Thurston’s Bounded image theorem

We are going to find $n$ such that if the image of $(\iota_{*}\circ\sigma_{m})^{n}$ is unbounded then $N$ contains a non-peripheral incompressible torus, contradicting our assumption. For that purpose we shall use the invariant $m$ introduced in [2].

Given a simple closed curve $d$ on a closed surface $S$ equipped with a hyperbolic metric $g$, we define

where $\mu(g)$ is a shortest marking for $(S,g)$, $\mu$ is a full marking, and the supremum of the first term in the maximum is taken over all incompressible subsurfaces $Y$ of $S$ whose boundaries $\partial Y$ contain $d$. See Definition 4.1 for more details.

It is not difficult to see that in the setting of Theorem 1.1, for a given sequence $\{m_{i}\}$ in $\mathcal{T}(\partial M)$, if the sequence $\{\sigma(m_{i})\}$ is unbounded, then there is a simple closed curve $d$ such that $m(\sigma(m_{i}),d,\mu)$ is unbounded (see Lemma 4.3). The core of our argument consists in showing, with the help of arguments from [3] and [2], that in this situation, there is a simple closed curve $d^{\prime}\subset\partial M$ such that $\{m(m_{i},d^{\prime},\mu)\}$ is unbounded and that $d\cup d^{\prime}$ bounds an essential annulus in $M$. Using this argument repeatedly, we build (when $\{(\iota_{*}\circ\sigma)^{n}(m_{i})\}$ is unbounded) an annulus in $N$ which goes through the interior of $M$ (viewed as a subset of $N$) $n$ times. If $n$ is large enough, this annulus must create an essential torus in $N$, and contradicts the assumption that $N$ is atoroidal.

Although this is the overall logic of the proof, in the following sections, we shall present the main steps in a different order. After setting up some preliminary definitions in Section 2, we shall discuss the topological part of the proof in Section 3. First we show that we can add some assumptions on the topology of $M$ which will simplify the arguments later on. Next, we study incompressible surfaces on $\partial M$ which can be extended multiple times through the characteristic submanifold of $M$ when it is viewed as a submanifold of $N$. This will give us an integer $n$ which appears in Theorem 1.1. In Section 4 we shall discuss the relation between the behaviour of the invariant $m$ defined above, and the convergence and divergence of Kleinian groups. In Section 5 we shall prove our key proposition, and obtain the curve $d^{\prime}$ mentioned above. Finally in Section 6 we shall put these pieces together to prove our main theorem.

An orientable irreducible compact 3-manifold which contains a non-peripheral incompressible surface is called a Haken manifold. We note that a compact irreducible 3-manifold with non-empty boundary is always Haken except for a 3-ball. We say that a Haken manifold is atoroidal when it does not contain a non-peripheral incompressible torus, and acylindrical when it does not contain a non-peripheral incompressible annulus. By the torus theorem for Haken manifolds ([25, 5, 6]), the former condition of the atoroidality is equivalent to the one that every monomorphism from $\mathbb{Z}\times\mathbb{Z}$ into the fundamental group is peripheral, i.e. is conjugate to the image of the fundamental group of a boundary component.

The Jaco-Shalen-Johannson theory [5, 6] tells us that in a Haken manifold, incompressible tori and incompressible annuli can stay only in a very restricted place. Let us state what the theory says in the case when a Haken manifold $M$ is atoroidal and boundary-irreducible.

For an orientable atoroidal Haken boundary-irreducible 3-manifold $M$, there exists a 3-submanifold $X$ of $M$ each of whose components is one of the following and which satisfies the following condition:

1. (a)

   An $I$-bundle whose associated $\partial I$-bundle coincides with its intersection with $\partial M$. Such a component is called a characteristic $I$-pair.

2. (b)

   A solid torus $V$ such that $V\cap\partial M$ consists of annuli which are incompressible on both $\partial V$ and $\partial M$. When $V\cap\partial M$ is connected, it winds around the core curve of $V$ more than once.

3. (c)

   A thickened torus $S^{1}\times S^{1}\times I$ at least one of whose boundary components lies on a component of $\partial M$.

Every properly embedded essential annulus (i.e. an incompressible annulus which is not homotopic into the boundary) is properly isotopic into $X$, and no component of $X$ is properly isotopic into another component.

Such $X$ is unique up to isotopy, and is called the characteristic submanifold of $M$. We note that in the case when $M$ has no torus boundary component, which is the assumption of our main theorem, a component of the last type (c) does not appear.

Thurston’s celebrated uniformisation theorem for Haken manifolds says that every atoroidal Haken manifold whose boundary consists of incompressible tori admits a hyperbolic structure of finite volume. More generally, he proved that every atoroidal Haken manifold, including the case when it has non-torus boundary components, admits a (minimally parabolic) convex hyperbolic structure of finite volume. The term ‘convex hyperbolic structure’ will be explained in the following subsection.

A Kleinian group is a discrete subgroup of ${\rm PSL}_{2}(\mathbb{C})$. In this paper, we always assume Kleinian groups to be torsion free, and finitely generated except for the case when we talk about geometric limits. For a Kleinian group $\Gamma$, we can consider the complete hyperbolic 3-manifold $\mathbb{H}^{3}/\Gamma$. The convex core of $\mathbb{H}^{3}/\Gamma$ is the smallest convex submanifold that is a deformation retract. The Kleinian group $\Gamma$ and the corresponding hyperbolic 3-manifold $\mathbb{H}^{3}/\Gamma$ are said to be geometrically finite when the convex core of $\mathbb{H}^{3}/\Gamma$ has finite volume. In particular, $\mathbb{H}^{3}/\Gamma$ is said to be convex compact, and $\Gamma$ to be convex cocompact if the convex core is compact. We also say that $\Gamma$ is minimally parabolic when every parabolic element in $\Gamma$ is contained in a rank-2 parabolic subgroup. Any convex cocompact Kleinian group is automatically minimally parabolic since it does not have parabolic elements.

A 3-manifold $M$ is said to have a hyperbolic structure when $\operatorname{Int}M$ is homeomorphic to $\mathbb{H}^{3}/\Gamma$ for a Kleinian group $\Gamma$, and we regard the pull-back of the hyperbolic metric to $\operatorname{Int}M$ as a hyperbolic structure on $M$. In particular if $\Gamma$ is taken to be geomerically finite or convex cocompact, we say that $M$ has a geometrically finite or convex compact hyperbolic structure. If $M$ admits a hyperbolic structure, then $M$ must be atoroidal.

The set of hyperbolic structures on $M$ modulo isotopy, which we denote by $\mathsf{AH}(M)$, can be identified with a subset of the set of faithful discrete representations of $\pi_{1}(M)$ into ${\rm PSL}_{2}(\mathbb{C})$ modulo conjugacy. We put on $\mathsf{AH}(M)$ a topology induced from the weak topology on the representation space. We regard an element of $\mathsf{AH}(M)$ both as a hyperbolic structure on $M$ and as a representation of $\pi_{1}(M)$ into ${\rm PSL}_{2}(\mathbb{C})$ depending on the situation.

A Kleinian group $G$ is said to be a quasi-conformal deformation of another Kleinian group $\Gamma$ if there is a quasi-conformal homeomorphism $f\colon\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ such that $G=f\Gamma f^{-1}$ as Möbius transformations on $\hat{\mathbb{C}}$. When $G$ is a quasi-conformal deformation of $\Gamma$, there is a diffeomorphism from $\mathbb{H}^{3}/\Gamma$ to $\mathbb{H}^{3}/G$ preserving the parabolicity in both directions, which induces an isomorphism between the fundamental groups coinciding with the isomorphism given by the conjugacy $G=f\Gamma f^{-1}$. We note that a quasi-conformal deformation of geometrically finite (resp. convex cocompact, minimally parabolic geometrically finite) group is again geometrically finite (resp. convex cocompact, minimally parabolic geometrically finite).

Let $M$ be a compact 3-manifold admitting a minimally parabolic geometrically finite hyperbolic structure $m$. Let $\mathsf{QH}(M)$ denote the set of all minimally parabolic geometrically finite hyperbolic structures on $M$ modulo isotopy, which is regarded as a subset of $\mathsf{AH}(M)$. Marden [9] showed that every minimally parabolic geometrically finite hyperbolic structures on $M$ is obtained as a quasi-conformal deformation of $m$. Therefore we call $\mathsf{QH}(M)$ the quasi-conformal deformation space. Furthermore, if $\partial M$ is incompressible, combined with the work of Ahlfors, Bers, Kra, Maskit and Sullivan, there is a parameterisation $q\colon\mathcal{T}(\partial M)\to\mathsf{QH}(M)$, where $\mathcal{T}(\partial M)$ denotes the Teichmüller space of $\partial M$, i.e. the direct product of the Teichmüller spaces of the components of $\partial M$. We shall refer to this map as the Ahlfors-Bers map.

In the case when $M$ is homeomorphic to $S\times[0,1]$ for a closed oriented surface $S$, the deformation spaces $\mathsf{AH}(M),\mathsf{QH}(M)$ are denoted by $\mathsf{AH}(S),\mathsf{QF}(S)$ respectively. The quasi-conformal deformation space $\mathsf{QF}(S)$ consists of quasi-Fuchsian representations of $\pi_{1}(S)$, i.e. quasi-conformal deformations of a Fuchsian representation, and is therefore called the quasi-Fuchsian space. The Ahlfors-Bers map can be expressed as $qf:\mathcal{T}(S)\times\mathcal{T}(\bar{S})\to\mathsf{QF}(S)$, where the second coordinate $\mathcal{T}(\bar{S})$ denotes the Teichmüller space of $S$ with orientation reversed, which is a more natural way for parametrisation since the boundary component $S\times\{1\}$ has the opposite orientation from the one given on $S\times\{0\}$ if we identify them with $S$ by dropping the second factor.

Now, let $M$ be an atoroidal Haken 3-manifold with non-empty incompressible boundary which does not contain a torus. Suppose that $M$ has a convex compact hyperbolic metric $m$, and let $S$ be a component of $\partial M$. Take a covering of $M$ associated with $\pi_{1}(S)\subset\pi_{1}(M)$, and lift the hyperbolic structure $m$ to the hyperbolic structure $\tilde{m}$ on $S\times[0,1]$. It is known (see [16, Proposition 7.1]) that the lifted structure $\tilde{m}$ is also convex cocompact, hence can be regarded as an element of $\mathsf{QF}(S)$. Therefore $\tilde{m}$ in turn corresponds to a point $(g_{S}(m),h_{S}(m))$ in $\mathcal{T}(S)\times\mathcal{T}(\bar{S})$. Let $S_{1},\dots,S_{k}$ be the components of $\partial M$ that are not tori, and we consider the point $h_{S_{i}}(m)\in\mathcal{T}(\bar{S}_{i})$ for each $i=1,\dots,k$. We define $\mathcal{T}(\bar{\partial}M)$ to be $\mathcal{T}(\bar{S}_{1})\times\dots\times\mathcal{T}(\bar{S}_{k})$. The map taking $g\in\mathcal{T}(\partial M)$ to $(h_{S_{1}}(q(g)),\dots,h_{S_{k}}(q(g)))\in\mathcal{T}(\bar{\partial}M)$ is called the skinning map, which we shall denote by $\sigma$.

Let $S$ be a connected compact orientable surface possibly with boundary, satisfying $\xi(S)=3g+n\geq 4$ where $g$ denotes the genus and $n$ denotes the number of the boundary components. The curve complex $\mathcal{CC}(S)$ of $S$ with $\xi(S)\geq 5$ is a simplicial complex whose vertices are isotopy classes of non-peripheral, non-contractible simple closed curves on $S$ such that $n+1$ vertices span an $n$-simplex when they are represented by pairwise disjoint simple closed curves. In the case when $\xi(S)=4$, we define $\mathcal{CC}(S)$ to be a graph whose vertices are isotopy classes of simple closed curves such that two vertices have smallest possible intersection. In the case when $S$ is an annulus, we define $\mathcal{CC}(S)$ to be a graph whose vertices are isotopy classes (relative to the endpoints) of non-peripheral simple arcs in $S$ such that two vertices are connected when they can be represented by arcs which are disjoint at their interiors. Masur-Minsky [11] proved that $\mathcal{CC}(S)$ is Gromov hyperbolic with respect to the path metric for any $S$.

A marking $\mu$ on $S$ consists of a pants decomposition of $S$, which is denoted by $\mathbf{base}(\mu)$ and whose components are called base curves, and a collection $\mathbf{t}(\mu)$ of simple closed curves, called transversals of $\mathbf{base}(\mu)$, such that each component of $\mathbf{base}(\mu)$ intersects at most one among them essentially. For two markings $\mu,\nu$ on $S$ and a subsurface $Y$, we define $d_{Y}(\mu,\nu)$ to be the distance between $\pi_{Y}(\mathbf{base}(\mu)\cup\mathbf{t}(\mu))$ and $\pi_{Y}(\mathbf{base}(\nu)\cup\mathbf{t}(\nu))$, where the projection $\pi_{Y}\colon\mathcal{CC}(S)\to\mathfrak{P}(\mathcal{CC}(Y))$ is obtained by taking the intersection of curves on $S$ with $Y$ and connecting the endpoints by arcs on $\operatorname{Fr}Y$ when the intersection contains arcs. In [12], a marking defined as such is called clean. In this paper, we only consider clean markings. A marking is called full when every base curve has a transversal. In general, for two sets of simple closed curves $a,b$ and a subsurface $Y$ of $S$, we define $d_{Y}(a,b)$ to be the distance in $\mathcal{CC}(Y)$ between $\pi_{Y}(a)$ and $\pi_{Y}(b)$ provided that both of them are non-empty. If one of them is empty, the distance is not defined.

For a point $m$ in $\mathcal{T}(S)$, its shortest marking, which is a full marking and is denoted by $\mu(m)$, has a shortest pants decomposition of $(S,m)$ as $\mathbf{base}(\mu(m))$, and $\mathbf{t}(\mu(m))$ consisting of shortest transversals, one for each component of $\mathbf{base}(\mu(m))$. When we talk about the distance $d_{Y}$ between two points in $\mathcal{T}(S)$ or between a point in $\mathcal{T}(S)$ and a marking, we identify points $m\in\mathcal{T}(S)$ with $\mu(m)$.

Let $M$ be an atoroidal boundary-irreducible Haken 3-manifold. Let $\{\rho_{i}\}$ be a sequence of faithful discrete representations of $\pi_{1}(M)$ into ${\rm PSL}_{2}(\mathbb{C})$. We define a geometric limit of $\{\rho_{i}(\pi_{1}(M))\}$ to be a Kleinian group $\Gamma$ such that every element $\gamma$ of $\Gamma$ is a limit of some sequence $\{g_{i}\in\rho_{i}(\pi_{1}(M))\}$, and every convergent sequence $\{\gamma_{i_{j}}\in\rho_{i_{j}}(\pi_{1}(M))\}$ has its limit in $\Gamma$.

Fixing a point $x\in\mathbb{H}^{3}$, and considering its projections $x_{i}$ in $\mathbb{H}^{3}/\rho_{i}(\pi_{1}(M))$ and $x_{\infty}$ in $\mathbb{H}^{3}/\Gamma$, the geometric convergence implies the existence of pointed Gromov-Hausdorff convergence of $((\mathbb{H}^{3}/\rho_{i}(\pi_{1}(M)),x_{i}))$ to $(\mathbb{H}^{3}/\Gamma,x_{\infty})$. This latter convergence means that there exist real numbers $r_{i}$ going to $\infty$, $K_{i}$ converging to $1$, and $K_{i}$-bi-Lipschitz diffeomorphisms $f_{i}$ (called approximate isometries) between $r_{i}$-balls $B_{r_{i}}(\mathbb{H}^{3}/\rho_{i}(\pi_{1}(M)),x_{i})$ and $B_{K_{i}r_{i}}(\mathbb{H}^{3}/\Gamma,x_{\infty})$. Suppose that $\{\rho_{i}\}$ converges to $\rho_{\infty}:\pi_{1}(M)\to{\rm PSL}_{2}(\mathbb{C})$ as representations and that $\{\rho_{i}(\pi_{1}(M))\}$ converges to $\Gamma$ geometrically. Then, $\rho_{\infty}(\pi_{1}(M))$ is a subgroup of the geometric limit $\Gamma$.

For an open irreducible 3-manifold $V$ with finitely generated fundamental group, a compact 3-dimensional submanifold $C$ in $V$ is called a compact core when the inclusion induces an isomorphism between their fundamental groups. The existence of compact cores was proved by Scott [19]. The case which interests us is when $V$ is a hyperbolic 3-manifold.

Let $\mathbb{H}^{3}/G$ be a hyperbolic 3-manifold associated with a finitely generated, torsion free Kleinian group $G$. By Margulis’s lemma, there is a positive constant $\varepsilon_{0}$ such that the set of points of $\mathbb{H}^{3}/G$ where the injectivity radii are less than $\varepsilon_{0}$ consists of a finite disjoint union of tubular neighbourhoods of closed geodesics of length less than $\varepsilon_{0}$, called Margulis tubes, and cusp neighbourhoods each of which is stabilised by a maximal parabolic subgroup of $G$, and whose quotient by its stabiliser is homeomorphic to $S^{1}\times\mathbb{R}^{2}$ when the stabiliser has rank 1, and to $S^{1}\times S^{1}\times\mathbb{R}$ when the stabiliser has rank 2. The former cusp neighbourhood is called a $\mathbb{Z}$-cusp neighbourhood, and the latter a torus cusp neighbourhood. The union of the cusp neighbourhoods is called the cuspidal part of $\mathbb{H}^{3}/G$. The complement of the cuspidal part is called the non-cuspidal part and is denoted by $(\mathbb{H}^{3}/G)_{0}$. Each boundary component of $(\mathbb{H}^{3}/G)_{0}$ is either an open annulus or a torus. By the relative compact core theorem by McCullough [13], there is a compact core $C_{G}\subset(\mathbb{H}^{3}/G)_{0}$ such that for each boundary component $B$ of $(\mathbb{H}^{3}/G)_{0}$, the intersection $C_{G}\cap B$ is a core annulus when $B$ is an open annulus, and is the entire $B$ when $B$ is a torus. We call such a compact core a relative compact core of $(\mathbb{H}^{3}/G)_{0}$.

Let $p\colon\mathbb{H}^{3}/\rho_{\infty}(\pi_{1}(M))\to\mathbb{H}^{3}/\Gamma$ be the covering map associated with the inclusion of $\rho_{\infty}(\pi_{1}(M))$ into the geometric limit $\Gamma$. Let $C$ be a relative compact core of $(\mathbb{H}^{3}/\rho_{\infty}(\pi_{1}(M)))_{0}$. Suppose that $\mathbb{H}^{3}/\Gamma$ has a torus cusp neighbourhood $T$. We say that $\mathbb{H}^{3}/\rho_{\infty}(\pi_{1}(M))$ wraps around $T$ when $p|C$ is homotoped to an immersion which goes around $T$ non-trivially, and hence cannot be homotoped to an embedding.

In this subsection, we shall show that to prove Theorem 1.1, we can put an extra assumption that all the characteristic $I$-pairs of $M$ are product bundles.

We consider an atoroidal Haken manifold as is given in Theorem 1.1. Let $p\colon\tilde{M}\to M$ be a finite-sheeted regular covering. Then $p$ induces the covering map between the boundaries $p_{\partial}\colon\partial\tilde{M}\to\partial M$. This map induces a proper embedding between Teichmüller spaces, $p_{\partial}^{*}\colon\mathcal{T}(\partial M)\to\mathcal{T}(\partial\tilde{M})$ which is obtained by pulling back the conformal structures by $p_{\partial}$. Also the involution $\iota$ lifts to an orientation-reversing involution $\tilde{\iota}\colon\partial\tilde{M}\to\partial\tilde{M}$ taking each component to another one. Since $\tilde{M}$ is also an atoroidal boundary-irreducible Haken manifold, we can consider the skinning map $\tilde{\sigma}\colon\mathcal{T}(\partial\tilde{M})\to\mathcal{T}(\bar{\partial}\tilde{M})$.

This result allows us to work on manifolds with topological features that will make the arguments simpler.

For some of our arguments we will need a stronger assumption than having only product bundles:

We are going to construct a cover with the properties (A) and (B) above. In order to do that, we need to examine how characteristic $I$-pairs are attached to other components of the characteristic submanifold. In the following proof of Lemma 3.4, it will turn out that there are two situations ((a) and (b) below) where the second condition of ‘strong untwistedness’ breaks down.

Lemmas 3.1 and 3.4 show that to prove Theorem 1.1, we have only to consider the case when $M$ is strongly untwisted.

Let $M$ be an atoroidal Haken manifold as in Theorem 1.1. Let $X$ be the characteristic submanifold of $M$. Assume that every $I$-bundle in $X$ is a product $I$-bundle.

It follows from the definition of characteristic submanifold that there is an isotopy which takes $V_{F}$ into the characteristic submanifold $X$. From now on, we assume that if $F$ is one-time vertically extendible then $F\subset X$ and $V_{F}\subset X$.

We note solid torus components in $X$ may add some complications in the case when $F$ is an annulus. If $F$ is contained in such a component of $X$, there may be more than one possible first elevation (even up to isotopy) and the $I$-bundles corresponding to two disjoint annuli may intersect (even up to isotopy).

We now define multiple elevations by induction.

We say that two multi-curves $c,d\subset\partial M$ intersect minimally if for every multicurves $c^{\prime},d^{\prime}$ homotopic to $c$ and $d$ respectively, $\sharp\{c\cap d\}\leq\sharp\{c^{\prime}\cap d^{\prime}\}$. Let $F,G\subset X\cap\partial M$ be two incompressible surfaces. We say that $F$ and $G$ intersect minimally if $\partial F$ intersects $\partial G$ minimally.

Since an $n$-time vertically extendible surface is $m$-vertically extendible for any $m\leq n$, we have $\Sigma^{n}\subset\Sigma^{m}$ up to isotopy.

In the next lemma we show that, when $N$ is atoroidal, $M$ cannot contain an $n$-time extendible surface for sufficiently large $n$. In the last section, this result will lead us to the constant $n$ of Theorem 1.1.

We first recall the definition of the invariant $m$ from [2], and see how it controls the behaviour of sequences of Fuchsian groups.

It follows from [2, Lemma 5.2] that two curves with unbounded $m$ cannot intersect:

The invariant $m$ is related to the divergence and convergence of a sequence by the following lemma:

We shall next establish a necessary condition on the invariant $m$ for algebraic convergence on a submanifold. We start with a fundamental result. Thurston proved in [23] the following which is the first half of the theorem often referred to as the ‘broken window only’ theorem. We note that the latter half of the broken window only theorem should need some rectification (see [17]) but is irrelevant to the present paper.

Using arguments from [2], we establish the following necessary condition for algebraic convergence on a submanifold.