Uniform spectral gap and orthogeodesic counting for strong convergence of Kleinian groups

In this subsection, we let $X$ denote an $n$-dimensional negatively pinched Hadamard manifold whose sectional curvatures lie between $-\kappa^{2}$ and $-1$ for some $\kappa\geq 1$. For any isometry $\gamma\in\operatorname{{\mathrm{I}som}}(X)$, we define its translation length $\tau(\gamma)$ as follows:

where $d_{X}$ is the Riemannian distance function in $X$. Based on the translation length, we can classify isometries in $X$ into 3 types; we call $\gamma$ loxodromic if $\tau(\gamma)>0$. In this case, the infimum is attained exactly when the points are on the axis of $\gamma$. The isometry $\gamma$ is called parabolic if $\tau(\gamma)=0$ and the infimum is not attained. The isometry $\gamma$ is elliptic if $\tau(\gamma)=0$ and the infimum is attained.

From now on, we consider torsion-free discrete isometry subgroups $\Gamma<\operatorname{{\mathrm{I}som}}(X)$, i.e. $\Gamma$ contains no elliptic elements. If $\Gamma<\operatorname{{\mathrm{I}som}}(\mathbb{H}^{n})$ is a torsion-free discrete isometry subgroup, we call it a Kleinian group. Given $0<\epsilon<\epsilon(n,\kappa)$, where $\epsilon(n,\kappa)$ is the Margulis constant depending on the dimension $n$ and the constant $\kappa$, let $\mathcal{T}_{\epsilon}(\Gamma)$ be the set consisting of all points $p\in X$ such that there exists an isometry $\gamma\in\Gamma$ with

It is an $\Gamma$-invariant set, and the quotient $\mathcal{T}_{\epsilon}(\Gamma)/\Gamma$ is the thin part of the quotient manifold $M=X/\Gamma$, denoted by $M^{<\epsilon}$.

A subgroup $P<\Gamma$ is called parabolic if the fixed point set of $P$ consists of a single point $\xi\in\partial_{\infty}X$, where $\partial_{\infty}X$ is the visual boundary of $X$. Note that $\mathcal{T}_{\epsilon}(P)\subset X$ is precisely invariant under $P$, i.e. $\operatorname{{\mathrm{s}tab}}_{\Gamma}(\mathcal{T}_{\epsilon}(P))=P$ [Bow95, Corollary 3.5.6]. By abuse of notation, we can regard $\mathcal{T}_{\epsilon}(P)$ as a subset of $M=X/\Gamma$, which is called a Margulis cusp. The union of all Margulis cusps consists of the cuspidal part of $M$, denoted by $\operatorname{{\mathrm{c}usp}}_{\epsilon}(M)$.

The limit set $\Lambda(\Gamma)$ of a discrete, torsion-free isometry subgroup $\Gamma<\operatorname{{\mathrm{I}som}}(X)$ is defined to be the set of accumulation points of a $\Gamma$-orbit $\Gamma(p)$ in $\partial_{\infty}X$ for any point $p\in X$. We call $\Gamma$ elementary if $\Lambda(\Gamma)$ is finite; Otherwise, we say $\Gamma$ is nonelementary. For any two points $\xi$ and $\eta$ in $\partial_{\infty}X$, we use $\xi\eta$ to denote the unique geodesic in $X$ connecting these two points. The convex hull of $\Lambda(\Gamma)\subset\partial_{\infty}X$ is the smallest closed convex subset in $X$ whose accumulation set is $\Lambda(\Gamma)$, denoted by $\operatorname{{\mathrm{H}ull}}(\Gamma)$. We let $C(M)=\operatorname{{\mathrm{H}ull}}(\Gamma)/\Gamma$ denote the convex core of quotient manifold $M=X/\Gamma$. For any constant $\epsilon>0$, we define the truncated core by

Given a constant $0<\epsilon<\epsilon(n,\kappa)$, a discrete isometry subgroup $\Gamma$ is geometrically finite if the truncated core $C(M)^{>\epsilon}$ is compact in $M=X/\Gamma$. If, in addition, $C(M)$ is compact, i.e. $\Gamma$ contains no parabolic isometries, then $\Gamma$ is called convex co-compact. Furthermore, if $\Gamma<\operatorname{{\mathrm{I}som}}(X)$ is geometrically finite, the parabolic fixed points in $\Lambda(\Gamma)$ are bounded, defined as follows:

Given a point $x\in X$ and a discrete isometry group $\Gamma\in\operatorname{{\mathrm{I}som}}(X)$, the Poincaré series is defined as

The critical exponent of $\Gamma$ is defined as

It is not hard to see that the definition of $\delta(\Gamma)$ is independent of the choice of $x$.

As in Section 2.1, we let $M=X/\Gamma$, where $X$ is a negatively pinched Hadamard manifold, and $\Gamma$ is a torsion-free discrete isometry subgroup. Define the Sobolev space $H^{1}(M)$ as the space obtained by the completion of $C^{\infty}_{0}(M)$ with respect to the norm $\|f\|=\sqrt{\int_{M}|f|^{2}+\int_{M}|\nabla f|^{2}}$. This space can be also defined as functions in $L^{2}(M)$ whose weak derivative (in the sense of distributions) is also in $L^{2}(M)$.

Given $f\in H^{1}(M)$, we define the Rayleigh quotient $R(f)$ of $f$ by

The Rayleigh quotient is closely related to the spectrum $Spec(M)$ of the negative Laplace operator. Namely, by posing the following minimization problem

we obtain a $L^{2}$ integrable smooth function $f$ satisfying $-\Delta f=\lambda f$.

We let $\lambda_{0}(M)$ denote the bottom of the spectrum, and we say that $\lambda\in Spec(M)$ is a small eigenvalue of $M$ if $\lambda<(n-1)^{2}/4$. Moreover, given a constant $\mu<(n-1)^{2}/4$, we define $Spec_{\mu}(M)$ as the collection (counting multiplicities) of eigenvalues of the negative Laplacian on $M$ less than or equal to $\mu$. The set of small eigenvalues is a finite set (see [Ham04]).

In the rest of the subsection, we list several properties of the bottom of the spectrum $\lambda_{0}(M)$. We will use these properties in Section 3 to prove the uniform spectral gap for strongly convergent sequences of geometrically finite groups $(\Gamma_{k}<\operatorname{{\mathrm{I}som}}(X))_{k\in\mathbb{N}}$.

If $X=\mathbb{H}^{n}$, we have the following result relating $\lambda_{0}(M)$ to the critical exponent $\delta(\Gamma)$.

Given a point $p\in\mathbb{H}^{n}$, and $\xi\in\partial_{\infty}\mathbb{H}^{n}$, the Busemann function $B(x,\xi)$ on $\mathbb{H}^{n}$ with respect to $p$ is defined by

where $\rho_{\xi}(t)$ is the unique geodesic ray from $p$ to $\xi$. The Busemann cocycle $\beta_{\xi}(x,y):\mathbb{H}^{n}\times\mathbb{H}^{n}\times\partial_{\infty}\mathbb{H}^{n}\rightarrow\mathbb{R}$ is defined by

For a discrete isometry subgroup $\Gamma<\operatorname{{\mathrm{I}som}}(\mathbb{H}^{n})$, there exists a family of finite measures $(\mu_{x})_{x\in\mathbb{H}^{n}}$ on $\partial_{\infty}\mathbb{H}^{n}$ whose support is the limit set $\Lambda(\Gamma)$ and satisfies the following conditions:

1. (1)

   It is $\Gamma$-invariant, i.e. $\gamma_{\ast}(\mu_{x})=\mu_{\gamma x}$.

2. (2)

   The Radon-Nikodym derivatives exist for all $x,y\in\mathbb{H}^{n}$, and for all $\xi\in\partial_{\infty}\mathbb{H}^{n}$ they satisfy

   $$\dfrac{d\mu_{x}}{d\mu_{y}}(\xi)=e^{-\delta(\Gamma)\beta_{\xi}(x,y)}.$$

Such family of measures is a family of Patterson-Sullivan density of dimension $\delta(\Gamma)$ for $\Gamma$. The Patterson-Sullivan measures have very nice properties when the group $\Gamma$ is geometrically finite.

The proof Theorem 2.6 relies heavily on the analysis of the Poincaré series of parabolic groups and its uniform convergence. This is also essential in the later proof of the convergence of Bowen-Margulis measures and the uniform counting formulas for orthogeodesics in the rest of the paper. For readers’ convenience, we list the analytic properties of the Poincaré series corresponding to parabolic groups in the section. The details can be found in [McM99, Section 6].

Let $L<\operatorname{{\mathrm{I}som}}(\mathbb{H}^{n})$ be a torsion-free elementary isometry subgroup, which is either a hyperbolic group, i.e. a cyclic group generated by a loxodromic isometry, or a parabolic group. Given $x\in\partial_{\infty}\mathbb{H}^{n}$ and $s\geq 0$, the absolute Poincaré series for $L$ is defined to be

where the derivative is measured in the spherical measure. Given any open subset $U\subset\partial_{\infty}\mathbb{H}^{n}$, define

Suppose that $(L_{k}<\operatorname{{\mathrm{I}som}}(\mathbb{H}^{n}))_{k\in\mathbb{N}}$ is a sequence of torsion-free elementary isometry subgroups which converges geometrically to a parabolic group $L<\operatorname{{\mathrm{I}som}}(\mathbb{H}^{n})$ with parabolic fixed point $c$, i.e., $L_{k}$ converges to $L$ in the Hausdorff topology on closed subsets of $\operatorname{{\mathrm{I}som}}(\mathbb{H}^{n})$. The Poincaré series for $(L_{k},s_{k})$ where $s_{k}\geq 0$ converges uniformly if for any compact subset $K\subset\partial_{\infty}\mathbb{H}^{n}\setminus\{c\}$ and $\epsilon>0$, there is a neighborhood $U$ of $c$ such that for all $x\in K$,

for $k\gg 0$ sufficiently large. By using the same argument of the proof of Theorem 6.1 in [McM99], we have the following:

The Bowen-Margulis measure is a measure defined on the unit tangent bundle $T^{1}\mathbb{H}^{n}$ of $\mathbb{H}^{n}$ in terms of the Patterson-Sullivan measures. One can identify the unit tangent bundle $T^{1}\mathbb{H}^{n}$ with the set of geodesic lines $l:\mathbb{R}\rightarrow\mathbb{H}^{n}$ such that the inverse map sends the geodesic line $l$ to its unit tangle vector $\dot{l}(0)$ at $t=0$. Given a point $x_{0}\in\mathbb{H}^{n}$, we can also identify $T^{1}\mathbb{H}^{n}$ with $\partial_{\infty}\mathbb{H}^{n}\times\partial_{\infty}\mathbb{H}^{n}\times\mathbb{R}$ via the Hopf’s parametrization:

where $v_{-},v_{+}$ are the endpoints at $-\infty$ and $\infty$ of the geodesic line defined by $v$ and $t$ is the signed distance of the closest point to $x_{0}$ on the geodesic line.

We let $\pi:T^{1}\mathbb{H}^{n}\rightarrow\mathbb{H}^{n}$ denote the basepoint projection. The geodesic flow on $T^{1}\mathbb{H}^{n}$ is the smooth one-parameter group of diffeomorphisms $(g^{t})_{t\in\mathbb{R}}$ of $T^{1}\mathbb{H}^{n}$ such that $g^{t}(l(s))=l(s+t)$, for all $l\in T^{1}\mathbb{H}^{n}$, and $s,t\in\mathbb{R}$. Similarly one can define the geodesic flow on $T^{1}M$ by replacing the geodesic lines $l$ by locally geodesic lines. The Kleinian group $\Gamma$ acts on $T^{1}\mathbb{H}^{n}$ via postcomposition, i.e. $\gamma\circ l$, and it commutes with the geodesic flow. For simplicity, we sometimes write $\delta(\Gamma)$ as $\delta$ if the context is clear in the rest of the paper.

Given the Patterson-Sullivan density $(\mu_{x})_{x\in\mathbb{H}^{n}}$ and a point $x_{0}\in\mathbb{H}^{n}$, one can define the Bowen-Margulis measure $\tilde{m}_{\rm{BM}}$ on $T^{1}\mathbb{H}^{n}$ given by

Here we introduce the notation $(v_{-}|v_{+})_{x_{0}}=\frac{1}{2}(\beta_{v_{-}}(y,x_{0})+\beta_{v_{+}}(y,x_{0}))$, where $y$ is any point in the geodesic joining $v_{-},v_{+}$. It is not hard to verify that $(v_{-}|v_{+})_{x_{0}}$ does not depend on $y$.

The Bowen-Margulis measure $\tilde{m}_{\rm{BM}}$ is independent of the choice of $x_{0}$, and it is invariant under both the action of the group $\Gamma$ and the geodesic flow. Hence, it descends to a measure $m_{\rm{BM}}$ on $T^{1}M$ invariant under the quotient geodesic flow, which is called the Bowen-Margulis measure on $T^{1}M$.

Another related measure we consider in the paper is the so called skinning measure. Let $D$ be a nonempty proper closed convex subset in $\mathbb{H}^{n}$. We denote its boundary by $\partial D$ and the set of points at infinity by $\partial_{\infty}D$. Let

be the closest point map. In particular, for points $x\in\mathbb{H}^{n}$, $P_{D}(x)$ is the point on $D$ which minimizes the distance function $d(y,x)$ for $y\in D$, and for points $\xi\in\partial_{\infty}\mathbb{H}^{n}\setminus\partial_{\infty}D$, $P_{D}(\xi)$ is the point $y\in D$ which minimizes the function $y\rightarrow\beta_{\xi}(y,x_{0})$ for a given $x_{0}$.

The outer unit normal bundle $\partial^{1}_{+}D$ of the boundary of $D$ is the topological submanifold of $T^{1}\mathbb{H}^{n}$ consisting of the geodesic lines $v:\mathbb{R}\rightarrow\mathbb{H}^{n}$ such that $P_{D}(v_{+})=v(0)$. Similarly, one can define the inner unit normal bundle $\partial^{1}_{-}D$ which consists of geodesic lines $v$ such that $P_{D}(v_{-})=v(0)$. Note that when $D$ is totally geodesic, $\partial^{1}_{+}D=\partial^{1}_{-}D$. Given the Patterson-Sullivan density $(\mu_{x})_{x\in\mathbb{H}^{n}}$, the outer skinning measure on $\partial^{1}_{+}D$ is the measure $\tilde{\sigma}^{+}_{D}$ defined by

Similarly, one can define the inner skinning measure $\tilde{\sigma}_{D}^{-}$ on $\partial^{1}_{-}D$ as follows:

For simplicity, we sometimes identify a precisely invariant subset $C\subset M=\mathbb{H}^{n}/\Gamma$ with its fundamental domain $\tilde{C}$ in the universal cover, and use the notation $\sigma^{\pm}_{\partial C}$ to denote the outer/inner skinning measure $\tilde{\sigma}^{\pm}_{\tilde{C}}$ on $\partial^{1}_{\pm}\tilde{C}$.

In this subsection, we first define strong convergence that admits disconnected limits. Suppose that $((M_{k},g_{k}))_{k\in\mathbb{N}}$ is a sequence of $n$-manifolds of pinched sectional curvature $-\kappa^{2}\leq K\leq-1$. We say that the sequence converges strongly to a (possibly disconnected) geometrically finite $n$-manifold $(N=\cup_{i}^{m}N_{i},g)$ if the following holds:

1. (1)

   There exist points $p_{k,i}\in M_{k}$, $p_{i}\in N_{i}$ so that $d(p_{k,i},p_{k,j})\rightarrow+\infty$ for $i\neq j$ and $(M_{k},p_{k,i})\rightarrow(N_{i},p_{i})$ geometrically, i.e., there exists an exhaustion $U_{1,i}\subset U_{2,i}\subset\ldots$ of relatively compact open sets of $(N_{i},p_{i})$ and smooth maps $\varphi_{k,i}:U_{k,i}\rightarrow M_{k}$ so that $\varphi_{k,i}(p_{i})=p_{k,i}$ and $\varphi_{k,i}^{*}g_{k}$ converges smoothly in compact sets to $g$.

2. (2)

   For any $\epsilon$, the truncated cores $C(M_{k})^{>\epsilon}$ converge to the disjoint union $\cup_{i}C(N_{i})^{>\epsilon}$. This means that for large $k$ we have $C(M_{k})^{>\epsilon}=\cup_{i=1}^{m}C(M_{k,i})^{>\epsilon}$ where $C(M_{k,i})^{>\epsilon}\subset Im(\varphi_{k,i})$ and $\varphi^{-1}_{k,i}(C(M_{k,i})^{>\epsilon})$ converges to $C(N_{i})^{>\epsilon}$ in the Hausdorff topology of compact sets in $N_{i}$.

The definition accommodates situations like Dehn drilling and pinching closed geodesics in hyperbolic 3-manifolds. The pinching case can result in disconnected limit manifolds. If $M_{k}$ and $N$ are hyperbolic manifolds, and $N$ is connected, this definition is equivalent to the one described in [McM99] for strong convergence. Because of this, in the cases when $N$ is connected we will simply omit the mention of possibly disconnected, as well as the sub-index $i$ from our notation.

Moreover, given a sequence $(M_{k})_{k\in\mathbb{N}}$ converging strongly to a possibly disconnected manifold $N$, we say that the sequence of functions $(f_{k}:M_{k}\rightarrow\mathbb{R})_{k\in\mathbb{N}}$ converges strongly to a function $f:N\rightarrow\mathbb{R}$ if, with the notation above, we have that for any basepoint $p_{i}$ the sequence $(f_{k}\circ\varphi_{k,i})_{k\in\mathbb{N}}$ converges smoothly in compact sets to $f$. Similarly, if $\Sigma_{k},\Sigma$ are smooth properly embedded submanifolds in $M_{k},N$, we say that $(\Sigma_{k})_{k\in\mathbb{N}}$ converges strongly to $\Sigma$ if $\varphi_{k,i}^{-1}(\Sigma_{k})$ converges smoothly in compact sets to $\Sigma$. Since for any fixed compact set in $N_{i}$ the maps $\varphi_{k,i}$ are embeddings for $k$ sufficiently large, we can define strong convergence of functions and submanifolds of $T^{1}M_{k}$ to $T^{1}M$ by composing the derivatives of $\varphi_{k,i}$ with the projections from $T^{*}M_{k}$ to $T^{1}M_{k}$.

Using the definition of strong convergence, we obtain a straightforward corollary:

In Section 3, we work on sequences of manifolds of negatively pinched curvature that converge strongly to (possibly disconnected) limit manifolds. Given an $n$-dimensional manifold $M$ with negatively pinched curvature (possibly disconnected) and a constant $\mu<(n-1)^{2}/4$, $Spec_{\mu}(M)$ is defined as the collection of eigenvalues of the negative Laplacian on $M$ less than $\mu$. If $M$ is disconnected, $Spec_{\mu}(M)$ agrees with the union of $Spec_{\mu}$ of each component of $M$ (counting multiplicity). Specifically, a function $f:M\rightarrow\mathbb{R}$ satisfies the equation $-\Delta f=\lambda f$ if and only if its restriction to each component of $M$ is either an eigenfunction with eigenvalue $\lambda$, or $0$. Moreover, while taking orthonormal eigenfunctions for $M$ we can consider that each eigenfunction has support in a unique component of $M$.

In Section 4 and Section 5, we focus on sequences of hyperbolic manifolds strongly converging to connected limit manifolds. Suppose now $M=\mathbb{H}^{n}/\Gamma$ is an $n$-dimensional hyperbolic manifold. As we stated in the Introduction, locally convex sets in $M$ are in 1-to-1 correspondence with $\Gamma$-precisely invariant convex sets in $\mathbb{H}^{n}$ by the projection map $\textup{Proj}:\mathbb{H}^{n}\rightarrow M$. In particular, we sometimes identify local convex sets with one of their lifts which are $\Gamma$-precisely invariant, and we don’t consider immersed locally convex sets, e.g. nonprimitive closed geodesics. For simplicity, we will omit the word locally and plainly denote the sets as convex.

We say that a convex set $D$ in $M$ is well-positioned if $\partial\tilde{D}$ is smooth, where $\tilde{D}$ denotes the lift of $D$ to $\mathbb{H}^{3}$, and $\sigma^{\pm}_{\partial D}$ has compact support.

Suppose that $(M_{k}=\mathbb{H}^{n}/\Gamma_{k})_{k\in\mathbb{N}}$ is a sequence of hyperbolic manifolds that converges strongly to a geometrically finite hyperbolic manifold $M=\mathbb{H}^{n}/\Gamma$. We say that well-positioned convex sets $D_{k}\subset M_{k}$ strongly converge to a well-positioned convex set $D\subset M$ if

1. (1)

   the boundary $\partial D_{k}$ converges strongly to $\partial D$, or equivalently, the lifts of $\varphi_{k}^{-1}(\partial D_{k})$ converge smoothly in compact sets to lifts of $\partial D$, where $\varphi_{k}:U_{k}\rightarrow M_{k}$ are the smooth maps in the definition of strong convergence of $(M_{k})_{k\in\mathbb{N}}$,

2. (2)

   $\bar{\pi}(supp(\sigma^{\pm}_{\partial D_{k}}))$ is contained in $\varphi_{k}\left(N_{1}(\bar{\pi}(supp(\sigma^{\pm}_{\partial D})))\right)$ for large $k$, where $\bar{\pi}:T^{1}M\rightarrow M$ and $N_{1}$ denotes the 1-neighborhood.