” with a speed that is at least linear ([6, p. 152]) as in Figure 3.2. See also [6, Ch. 4, Fig. 1] for the illustration of sharp actions in the general setting.

Definition 3.15** (Kassel-Kobayashi [6, Def. 4.2]).**

Let $c\in(0,1]$ and $C\geq 0$. A discrete subgroup $\Gamma\subset G$ is called $(c,C)$-sharp for $\mathrm{AdS}^{3}$ if for any $\gamma=(\alpha,\beta)\in\Gamma$,

[TABLE]

If $\Gamma$ is $(c,C)$-sharp for some $c$ and $C$, then $\Gamma$ is called sharp for $\mathrm{AdS}^{3}$.

Kassel [5, Thm. 0.2.13] and Guériataud-Kassel [3, Thm. 1.8] proved that any finitely generated discontinuous group $\Gamma\subset G$ for $\mathrm{AdS}^{3}$ is sharp. However, our discontinuous group $\Gamma_{\nu}(a_{-},a_{+},r,R)$ is infinitely generated, and actually, $\Gamma_{\nu}(a_{-},a_{+},r,R)$ may not be sharp for $\mathrm{AdS}^{3}$. The next proposition gives a necessary and sufficient condition for the sharpness of the action of $\Gamma_{\nu}(a_{-},a_{+},r,R)$ on $\mathrm{AdS}^{3}$ for $\nu\gg 0$:

Proposition 3.16**.**

Let $(a_{-},a_{+},r,R)$ be a quadruple of sequences satisfying Assumptions 1–3 and the condition (3.27). We set

[TABLE]

Then $0\leq A\leq 1$. Moreover, take $\nu\in\mathbb{N}$ as in Proposition 3.12. Then the following hold for the discontinuous group $\Gamma_{\nu}\equiv\Gamma_{\nu}(a_{-},a_{+},r,R)$ for $\mathrm{AdS}^{3}$:

1. $(1)$

if $c>A/\sqrt{2(1+(1-A)^{2})}$, then $\Gamma_{\nu}$ is not $(c,C)$-sharp for any $C\geq 0$; 2. $(2)$

Assume $0<c<A/\sqrt{2(1+(1-A)^{2})}$. Then $\Gamma_{\nu}$ is $(c,0)$-sharp if for any $k\geq\nu$, we have

[TABLE]

where $B\in(0,A)$ is defined by $c=B/\sqrt{2(1+(1-B)^{2})}$.

In particular, $\Gamma_{\nu}$ is sharp for $\nu\gg 0$ if and only if $A\neq 0$.

Proof.

Take $\nu\in\mathbb{N}$ as in Proposition 3.12. Then $R(k)>r(k)$ and $a_{+}(k)>a_{-}(k)$ for any integer $k\geq\nu$ by the inequality (3.3). Moreover, $a_{-}(k)>\max\{R(k),1\}$ by Lemma 3.10 $(1)$. Hence we have $a_{-}(k)a_{+}(k)>R(k)>r(k)$ and thus

[TABLE]

In particular, we obtain $0\leq A\leq 1$.

1. $(1)$

Recall that $\Gamma_{\nu}$ is generated by $\{(\alpha_{k},\beta_{k})\mid k=\nu,\nu+1,\ldots\}$. We claim:

[TABLE]

If we could show this claim, then (1) is obvious. Note $\lim_{k\to\infty}\sqrt{\|\alpha_{k}\|^{2}+\|\beta_{k}\|^{2}}=\infty$ since the map $\|\cdot\|\colon\mathrm{SL}(2,\mathbb{R})\rightarrow[0,+\infty]$ is proper and since $\{(\alpha_{k},\beta_{k})\mid k=\nu,\nu+1,\ldots\}$ is an infinite discrete subset of $G$.

We now prove (3.32). By Proposition 3.8 (1), for $k\geq\nu$,

[TABLE]

Since $\alpha_{k}=r(k)^{-1}\begin{pmatrix}a_{+}(k)&-(a_{-}(k)a_{+}(k)+r(k)^{2})\\ 1&-a_{-}(k)\end{pmatrix}$ by the definition (3.2), we have

[TABLE]

where the first and second inequalities follow from Lemma 3.11 (5) and the definition (2.1), respectively. Similarly, we have

[TABLE]

Here we put

[TABLE]

Then, by (3.33)–(3.35), we have

[TABLE]

Here $0<a_{k}<1$ by (3.31) and $\lim_{k\to\infty}b_{k}=0$ by the condition (3.27) since $0<b_{k}\leq\varepsilon(\nu)(2\log(R(k)/r(k)))^{-1}$. Hence, by applying Lemma 3.17 below to $a_{k}$ and $b_{k}$, we obtain the inequality (3.32). Thus (1) is proved. 2. $(2)$

Suppose $0<c<A/\sqrt{2(1+(1-A)^{2})}$. Take $\nu\in\mathbb{N}$ as in Proposition 3.12 and assume that the inequalities (3.30) hold for any integer $k\geq\nu$. Then we note $\varepsilon(\nu)<1$. Setting $2\xi:=A-B(>0)$, for any integer $k\geq\nu$, we have

[TABLE]

by (3.30). Let $\gamma\neq e$ be an arbitrary element of $F_{\nu}^{\infty}$, and $m:=\ell(\gamma)$ the word length of $\gamma$. We write $\gamma=\gamma_{k_{1}}^{s_{1}}\cdots\gamma_{k_{m}}^{s_{m}}$ for the reduced expression where $s_{1},\ldots,s_{m}\in\{1,-1\}$ and $k_{1},\ldots,k_{m}\geq\nu$. By Proposition 3.8 (2), we have

[TABLE]

Here the second, third, and fourth inequalities follow from (3.36), (3.37), and Proposition 3.8 (1), respectively. Then $\|\rho(\gamma)\|<(1-B)\|j(\gamma)\|$ and thus

[TABLE]

Hence $\Gamma_{\nu}$ is $(c,0)$-sharp, which proves (2).

∎

In the above proof, we have used the following elementary lemma:

Lemma 3.17**.**

Let $a=\{a_{k}\}_{k\in\mathbb{N}}\in[0,1]^{\mathbb{N}}$ and $b=\{b_{k}\}_{k\in\mathbb{N}}\in[0,\infty]^{\mathbb{N}}$. If $\displaystyle\lim_{k\to\infty}b_{k}=0$, then we have

[TABLE]

Proof.

We note that the continuous function $f(x):=x(1+(1-x)^{2})^{-\frac{1}{2}}$ is monotone increasing on the interval $[0,2]$. In particular, we have $\displaystyle\liminf_{k\to\infty}f(a_{k})=f(\liminf_{k\to\infty}a_{k})$. Hence, to prove our claim, it suffices to show

[TABLE]

For this purpose, take an arbitrary $0<\epsilon<1$. By $\lim_{k\to\infty}b_{k}=0$, we have $0\leq b_{k}\leq\epsilon(1+\epsilon)^{-1}(<1)$ for any sufficiently large integer $k$. Then the inequalities

[TABLE]

holds. Indeed, since $(1+\epsilon)^{-1}\leq 1-b_{k}\leq 1$, we have

[TABLE]

and

[TABLE]

Thus the inequality (3.39) holds.

Here we take $M>0$ such that $f(y)\leq f(x)+M(y-x)$ for any $0\leq x\leq y\leq 2$. Then we have $f(a_{k})\leq f(a_{k}+b_{k})\leq f(a_{k})+Mb_{k}$ since $0\leq a_{k}\leq a_{k}+b_{k}\leq 2$. By combining this with the inequalities (3.39), we have

[TABLE]

Taking the limit as $k\to\infty$, we obtain

[TABLE]

Since $0<\epsilon<1$ is arbitrary, the equality (3.38) holds. This proves our claim. ∎

Remark 3.18**.**

In Proposition 3.16, we did not treat the case where $c$ takes the critical value $c(a_{-},a_{+},r,R):=A(2(1+(1-A)^{2}))^{-\frac{1}{2}}$. At this critical value, the discontinuous group $\Gamma_{\nu}(a_{-},a_{+},r,R)$ may be $(c,C)$-sharp for all $C\geq 0$, and may not be $(c,C)$-sharp for all $C\geq 0$. We give such examples below.

Let $\delta=\pm 1$ and $b\geq 0$. We define the following quadruple $(a_{-},a_{+},r,R_{\delta})$:

[TABLE]

for $k\geq 2$. This quadruple satisfies Assumptions 1–3 and the condition (3.27). Then the critical value $c\equiv c(a_{-},a_{+},r,R_{\delta})$ is $(2(b^{2}+(b+1)^{2}))^{-\frac{1}{2}}$ because

[TABLE]

Take $\nu\in\mathbb{N}$ as in Proposition 3.12. Let $\Gamma_{\nu,\delta}$ be the discontinuous group $\Gamma_{\nu}(a_{-},a_{+},r,R_{\delta})$ for $\mathrm{AdS}^{3}$. Then, the following hold:

1. $(1)$

if $\delta=1$, then $\Gamma_{\nu,\delta}$ is $(c,0)$-sharp for $\nu\gg 0$, and thus $(c,C)$-sharp for all $C\geq 0$; 2. $(2)$

if $\delta=-1$, then $\Gamma_{\nu,\delta}$ is not $(c,C)$-sharp for all $C\geq 0$.

Indeed, suppose $\delta=1$. Then we may and do take $\nu\gg 0$ such that for all $k\geq\nu$,

[TABLE]

Then the inequality $(b+1)^{-1}\|j(\gamma)\|\leq\|j(\gamma)\|-\|\rho(\gamma)\|$ for any $\gamma\in F_{\nu}^{\infty}$ follows from (1) and (2) of Proposition 3.8 by an argument similar to the proof of Proposition 3.16 (2). Hence $\|\rho(\gamma)\|\leq b(1+b)^{-1}\|j(\gamma)\|$ and thus (1) holds since

[TABLE]

On the other hand, suppose $\delta=-1$. Since $\varepsilon(\nu)<1$, we have $|\|\alpha_{k}\|-\|\beta_{k}\||\leq 2(k-\log\log k)+1$, $\|\alpha_{k}\|\geq 2(b+1)k$, and $\|\beta_{k}\|\geq 2bk$ for any integer $k\geq\nu$ by (3.33), ($(1)$), and (3.35), respectively. Thus (2) follows readily from

[TABLE]

Example 3.19**.**

The discontinuous groups $\Gamma_{\nu}(a_{-},a_{+},r,R)$ associated to the quadruples $(a_{-},a_{+},r,R)$ in Table 3.1 are all non-sharp by Proposition 3.16.

4 Construction of $\Gamma$ with large counting

Let $G=\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$. In this section, we explain the construction of a discontinuous group $\Gamma\subset G$ for which the asymptotic growth of the counting $N_{\Gamma}(x,R)$ is as rapid as we wish, and thus complete the proof of Theorem 1.2.

4.1 Construction of $\Gamma$ and a lower bound of

$N_{\Gamma}(o,R)$

To construct a discontinuous group satisfying the desired property, we use the following lemma which is a special case of Proposition 3.13:

Lemma 4.1**.**

Let $p(t)$ be a $C^{2}$-function defined for sufficiently large $t$, say $t\geq\nu_{0}$, such that $p(t)>0$, $p^{\prime}(t)>0$, $p^{\prime\prime}(t)<0$ and $\lim_{t\to\infty}p(t)=\infty$, and $q(t)$ a continuous function satisfying $\lim_{t\to\infty}q(t)=\infty$. Let $\Gamma_{\nu}(p,q)$ denote the group $\Gamma_{\nu}(a_{-},a_{+}.r,R)$ associated to the sequences

[TABLE]

for $k\geq\nu_{0}$. Then $\Gamma_{\nu}(p,q)$ is a discontinuous group for $\mathrm{AdS}^{3}$ if $\nu\gg 0$.

Proof of Lemma 4.1.

Let us check that our $(a_{-},a_{+},r,R)$ satisfies Assumptions 1–3 and the condition (3.27). If we could check this, it follows from Proposition 3.13 that $\Gamma_{\nu}(p,q)$ is a discontinuous group for $\mathrm{AdS}^{3}$ for sufficiently large $\nu\in\mathbb{N}$, which proves our claim.

Since $\lim_{t\to\infty}q(t)=\infty$, we may assume $q(t)>4$ for any $t\geq\nu_{0}$. We note that the derivative $p^{\prime}(x)$ is monotone decreasing because $p^{\prime\prime}(x)<0$. By the mean value theorem, for any integer $k\geq\nu_{0}$, we have

[TABLE]

[TABLE]

Thus Assumption 1 is verified. Assumption 2 is obviously satisfied.

Take any $k,\ell\geq\nu_{0}$ and $\delta,\epsilon\in\{+,-\}$ such that $(k,\delta)\neq(\ell,\epsilon)$. Since the function $p(x)$ is monotone increasing,

[TABLE]

where the second inequality follows from the mean value theorem. Hence we get

[TABLE]

and thus Assumption 3 is verified since $\lim_{k\to\infty}q(k)=\infty$. The condition (3.27) is also satisfied since $\lim_{k\to\infty}R(k)/r(k)=\lim_{k\to\infty}p(k)=\infty$. This completes the proof. ∎

The next proposition gives a lower bound of the counting $N_{\Gamma}(o,R)$ of the orbit through the base point $o$ of the discontinuous group $\Gamma=\Gamma_{\nu}(p,q)$.

Proposition 4.2**.**

Let $\Gamma_{\nu}\equiv\Gamma_{\nu}(p,q)$ be the discontinuous group for $\mathrm{AdS}^{3}$ associated to a pair of functions $(p(t),q(t))$ as in Lemma 4.1. Here we have taken $\nu\in\mathbb{N}$ as in Proposition 3.12. Then

[TABLE]

Proof.

Since we have taken $\nu\in\mathbb{N}$ as in Proposition 3.12, we get $\log p(k)=\log(R(k)/r(k))>1$ for any $k\geq\nu$ and thus $p(k)\geq 2$. We also recall from Definition 3.1 that the group $\Gamma_{\nu}$ is generated by $\{(\alpha_{k},\beta_{k})\mid k=\nu,\nu+1,\ldots\}$.

By the definitions (3.2) and (4.1), for $k\geq\nu$, we have

[TABLE]

whence the pseudo-distance of $\alpha_{k}^{-1}\beta_{k}\in\mathrm{SL}(2,\mathbb{R})$ is computed by (2.1):

[TABLE]

We then observe for any $R>0$:

[TABLE]

since $2\cosh R>e^{R}$. Since $\Gamma_{\nu}$ acts freely on $\mathrm{AdS}^{3}$, we deduce

[TABLE]

Recall that $p(t)$ is monotone increasing. Hence we conclude $N_{\Gamma_{\nu}}(o,4\log p(t))\geq\#\{k\in\mathbb{N}\mid\nu\leq k\leq t\}>t-\nu$ for all $t\geq\nu$. ∎

Example 4.3**.**

Let $\Gamma_{\nu}\equiv\Gamma_{\nu}(a_{-},a_{+},r,R)$ be the discontinuous group for $\mathrm{AdS}^{3}$ associated to the quadruple $(a_{-},a_{+},r,R)$ in Table 3.1 (3). Here we have taken $\nu\in\mathbb{N}$ as in Proposition 3.12. By an argument similar to the proof of Proposition 4.2, we have

[TABLE]

4.2 Proof of Theorem 1.2

Let us prove Theorem 1.2. We begin with a lemma which reduces the estimate of $N_{\Gamma}(x,R)$ to the case $x=o$.

Lemma 4.4**.**

Let $\Gamma$ be a discontinuous group for $\mathrm{AdS}^{3}$. For $g=(\alpha,\beta)\in G$, $x\in\mathrm{AdS}^{3}$, and $R>0$, we have

[TABLE]

Proof.

By Lemma 2.2, we have $\|gx\|=\|\alpha x\beta^{-1}\|\leq\|x\|+(\|\alpha\|+\|\beta\|)$ and thus $gB(R)\subset B(R+(\|\alpha\|+\|\beta\|))$. Similarly, we have $g^{-1}B(R-(\|\alpha\|+\|\beta\|))\subset B(R)$. Thus we obtain

[TABLE]

The assertion follows immediately from these inclusion relations. ∎

We also use the following elementary lemma.

Lemma 4.5**.**

Given an increasing function $f\colon\mathbb{R}\to\mathbb{R}_{>0}$, there exists a strictly increasing, convex, and $C^{2}$ function $F\colon\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{>0}$ satisfying $F>f$ on $\mathbb{R}_{\geq 0}$ and $\lim_{x\to\infty}F(x)=\infty$.

Proof of Lemma 4.5.

We can construct such a function $F$ on $[0,n]$ by induction on $n\in\mathbb{N}$, applying the following obvious claim to

[TABLE]

Claim. Given an interval $[s,t]$ and constants $a,b,c>0$, there exists a convex, strictly increasing, and $C^{2}$ function $h\colon[s,t]\rightarrow\mathbb{R}_{>0}$ satisfying

[TABLE]

∎

We are ready to prove Theorem 1.2.

Proof of Theorem 1.2.

Since the $G$-action on $\mathrm{AdS}^{3}$ is transitive, we may and do assume $x=o$ by Lemma 4.4. Moreover, by Lemma 4.5, it suffices to consider the case where $f(t)$ is a $C^{2}$-function such that $f(t)>0$, $f^{\prime}(t)>0$, $f^{\prime\prime}(t)>0$ for any $t>0$ and $\lim_{t\to\infty}f(t)=\infty$.

Let $p(t)$ be the inverse function of $sf(s)$ defined for $t\gg 0$. Then $p(t)$ is a $C^{2}$-function such that $p(t)>0$, $p^{\prime}(t)>0$, $p^{\prime\prime}(t)<0$ for $t\gg 0$, and $\lim_{t\to\infty}p(t)=\infty$. Take an arbitrary continuous function $q(t)$ satisfying $\lim_{t\to\infty}q(t)=\infty$. Then $\Gamma_{\nu}(p,q)$ is a discontinuous group for $\mathrm{AdS}^{3}$ if $\nu\gg 0$ by Lemma 4.1. Take $\nu\in\mathbb{N}$ as in Proposition 3.12 and set $\Gamma\equiv\Gamma_{f,o}:=\Gamma_{\nu}(p,q)$. By Proposition 4.2, $N_{\Gamma}(o,R)\geq e^{R/4}f(e^{R/4})-\nu$ for all $R\geq 4\log p(\nu)$. Thus $\lim_{R\to\infty}N_{\Gamma}(o,R)/f(R)=\infty$ and the proof of Theorem 1.2 is completed. ∎

5 Application to the spectral analysis

In this section, we complete the proofs of Theorems 1.1 and 1.5.

Associated to a quadruple $(a_{-},a_{+},r,R)$ of sequences, we have defined in Section 3 the subgroup $\Gamma_{\nu}\equiv\Gamma_{\nu}(a_{-},a_{+},r,R)$ of $G=\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$, and proved that $\Gamma_{\nu}$ is a discontinuous group for $X=\mathrm{AdS}^{3}$ when $\nu\in\mathbb{N}$ is sufficiently large if Assumptions 1–3 and the condition (3.27) are satisfied. Then the quotient space $X_{\Gamma_{\nu}}:=\Gamma_{\nu}\backslash X$ admits an anti-de Sitter structure via the covering $X\rightarrow X_{\Gamma_{\nu}}$. Let $\square_{X_{\Gamma_{\nu}}}$ be the Laplacian on $X_{\Gamma_{\nu}}$, which is not an elliptic operator, but a hyperbolic operator because $X_{\Gamma_{\nu}}$ is a Lorentzian manifold. Our interest here is the discrete spectrum $\mathrm{Spec}_{d}(\square_{X_{\Gamma_{\nu}}})$ of the Laplacian $\square_{X_{\Gamma_{\nu}}}$.

In this section, we give a sufficient condition on $(a_{-},a_{+},r,R)$ for the exponential growth condition (1.1) of the counting $N_{\Gamma_{\nu}}(x,R)$. By the criterion of sharpness in Section 3.4, we give non-sharp $\Gamma_{\nu}$ such that the counting $N_{\Gamma_{\nu}}(x,R)$ has at most an exponential growth uniformly on $x\in X$. Moreover, we use this counting result to construct an infinite subset of $\mathrm{Spec}_{d}(\square_{X_{\Gamma_{\nu}}})$ for such $\Gamma_{\nu}$. We show:

Theorem 5.1**.**

Let $(a_{-},a_{+},r,R)$ be a quadruple of sequences satisfying Assumptions 1-3, and $\nu\in\mathbb{N}$ as in Proposition 3.12. If there exist $A,a>0$ such that $\nu\geq 1+(1-\log A)/a$ and $R(k)/r(k)\geq Ae^{ak}$ for any integer $k\geq\nu$, then the following claims hold:

1. $(1)$

$N_{\Gamma_{\nu}}(x,R)\leq 2^{2R/a}$* for any $x\in X$ and any $R>0$;* 2. $(2)$

there exists $m_{0}=m_{0}(\Gamma_{\nu})>0$ such that

[TABLE]

Example 5.2**.**

The quadruple $(a_{-},a_{+},r,R)$ in Table 3.1 (1) apply to Theorem 5.1.

Postponing the proof of Theorem 5.1, we give proofs of Theorems 1.1 and 1.5.

Proof of Theorems 1.1

and 1.5.

The discontinuous group $\Gamma_{\nu}(a_{-},a_{+},r,R)$ associated to Example 5.2 is non-sharp by Proposition 3.16. Applying $(1)$ and $(2)$ of Theorem 5.1, we get Theorems 1.1 and 1.5, respectively. ∎

To prove Theorem 5.1 $(2)$, namely, to construct an infinite subset of the discrete spectrum, we use the following fact, established by Kassel-Kobayashi [6], and Theorem 5.1 $(1)$ imply Theorem 5.1 $(1)$ immediately:

Fact 5.3** ([6, Thm. 3.8 (1) and its proof]).**

Let $\Gamma$ be a discontinuous group for $X=\mathrm{AdS}^{3}$ satisfying the exponential growth condition (1.1). Then there exists $m_{0}(\Gamma)>0$ such that

[TABLE]

Remark 5.4**.**

Kassel-Kobayashi constructed in [6, Cor. 9.10] an infinite subset of the discrete spectrum $\mathrm{Spec}_{d}(\square_{X_{\Gamma}})$ which is stable under small deformations of a sharp discontinuous group $\Gamma$ in $G$. Then does there exist a stable discrete spectrum under small deformations of $\Gamma_{\nu}(a_{-},a_{+},r,R)$? Since $\Gamma_{\nu}(a_{-},a_{+},r,R)$ is infinitely generated, we need to treat deformation carefully.

A small deformation of $\Gamma_{\nu}(a_{-},a_{+},r,R)$ in $G$ is neither injective nor discrete in general. Therefore, there exists no stable discrete spectrum in the sense of Kassel-Kobayashi [6]. However, the following question looks reasonable: does there exist an infinite subset of the discrete spectrum which is stable under small deformations of $\Gamma_{\nu}(a_{-},a_{+},r,R)$ which act on $\mathrm{AdS}^{3}$ properly discontinuously?

We prepare some results needed for the proof of Theorem 5.1 $(1)$. The following lemma was proved in [6] in the general setting where $X$ is a reductive symmetric space. Since it plays a crucial role in proving Theorem 5.1, we give an elementary proof for $X=\mathrm{AdS}^{3}$ for the convenience of the reader.

Lemma 5.5** ([6, Lem. 4.4 and 4.17]).**

For $(\alpha,\beta)\in G$ and $x\in X=\mathrm{AdS}^{3}$,

[TABLE]

Proof.

By Lemma 2.2, we have

[TABLE]

Summing up (5.1) and (5.2), we have $\|(\alpha,\beta)x\|+\|x\|\geq\left|\|\alpha\|-\|\beta\|\right|$. ∎

Fact 5.6** ([6, Def–Lem. 4.20]).**

Let $\Gamma$ be a discontinuous group for $X=\mathrm{AdS}^{3}$. Then, the set

[TABLE]

is a fundamental domain of $X$ for the action of $\Gamma$. In particular, $\Gamma\mathcal{D}_{X_{\Gamma}}=X$.

Therefore, we may assume $x\in\mathcal{D}_{X_{\Gamma}}$ to study $N_{\Gamma}(x,R)$.

Lemma 5.7** (cf. [6, Lem. 4.21]).**

For any $x\in\mathcal{D}_{X_{\Gamma}}$ and any $\gamma=(\alpha,\beta)\in\Gamma$,

[TABLE]

Proof.

Let $\gamma=(\alpha,\beta)\in\Gamma$ and $x\in\mathcal{D}_{X_{\Gamma}}$. Then we have

[TABLE]

by the definition of $\mathcal{D}_{X_{\Gamma}}$ and Lemma 5.5, and thus Lemma 5.7 holds. ∎

Lemma 5.8**.**

We set $\displaystyle S(R):=\bigcup_{m=1}^{\infty}\left\{(k_{1},\ldots,k_{m})\in\mathbb{N}_{+}^{m}\middle|\sum_{i=1}^{m}k_{i}\leq R\right\}$ for any $R\in\mathbb{N}$. Then the cardinality of $S(R)$ equals $2^{R}-1$.

Proof.

For $(k_{1},\ldots,k_{m})\in S(R)$, we define the binary number $1\underbrace{0\ldots 0}_{k_{1}-1}\ldots 1\underbrace{0\ldots 0}_{k_{m}-1}$, which induces a bijection $S(R)\xrightarrow{\sim}\mathbb{Z}\cap[1,2^{R}-1]$. Hence $\#S(R)=2^{R}-1$. ∎

We are ready to prove Theorem 5.1.

Proof of Theorem 5.1.

We now prove $(1)$. Let $\Gamma_{\nu}\equiv\Gamma_{\nu}(a_{-},a_{+},r,R)$. Take $\nu\in\mathbb{N}$ as in Proposition 3.12 and assume $\nu\geq 1+(1-\log A)/a$. Then recall $\varepsilon(\nu)(<1)\leq 2$. Moreover, take $\gamma\in F^{\infty}_{\nu}\smallsetminus\{e\}$. Let $m:=\ell(\gamma)$ be the word length of $\gamma$, and we write $\gamma=\gamma_{k_{1}}^{s_{1}}\cdots\gamma_{k_{m}}^{s_{m}}$ for the reduced expression where $s_{1},\ldots,s_{m}\in\{1,-1\}$ and $k_{1},\ldots,k_{m}\geq\nu(\geq 2)$. By Proposition 3.8 (1),

[TABLE]

To prove $(1)$, we may and do assume $x\in\mathcal{D}_{X_{\Gamma_{\nu}}}$ by Fact 5.6. Suppose $(j(\gamma),\rho(\gamma))x\in B(R)$. Then $\left|\|j(\gamma)\|-\|\rho(\gamma)\|\right|/2\leq R$ by Lemma 5.7. Hence $(k_{1}-(\nu-1),\ldots,k_{m}-(\nu-1))\in S(\lfloor R/a\rfloor)$ by the inequality (5), where $\lfloor R\rfloor$ is the largest integer less than or equal to $R$ for $R\in\mathbb{R}$, and in particular, $m\leq R/a$. The number of such $(k_{1},\ldots,k_{m})$ is at most $2^{R/a}-1$ by Lemma 5.8 and thus the number of such $\gamma=\gamma_{k_{1}}^{s_{1}}\cdots\gamma_{k_{m}}^{s_{m}}(\neq e)$ with $s_{1},\ldots,s_{m}\in\{1,-1\}$ is at most $(2^{R/a}-1)2^{R/a}$. Hence we obtain $N_{\Gamma_{\nu}}(x,R)\leq(2^{R/a}-1)2^{R/a}+1\leq 2^{2R/a}$, which proves $(1)$. The assertion $(2)$ follows from $(1)$ and Fact 5.3. ∎

Appendix A Proof of the formula (1.3)

The formula (1.3) is probably well known, but we give a proof for the reader’s convenience. It suffices to show the following proposition:

Proposition A.1**.**

Let $\Gamma$ be a discrete group of isometries of a complete Riemannian manifold $X$. We write $B(x;R)$ for the ball of radius $R$ centered at $x\in X$. Fix a point $x_{0}\in X$. Then,

[TABLE]

Proof.

Let $x\in X$. We denote by $d_{X}(\cdot,\cdot)$ the distance of $X$, and set $3c:=\inf\{d_{X}(x,y)\mid y\in\Gamma x\smallsetminus\{x\}\}$. By [4, Ch. IV, Thm. 2.2], the orbit $\Gamma x$ of the discrete group $\Gamma$ is discrete, hence we have $c>0$. By the triangle inequality, the inclusion relation

[TABLE]

holds. We note that the left hand side of (A.1) is a disjoint union by the definition of $c$. Since $\mathrm{vol}(B(y;c))=\mathrm{vol}(B(x;c))$ for any $y\in\Gamma x$, we obtain the desired inequality taking the volumes of the both hand sides of (A.1). ∎

Acknowledgements

The author would like to express his sincere gratitude to Professor Toshiyuki Kobayashi for his support and encouragement. He would also like to show his appreciation to Dr. Yosuke Morita for his helpful comments. Thanks are also due to an anonymous referee for comments and suggestions that improved this paper. This work was supported by JSPS KAKENHI Grant Number 18J20157 and the Program for Leading Graduate Schools, MEXT, Japan.