3.1.1 Initial data

Modulo good pants constructions and lifting to finite covers, our construction depends (only) on the following choice of auxiliary data.

1. – A geometric triangulation of $N$ : that is, a division of the target hyperbolic 3-manifold $N$ into a simplicial complex of which the simplices are all totally geodesic. When a positive constant $\unicode[STIX]{x1D705}$ is given, we require that the length of edges are at most $10^{-2}\unicode[STIX]{x1D705}$ , or at most $10^{-2}$ if $\unicode[STIX]{x1D705}$ exceeds $1$ .

2. – A homological lift, by which we mean a homomorphism of homology:

   $$\begin{eqnarray}i_{\ast }:H_{1}(N;\mathbb{Z})\rightarrow H_{1}(M;\mathbb{Z}),\end{eqnarray}$$
   together with an embedding of the $1$ -skeleton:
   $$\begin{eqnarray}i_{1}:N^{(1)}\rightarrow M.\end{eqnarray}$$
   We require $i_{1}$ to realize $i_{\ast }$ in the sense that the induced homomorphism $i_{1\ast }:H_{1}(N^{(1)};\mathbb{Z})\rightarrow H_{1}(M;\mathbb{Z})$ factors through $i_{\ast }$ .

3. – A pair of trivializations of the $\text{SO}(3)$ -principal bundles:

   $$\begin{eqnarray}t_{N}:\text{SO}(N)\rightarrow N\times \text{SO}(3)\end{eqnarray}$$
   and
   $$\begin{eqnarray}t_{M}:\text{SO}(M)\rightarrow M\times \text{SO}(3).\end{eqnarray}$$

A collection of initial data can be easily prepared as required. Specifically, a geometric triangulation can be constructed in the following way: first take a Dirichlet polyhedron $P$ in the universal cover $\mathbb{H}^{3}$ of $N$ . The combinatorial barycentric subdivision of $P$ can be realized as a geometric triangulation of $P$ , and the barycenters can be chosen to match up under side-pairing. Then we have an induced geometric triangulation of $N$ as claimed. By iterating barycentric subdivisions, we can bound the edge length by any required positive constant $10^{-2}\unicode[STIX]{x1D705}$ . A default option for a homological lift could be the trivial homomorphism together with an embedding of the $1$ -skeleton of $N$ into a topologically embedded ball of $M$ . Choosing trivializations of the frame bundles amounts to finding a field of frames over each base manifold, which is always possible as the tangent bundle of any orientable closed $3$ -manifold is trivial.

3.1.2 Geometric notations

Having chosen the geometric triangulation of $N$ , we enumerate the vertices of $N$ by

$$\begin{eqnarray}V_{N}=\{n_{1},n_{2},\ldots ,n_{l}\}.\end{eqnarray}$$

When there is an edge between $n_{i}$ and $n_{j}$ , denote the oriented edge from $n_{i}$ to $n_{j}$ by $e_{ij}$ , so $e_{ji}$ is the orientation-reversal of the same edge. When there is a $2$ -simplex spanned by a triple of vertices $(n_{i},n_{j},n_{k})$ , denote the marked $2$ -simplex by $\unicode[STIX]{x1D6E5}_{ijk}$ .

Associated to any marked $2$ -simplex $\unicode[STIX]{x1D6E5}_{ijk}$ there is a frame $\mathbf{F}_{ijk}$ based at $n_{i}$ defined as follows. For any oriented edge $e_{ij}$ , denote by $\vec{v}_{ij}$ the unit tangent vector of $e_{ij}$ based at $n_{i}$ pointing to  $n_{j}$ . For any marked $2$ -simplex $\unicode[STIX]{x1D6E5}_{ijk}$ , denote by

$$\begin{eqnarray}\vec{n}_{ijk}=\frac{\vec{v}_{ij}\times \vec{v}_{ik}}{|\vec{v}_{ij}\times \vec{v}_{ik}|}\end{eqnarray}$$

the distinguished normal vector of $\unicode[STIX]{x1D6E5}_{ijk}$ at $n_{i}$ . The cross product does not vanish since $\unicode[STIX]{x1D6E5}_{ijk}$ is totally geodesic in $N$ . Then the frame associated to $\unicode[STIX]{x1D6E5}_{ijk}$ based at $n_{i}$ is defined by

$$\begin{eqnarray}\mathbf{F}_{ijk}=(\vec{v}_{ij},\vec{n}_{ijk}).\end{eqnarray}$$

In this paper, we only write the first two vectors for a frame, such as $\mathbf{F}=(\vec{v},\vec{n})$ , implicitly meaning that the third vector is $\vec{v}\times \vec{n}$ . We also introduce the notation

$$\begin{eqnarray}-\mathbf{F}_{ijk}=(-\vec{v}_{ij},-\vec{n}_{ijk}).\end{eqnarray}$$

Note that if we write down all three vector components in a frame, $-\mathbf{F}_{ijk}$ only changes sign for the first two vectors from that of $\mathbf{F}_{ijk}$ , not for the third. Observe that $-\mathbf{F}_{jik}$ is the parallel transport of $\mathbf{F}_{ijk}$ from $n_{i}$ to $n_{j}$ along $e_{ij}$ . Collectively we write

$$\begin{eqnarray}F_{N}=\{\pm \mathbf{F}_{ijk}:\unicode[STIX]{x1D6E5}_{ijk}\text{ a marked 2-simplex of }N\}\subset \text{SO}(N)|_{V_{N}}.\end{eqnarray}$$

With the chosen homological lift and the trivialization of frame bundles, we define a morphism of $\text{SO}(3)$ -principal bundles

$$\begin{eqnarray}\mathbf{i}_{1}:\text{SO}(N)|_{N^{(1)}}\rightarrow \text{SO}(M)\end{eqnarray}$$

over a base map $i_{1}:N^{(1)}\rightarrow M$ by the following commutative diagram:

(1)

Via the bundle morphism $\mathbf{i}_{1}$ , we define objects associated to $M$ accordingly, namely,

$$\begin{eqnarray}m_{k}=i_{1}(n_{k})\end{eqnarray}$$

and

$$\begin{eqnarray}\mathbf{F}_{ijk}^{M}=(\vec{v}_{ij}^{M},\vec{n}_{ijk}^{M})=\mathbf{i}_{1}(\mathbf{F}_{ijk}).\end{eqnarray}$$

Collectively we write

$$\begin{eqnarray}V_{M}=\{m_{1},m_{2},\ldots ,m_{l}\}\subset M,\end{eqnarray}$$

and

$$\begin{eqnarray}F_{M}=\mathbf{i}_{1}(F_{N})\,\subset \,\text{SO}(M)|_{V_{M}}.\end{eqnarray}$$

Observe that, since $\mathbf{i}_{1}$ commutes with the right $\text{SO}(3)$ -multiplication, $-\mathbf{F}_{ijk}^{M}$ equals $\mathbf{i}_{1}(-\mathbf{F}_{ijk})$ .

Denote by $\unicode[STIX]{x1D703}_{ijk}$ the external angle of the geodesic $2$ -simplex $\unicode[STIX]{x1D6E5}_{ijk}$ in $N$ at the vertex $n_{i}$ . In other words, it is the geometric angle formed by the vectors $-\vec{v}_{ij}$ and $\vec{v}_{ik}$ , valued in the interval $(0,\unicode[STIX]{x1D70B})$ . Because there are only finitely many $2$ -simplices $\unicode[STIX]{x1D6E5}_{ijk}$ , there is a uniform bound constant

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{0}=\min \{\unicode[STIX]{x1D703}_{ijk},\unicode[STIX]{x1D70B}-\unicode[STIX]{x1D703}_{ijk}:\unicode[STIX]{x1D6E5}_{ijk}\text{ any 2-simplex of }N\}\in (0,\unicode[STIX]{x1D70B})\end{eqnarray}$$

such that

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{ijk}\in [\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D70B}-\unicode[STIX]{x1D719}_{0}].\end{eqnarray}$$

3.1.3 Homological instruction

Our construction follows the homological instruction given by the homomorphism defined in the following proposition.

Proposition 3.1. There exists a homomorphism

$$\begin{eqnarray}\mathbf{j}_{\ast }:H_{1}(\text{SO}(N),F_{N};\,\mathbb{Z})\rightarrow H_{1}(\text{SO}(M),F_{M};\,\mathbb{Z})\end{eqnarray}$$

that is uniquely associated to the bundle morphism $\mathbf{i}_{1}$ , such that the following diagrams of homomorphisms are commutative:

(2)

and

(3)

The horizontal homomorphisms are those of the long exact sequences for pairs and the vertical homomorphisms involving $\mathbf{i}_{1}$ are induced by restrictions.

Proof. By definition, there is an induced homomorphism

$$\begin{eqnarray}\mathbf{i}_{1\ast }:H_{1}(\text{SO}(N)|_{N^{(1)}},F_{N};\,\mathbb{Z})\rightarrow H_{1}(\text{SO}(M),F_{M};\,\mathbb{Z}).\end{eqnarray}$$

The relative homology $H_{2}(\text{SO}(N),\text{SO}(N)|_{N^{(1)}};\mathbb{Z})$ is generated by the $2$ -simplices of $N$ , or more precisely, by the $t_{N}$ -horizontal $2$ -simplices of $\text{SO}(N)$ . The assumption that $i_{1}:N^{(1)}\rightarrow M$ induces a homomorphism $i_{\ast }:H_{1}(N;\mathbb{Z})\rightarrow H_{1}(M;\mathbb{Z})$ forces boundaries of $t_{N}$ -horizontal $2$ -simplices of $\text{SO}(N)$ to be sent via $\mathbf{i}_{1}$ to $t_{M}$ -horizontal boundaries in $\text{SO}(M)$ , by the diagram (1). Since $F_{M}$ is zero-dimensional, the restriction of $\mathbf{i}_{1\ast }$ to $\unicode[STIX]{x2202}_{\ast }H_{2}(\text{SO}(N),\text{SO}(N)|_{N^{(1)}};\,\mathbb{Z})$ is hence trivial. Moreover, the inclusion induces an epimorphism $H_{1}(\text{SO}(N)|_{N^{(1)}},F_{N};\,\mathbb{Z})\rightarrow H_{1}(\text{SO}(N),F_{N};\,\mathbb{Z})$ . It follows from the long exact sequence for the triple $(\text{SO}(N),\text{SO}(N)|_{N^{(1)}},F_{N})$ that $\mathbf{i}_{1\ast }$ descends to a unique homomorphism $\mathbf{j}_{\ast }$ as claimed. The commutative diagrams are immediate consequences of the naturality of the long exact sequence, for the map of pairs $\mathbf{i}_{1}:(\text{SO}(N)|_{N^{(1)}},F_{N})\rightarrow (\text{SO}(M),F_{M})$ , and the way $\mathbf{j}_{\ast }$ is induced.◻

3.2.1 Wiring in $M$ with the $1$ -skeleton of $N$

Corresponding to each edge $e_{ij}$ from $n_{i}$ to $n_{j}$ of $N$ , we create an oriented geodesic arc $e_{ij}^{M}$ from $m_{i}$ to $m_{j}$ invoking the enhanced connection principle. To avoid ambiguity, suppose that $i<j$ , and take $k$ to be the smallest index such that $N$ has a $2$ -simplex $\unicode[STIX]{x1D6E5}_{ijk}$ with its vertices being $n_{i}$ , $n_{j}$ and $n_{k}$ . We only construct $e_{ij}^{M}$ with respect to such triples of vertices $(n_{i},n_{j},n_{k})$ , as follows.

Because of the diagram (2), the relative homology class

has boundary $[-\mathbf{F}_{jik}^{M}]-[\mathbf{F}_{ijk}^{M}]$ in $H_{0}(\mathbf{F}_{ijk}^{M}\cup (-\mathbf{F}_{jik}^{M});\,\mathbb{Z})$ , which can be canonically identified as a submodule of $H_{0}(F_{M};\,\mathbb{Z})$ . By the enhanced connection principle (Theorem 2.4), we construct an oriented geodesic arc $e_{ij}^{M}$ in $M$ from $m_{i}$ to $m_{j}$ with the following description.

1. – The length of $e_{ij}^{M}$ is $(\unicode[STIX]{x1D6FF}/L)$ -close to $L$ , the initial direction of $e_{ij}^{M}$ is $(\unicode[STIX]{x1D6FF}/L)$ -close to the first vector of $\mathbf{F}_{ijk}^{M}$ and the terminal direction of $e_{ij}^{M}$ is $(\unicode[STIX]{x1D6FF}/L)$ -close to the first vector of $-\mathbf{F}_{jik}^{M}$ .

2. – The parallel transport of $\mathbf{F}_{ijk}^{M}$ along $e_{ij}^{M}$ is $(\unicode[STIX]{x1D6FF}/L)$ -close to $-\mathbf{F}_{jik}^{M}$ , and there is a unique shortest path in $\text{SO}(M)|_{m_{j}}$ between these two frames.

3. – In $H_{1}(\text{SO}(M),\mathbf{F}_{ijk}^{M}\cup (-\mathbf{F}_{jik}^{M});\,\mathbb{Z})$ ,

   

Construct $e_{ij}^{M}$ accordingly for all oriented edges $e_{ij}$ , $i<j$ , with respect to the smallest possible $k$ as above. We define $e_{ji}^{M}$ to be the orientation-reversal of $e_{ij}^{M}$ .

For each edge in $N^{(1)}$ , we want to construct exactly one edge in $Z$ corresponding to it. However, in the above construction, we made some extra choices: for each edge in $N$ connecting vertices $n_{i}$ and $n_{j}$ , we first order $i$ and $j$ such that $i<j$ , then take the smallest $k$ such that $N$ has the $2$ -simplex $\unicode[STIX]{x1D6E5}_{ijk}$ . One could certainly construct another oriented geodesic arc $e_{ij}^{M}$ with respect to a different available $k$ , or construct $e_{ij}^{M}$ for $i>j$ directly, satisfying the same description. However, such extra work is unnecessary, and not very helpful. The point is that the oriented geodesics we have constructed work for all other choices.

Proof. It is easy to see that the first two items in the description remain true if we take any other $2$ -simplex $\unicode[STIX]{x1D6E5}_{ijk^{\prime }}$ or consider $e_{ji}$ , by the fact that $\mathbf{i}_{1}$ commutes with the right $\text{SO}(3)$ -action. So our goal is to verify the third item of the description, and it amounts to proving the following statement: for an edge $e_{ij}$ in the triangulation of $N$ from $n_{i}$ to $n_{j}$ , and a geodesic arc $e_{ij}^{M}$ in $M$ from $m_{i}$ to $m_{j}$ , suppose that the following equality of relative homology classes holds for some marked $2$ -simplex $\unicode[STIX]{x1D6E5}_{ijk}$ of $N$ :

(4)

Then for any other marked $2$ -simplex $\unicode[STIX]{x1D6E5}_{ijk^{\prime }}$ of $N$ ,

(5)

and

(6)

We first verify the identity (5). Denote by

$$\begin{eqnarray}A(\unicode[STIX]{x1D703})=\left[\begin{array}{@{}ccc@{}}1 & 0 & 0\\ 0 & \cos \unicode[STIX]{x1D703} & \sin \unicode[STIX]{x1D703}\\ 0 & -\!\sin \unicode[STIX]{x1D703} & \cos \unicode[STIX]{x1D703}\end{array}\right]\end{eqnarray}$$

the $3\times 3$ orthogonal matrix which represents a rotation about the first vector. Since the first vectors in both $\mathbf{F}_{ijk}$ and $\mathbf{F}_{ijk^{\prime }}$ are $\vec{v}_{ij}$ , we have

$$\begin{eqnarray}\mathbf{F}_{ijk^{\prime }}=\mathbf{F}_{ijk}\cdot A(\unicode[STIX]{x1D703}_{0}).\end{eqnarray}$$

Here $\unicode[STIX]{x1D703}_{0}$ is the dihedral angle between $\unicode[STIX]{x1D6E5}_{ijk}$ and $\unicode[STIX]{x1D6E5}_{ijk^{\prime }}$ . Since $2$ -simplices are totally geodesic in $N$ , we have

$$\begin{eqnarray}-\mathbf{F}_{jik^{\prime }}=(-\mathbf{F}_{jik})\cdot A(\unicode[STIX]{x1D703}_{0}).\end{eqnarray}$$

Similar to the notation of relative homology classes given by parallel transportations, we use

to denote the relative homology class in $H_{1}(\text{SO}(N),F_{N};\,\mathbb{Z})$ which is given by the path $t\mapsto \mathbf{F}_{ijk}\cdot A(\unicode[STIX]{x1D703}_{0}t)$ with the initial point $\mathbf{F}_{ijk}$ and the terminal point $\mathbf{F}_{ijk^{\prime }}$ .

Note that the parallel transportation along a fixed geodesic commutes with the right $\text{SO}(3)$ -action on the frame bundle. By considering the parallel transportation of $\mathbf{F}_{ijk}\cdot A(\unicode[STIX]{x1D703}_{0}t)$ along $e_{ij}$ for all $t\in [0,1]$ , we get a map $I\times I\rightarrow \text{SO}(N)$ such that the relative homology class of its boundary in $H_{1}(\text{SO}(N),F_{N};\,\mathbb{Z})$ is

(7) $$\begin{eqnarray}[\mathbf{F}_{ijk},-\mathbf{F}_{jik}]+[-\mathbf{F}_{jik},-\mathbf{F}_{jik^{\prime }}]-[\mathbf{F}_{ijk^{\prime }},-\mathbf{F}_{jik^{\prime }}]-[\mathbf{F}_{ijk},\mathbf{F}_{ijk^{\prime }}]=0.\end{eqnarray}$$

Here we have suppressed the understood paths to abbreviate the expression.

Passing to $M$ , we again observe that the identification $\mathbf{i}_{1}:\text{SO}(N)|_{V_{N}}\rightarrow \text{SO}(M)|_{V_{M}}$ commutes with the right $\text{SO}(3)$ -action. Therefore,

$$\begin{eqnarray}\mathbf{F}_{ijk^{\prime }}^{M}=\mathbf{F}_{ijk}^{M}\cdot A(\unicode[STIX]{x1D703}_{0})\quad \text{and}\quad -\mathbf{F}_{jik^{\prime }}^{M}=(-\mathbf{F}_{jik}^{M})\cdot A(\unicode[STIX]{x1D703}_{0}).\end{eqnarray}$$

Let $-\bar{\mathbf{F}}_{jik}^{M}$ be the parallel transportation of $\mathbf{F}_{ijk}^{M}$ along $e_{ij}^{M}$ . Note that $-\bar{\mathbf{F}}_{jik}^{M}$ is not exactly $-\mathbf{F}_{jik}^{M}$ , but there exists a $(\unicode[STIX]{x1D6FF}/L)$ -short geodesic $\unicode[STIX]{x1D6FE}:I\rightarrow \text{SO}(3)$ with $\unicode[STIX]{x1D6FE}(0)=I_{3}$ such that $-\mathbf{F}_{jik}^{M}=-\bar{\mathbf{F}}_{jik}^{M}\cdot \unicode[STIX]{x1D6FE}(1)$ . We can define a map $I\times I\rightarrow \text{SO}(M)$ as follows. For every $t\in [0,1]$ , the $t$ -slice is the concatenation of a path defined by the parallel transportation of $\mathbf{F}_{ijk}^{M}\cdot A(\unicode[STIX]{x1D703}_{0}t)$ along $e_{ij}^{M}$ (ending as $(-\bar{\mathbf{F}}_{jik}^{M})\cdot A(\unicode[STIX]{x1D703}_{0}t)$ ), and the path defined by the $\text{SO}(3)$ -action of $A(\unicode[STIX]{x1D703}_{0}t)^{-1}\cdot \unicode[STIX]{x1D6FE}\cdot A(\unicode[STIX]{x1D703}_{0}t)$ (ending as $(-\mathbf{F}_{jik}^{M})\cdot A(\unicode[STIX]{x1D703}_{0}t)$ ).

The relative homology class of the boundary of this mapped-in square, in $H_{1}(\text{SO}(M),F_{M};\,\mathbb{Z})$ , is

(8) $$\begin{eqnarray}[\mathbf{F}_{ijk}^{M},-\mathbf{F}_{jik}^{M}]+[-\mathbf{F}_{jik}^{M},-\mathbf{F}_{jik^{\prime }}^{M}]-[\mathbf{F}_{ijk^{\prime }}^{M},-\mathbf{F}_{jik^{\prime }}^{M}]-[\mathbf{F}_{ijk}^{M},\mathbf{F}_{ijk^{\prime }}^{M}]=0.\end{eqnarray}$$

The suppressed path labels are $e_{ij}^{M}$ , $A(\unicode[STIX]{x1D703}_{0}t)$ , $e_{ij}^{M}$ and $A(\unicode[STIX]{x1D703}_{0}t)$ accordingly.

By the diagram (3) and the fact that $\mathbf{i}_{1}:\text{SO}(N)|_{V_{N}}\rightarrow \text{SO}(M)|_{V_{M}}$ commutes with the right $\text{SO}(3)$ -action, we have

(9)

As $\mathbf{j}_{\ast }$ is a homomorphism, by the equalities (7), (8) and (9), and the assumed identity (4), we obtain

This verifies the claimed identity (5).

The verification of the identity (6) is similar, so we only give a sketch here.

Denote by

$$\begin{eqnarray}C(\unicode[STIX]{x1D703})=\left[\begin{array}{@{}ccc@{}}\cos \unicode[STIX]{x1D703} & \sin \unicode[STIX]{x1D703} & 0\\ -\!\sin \unicode[STIX]{x1D703} & \cos \unicode[STIX]{x1D703} & 0\\ 0 & 0 & 1\end{array}\right]\end{eqnarray}$$

the $3\times 3$ orthogonal matrix which represents a rotation about the third vector. By considering the parallel transportation of $\mathbf{F}_{ijk}\cdot C(\unicode[STIX]{x1D70B}t)$ along $e_{ij}$ , we get a map $I\times I\rightarrow \text{SO}(N)$ , of which the relative homology class of the boundary in $H_{1}(\text{SO}(N),F_{N};\,\mathbb{Z})$ is

(10) $$\begin{eqnarray}[\mathbf{F}_{ijk},-\mathbf{F}_{jik}]+[-\mathbf{F}_{jik},\mathbf{F}_{jik}]-[-\mathbf{F}_{ijk},\mathbf{F}_{jik}]-[\mathbf{F}_{ijk},-\mathbf{F}_{ijk}]=0.\end{eqnarray}$$

The suppressed path labels are $e_{ij}$ , $C(\unicode[STIX]{x1D70B}t)$ , $e_{ij}$ and $C(\unicode[STIX]{x1D70B}t)$ accordingly. Note that

holds.

As for the above situation, there is a vanishing relative homology class in $H_{1}(\text{SO}(M),F_{M};\mathbb{Z})$ , which is the boundary of a likewise mapped-in square in $\text{SO}(M)$ . We proceed in a similar fashion as before to complete the verification, with the only difference explained as follows.

While the paths in $\text{SO}(N)$ giving $[(-\mathbf{F}_{ijk})\xrightarrow[{}]{e_{ij}}\mathbf{F}_{jik}]$ and $[\mathbf{F}_{jik}\xrightarrow[{}]{e_{ji}}(-\mathbf{F}_{jik})]$ are exactly the inverse of each other, the paths in $\text{SO}(M)$ giving $[(-\mathbf{F}_{ijk}^{M})\xrightarrow[{}]{e_{ij}^{M}}\mathbf{F}_{jik}^{M}]$ and $[\mathbf{F}_{jik}^{M}\xrightarrow[{}]{e_{ji}^{M}}(-\mathbf{F}_{jik}^{M})]$ may differ by a uniform short distance along the parallel transportation part. However, this does not affect the fact that the relative homology classes in $H_{1}(\text{SO}(M),F_{M};\mathbb{Z})$ are the negative of each other, which can again be shown by a mapped-in square argument. So by comparing the equalities and using the definition that $e_{ij}^{M}$ is the orientation-reversal of $e_{ji}^{M}$ , one will obtain

which verifies the identity (6).◻

We attach edges to the analogue $0$ -skeleton $Z^{0}$ to build the analogue $1$ -skeleton $Z^{1}$ , so that $Z^{1}$ is naturally isomorphic to the $1$ -skeleton of $N$ . The geodesic arcs $e_{ij}^{M}$ that we have constructed give rise to a distinguished immersion:

$$\begin{eqnarray}Z^{1}{\looparrowright}M.\end{eqnarray}$$

3.2.2 Killing triangles with good surfaces

Corresponding to each $2$ -simplex $\unicode[STIX]{x1D6E5}_{ijk}$ spanned by a triple of vertices $(n_{i},n_{j},n_{k})$ of $N$ , we create an immersed nearly totally geodesic subsurface of $M$ with connected cornered boundary, $S_{ijk}^{M}$ , such that $\unicode[STIX]{x2202}S_{ijk}^{M}$ is the cyclic concatenation of the segments $e_{ij}^{M}$ , $e_{jk}^{M}$ and $e_{ki}^{M}$ . Combinatorially $S_{ijk}^{M}$ is built as a panted surface collared by a three-spike crown (an annulus with three corners on the outer boundary). Since these immersed subsurfaces of $M$ should correspond to the $2$ -simplices of $N$ regardless of the marking, we only need to construct $S_{ijk}^{M}$ for those marked $\unicode[STIX]{x1D6E5}_{ijk}$ such that $i<j<k$ , as follows.

Denote by $\unicode[STIX]{x1D6FE}_{ijk}$ the unique oriented closed geodesic of $M$ which is freely homotopic to the cyclic concatenation of the three oriented geodesic arcs $e_{ij}^{M}$ , $e_{jk}^{M}$ and $e_{ki}^{M}$ . The following lemma describes the shape and framing homology class of $\unicode[STIX]{x1D6FE}_{ijk}$ .

We denote by

$$\begin{eqnarray}I(\unicode[STIX]{x1D703})=2\log (\sec (\unicode[STIX]{x1D703}/2))\end{eqnarray}$$

the limit inefficiency associated to a bending angle $\unicode[STIX]{x1D703}\in (0,\unicode[STIX]{x1D70B})$ (see [Reference Liu and MarkovicLM15, § 4.2] for the geometric meaning of this quantity). Recall that we have introduced a uniform bound $\unicode[STIX]{x1D719}_{0}\in (0,\unicode[STIX]{x1D70B})$ depending only on the chosen geometric triangulation of $N$ , so that all the $2$ -simplex external angles $\unicode[STIX]{x1D703}_{ijk}$ of $N$ lie in the interval $[\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D70B}-\unicode[STIX]{x1D719}_{0}]$ . Recall also the $\mathbb{Z}_{2}$ -valued panted-cobordism invariant $\unicode[STIX]{x1D70E}$ from Theorem 2.3.

Proof. The first item of part (ii), about the geometry of $\unicode[STIX]{x1D6FE}_{ijk}$ , follows directly from the formulas of [Reference Liu and MarkovicLM15, Lemma 4.8]. Note that we have been working under the hypothesis that the parameters $\unicode[STIX]{x1D6FF}$ and $L$ are suitably given. In particular, we see that $\unicode[STIX]{x1D6FE}_{ijk}$ is homotopically nontrivial.

For part (i) about the geometry of the immersed annulus, the situation is very similar to [Reference SunSun15b, Lemma 3.5] and the discussion thereafter, so we give a very brief sketch. The free homotopy between the cyclic concatenation of $e_{ij}^{M}$ , $e_{jk}^{M}$ and $e_{ki}^{M}$ and $\unicode[STIX]{x1D6FE}_{ijk}$ can be realized by a map of an annulus. To be specific, we may triangulate the annulus by adding three vertices $p_{i}$ , $p_{j}$ and $p_{k}$ on $\unicode[STIX]{x1D6FE}$ , and adding six edges, say, $p_{i}m_{i}$ , $p_{i}m_{j}$ , $p_{j}m_{j}$ , $p_{j}m_{k}$ , $p_{k}m_{k}$ and $p_{k}m_{i}$ . The mapped-in annulus can be homotoped relative to the boundary so that the triangulation is totally geodesic on each simplex, and the edges $p_{i}m_{i}$ , $p_{j}m_{j}$ and $p_{k}m_{k}$ are perpendicular to $\unicode[STIX]{x1D6FE}_{ijk}$ . Then all the claimed estimates follow directly from the formulas of [Reference Liu and MarkovicLM15, Lemma 4.8].

It remains to show the homological properties about $\unicode[STIX]{x1D6FE}_{ijk}$ , namely, the last two items of part (ii). The homology class of $\unicode[STIX]{x1D6FE}_{ijk}$ can be seen easily from the construction of $Z^{1}{\looparrowright}M$ . In fact, by projecting paths of frames in the base manifold, it is clear that the cycle $e_{ij}^{M}+e_{jk}^{M}+e_{ki}^{M}$ represents the class $i_{\ast }[\unicode[STIX]{x2202}\unicode[STIX]{x1D6E5}_{ijk}]$ in $H_{1}(M;\mathbb{Z})$ , where the latter equals zero.

To prove the vanishing of $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6FE}_{ijk})$ , we need to explain the constructive definition of $\unicode[STIX]{x1D70E}$ in more detail. For any $(R,\unicode[STIX]{x1D716})$ -good curve $\unicode[STIX]{x1D6FE}$ of $M$ , recall that a canonical lifting of $\unicode[STIX]{x1D6FE}$ is a loop of frames in $\text{SO}(M)$ defined as follows: take a point $p\in \unicode[STIX]{x1D6FE}$ and a frame $\mathbf{F}=(\vec{v}_{p},~\vec{n}_{p},~\vec{v}_{p}\times \vec{n}_{p})$ based at $p$ such that $\vec{v}_{p}$ is tangent to $\unicode[STIX]{x1D6FE}$ ; starting from $\mathbf{F}$ , draw a path of frames by parallel transporting $\mathbf{F}$ along $\unicode[STIX]{x1D6FE}$ for one round, arriving at a frame $\mathbf{F}^{\prime }$ over the same point $p$ ; continue by making a full rotation of $\mathbf{F}^{\prime }$ (counterclockwise) about its first vector; close the path by joining $\mathbf{F}^{\prime }$ to $\mathbf{F}$ by the shortest path in $\text{SO}(M)|_{p}$ . When $\unicode[STIX]{x1D6FE}$ is null-homologous in $H_{1}(M;\mathbb{Z})$ , any canonical lifting of $\unicode[STIX]{x1D6FE}$ defines one and the same class in the fiber submodule $H_{1}(\text{SO}(3);\mathbb{Z})\cong \mathbb{Z}_{2}$ of $H_{1}(\text{SO}(M);\mathbb{Z})$ . In this case, the invariant $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6FE})$ can be explicitly identified with the class of the canonical lifting of  $\unicode[STIX]{x1D6FE}$ .

In the following, we denote the unique nontrivial element of the fiber submodule $H_{1}(\text{SO}(3);\mathbb{Z})$ by $c$ . The vanishing of $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6FE})$ is equivalent to the identity

(11)

for some (hence any) frame $\mathbf{F}$ at a point $p\in \unicode[STIX]{x1D6FE}_{ijk}$ whose first vector is tangent to $\unicode[STIX]{x1D6FE}_{ijk}$ . The label of the arrow is considered as the closed path with both endpoints the point $p$ .

To prove the identity (11), we start by working with geodesic $2$ -simplices of $N$ . Denote by

$$\begin{eqnarray}B(\unicode[STIX]{x1D703})=\left[\begin{array}{@{}ccc@{}}\cos \unicode[STIX]{x1D703} & 0 & \sin \unicode[STIX]{x1D703}\\ 0 & 1 & 0\\ -\!\sin \unicode[STIX]{x1D703} & 0 & \cos \unicode[STIX]{x1D703}\end{array}\right]\end{eqnarray}$$

the $3\times 3$ matrix representing the rotation about the second vector of an angle $\unicode[STIX]{x1D703}$ . It can be observed in $N$ that, in $H_{1}(\text{SO}(N);\mathbb{Z})$ ,

(12)

One way to see this is by a continuation argument. The left-hand side is invariant if the vertices $(n_{i},n_{j},n_{k})$ of $\unicode[STIX]{x1D6E5}_{ijk}$ vary continuously in $N$ . Shrinking all the edge lengths towards zero, the sum of the external angles of $\unicode[STIX]{x1D6E5}_{ijk}$ tends to $2\unicode[STIX]{x1D70B}$ . In the limit, the cycle of frames represented by the left-hand side is a full rotation of a frame, say $\mathbf{F}_{ijk}$ , about its second vector, yielding the homology class $c\in H_{1}(\text{SO}(N);\mathbb{Z})$ . Therefore the identity (12) holds in general.

By applying the homomorphism $\mathbf{j}_{\ast }$ , we have in $H_{1}(\text{SO}(M);\mathbb{Z})$ :

(13)

Using the nearly totally geodesic annulus of part (i), the left-hand side of the identity (13) can be identified with the left-hand side of the identity (11). To visualize a homotopy between the loops of frames, assume for the moment that the annulus was genuinely totally geodesic. By foliating the annulus with smooth curves in its interior, the homotopy can be seen as the field of frames over the annulus, such that, at any point, the first vector is tangent to the leaf, and the second vector is perpendicular to the annulus. The continuity near the cornered boundary can be ensured as the frames are parallel transported along the edges and rotated counterclockwise about the (instant) second vector of the external angle by the $B$ -matrix at the corners. In our actual situation, the annulus is nearly totally geodesic, so a homotopy should be given as an approximation to the above picture. With this picture in mind, it is possible to check the equality explicitly by dividing the annulus into geodesic $2$ -simplices, as we have done, and discretizing the described homotopy. We omit the details since the procedure is straightforward.

Therefore, we have verified the identity (11), and hence the claimed vanishing of $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6FE}_{ijk})$ .◻

Corresponding to each $2$ -simplex $\unicode[STIX]{x1D6E5}_{ijk}$ spanned by a triple of vertices $(n_{i},n_{j},n_{k})$ of $N$ , only for those with $i<j<k$ do we take the corresponding oriented good curve $\unicode[STIX]{x1D6FE}_{ijk}$ of $M$ as guaranteed by Lemma 3.2. By part (i) of Lemma 3.2, we may enrich $\unicode[STIX]{x1D6FE}_{ijk}$ with a point $p\in \unicode[STIX]{x1D6FE}_{ijk}$ and an auxiliary frame $\mathbf{F}_{p}=(\vec{v}_{p},\vec{n}_{p})$ . To be specific, we take $\vec{v}_{p}$ to be the direction vector of $\unicode[STIX]{x1D6FE}_{ijk}$ at $p$ , and require that $\vec{n}_{p}$ is $(10\unicode[STIX]{x1D6FF}/L)$ -close to the parallel transport of the $\vec{n}$ -vector of $\mathbf{F}_{ijk}^{M}$ along a shortest possible path on the asserted immersed nearly totally geodesic annulus. By part (ii) of Lemma 3.2, we invoke Theorem 2.3 to construct an immersed oriented nearly totally geodesic $(R,\unicode[STIX]{x1D716})$ -panted subsurface bounded by $\unicode[STIX]{x1D6FE}_{ijk}$ , requiring the normal vector there to be $(\unicode[STIX]{x1D6FF}/L)$ -close to $\vec{n}_{p}$ . Moreover, the panted subsurface can be constructed to be fat in the sense that every properly embedded essential arc of the panted subsurface has length at least $R$ under the immersion into $M$ (see [Reference SunSun15a, § 3.1] and [Reference LiuLiu16, Lemma 4.2]). Glue up the constructed panted subsurface with the annulus of Lemma 3.2(i) along $\unicode[STIX]{x1D6FE}_{ijk}$ . We denote the resulting subsurface as

$$\begin{eqnarray}S_{ijk}^{M}{\looparrowright}M.\end{eqnarray}$$

This is our claimed counterpart in $M$ of the $2$ -simplex $\unicode[STIX]{x1D6E5}_{ijk}$ of $N$ .

We attach the abstract surfaces $S_{ijk}^{M}$ to $Z^{1}$ by the obvious attaching map of $\unicode[STIX]{x2202}S_{ijk}^{M}=e_{ij}^{M}\cup e_{jk}^{M}\cup e_{ki}^{M}$ . The resulting abstract $2$ -complex is our claimed $Z$ , coming with a naturally induced immersion:

$$\begin{eqnarray}Z{\looparrowright}M.\end{eqnarray}$$

3.2.3 Quasiconvexity and convex core

We verify that the immersion $Z{\looparrowright}M$ induces an embedding $\unicode[STIX]{x1D70B}_{1}(Z)\rightarrow \unicode[STIX]{x1D70B}_{1}(M)$ , and describe the topology of the cover of $M$ corresponding to (any conjugate of) $\unicode[STIX]{x1D70B}_{1}(Z)$ . For the latter, we introduce a compact $3$ -manifold ${\mathcal{Z}}$ as a topological model, which can be constructed as follows.

Note that the geodesic triangulation of $N$ induces a topological handle-decomposition of $N$ of which the simplices correspond naturally to the handles. Take the union ${\mathcal{N}}^{(1)}$ of the $0$ -handles and the $1$ -handles, which is just a regular neighborhood of the $1$ -skeleton of $N$ . The $2$ -handles $\unicode[STIX]{x1D6E5}_{ijk}\times [0,1]$ are attached to $\unicode[STIX]{x2202}{\mathcal{N}}^{(1)}$ along disjoint annuli $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E5}_{ijk}\times [0,1]$ . We copy ${\mathcal{N}}^{(1)}$ to be ${\mathcal{Z}}^{1}$ and attach to $\unicode[STIX]{x2202}{\mathcal{Z}}^{1}$ analogue $2$ -handles $S_{ijk}^{M}\times [0,1]$ along $\unicode[STIX]{x2202}S_{ijk}^{M}\times [0,1]$ accordingly. Since there is a natural combinatorial identification of $\unicode[STIX]{x2202}S_{ijk}^{M}$ with $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E5}_{ijk}$ by our construction of $S_{ijk}^{M}$ , the attaching map can be defined via the identification. The result is an abstract compact $3$ -manifold ${\mathcal{Z}}$ with a spine homeomorphic to the $2$ -complex $Z$ .

Recall that for a Kleinian group $\unicode[STIX]{x1D6E4}$ acting on $\mathbb{H}^{3}$ , the convex hull of $\unicode[STIX]{x1D6E4}$ is the convex hull of the limit set in $\mathbb{H}^{3}$ , and the convex core of $\mathbb{H}^{3}/\unicode[STIX]{x1D6E4}$ is the quotient of the convex hull by the action of  $\unicode[STIX]{x1D6E4}$ .

We close this subsection by pointing out the range of the parameters $(\unicode[STIX]{x1D6FF},L)$ and $\unicode[STIX]{x1D716}$ that we need for the above construction. After fixing a choice for the set of initial data, we require that a constant $0<\unicode[STIX]{x1D716}<1$ be chosen less than $\text{inj}(M)$ and $10^{-1}\unicode[STIX]{x1D719}_{0}$ , and take $\unicode[STIX]{x1D6FF}$ to be $10^{-1}\unicode[STIX]{x1D716}$ . The sufficiently large constant $L>1$ can be chosen, according to $\unicode[STIX]{x1D6FF}$ and the geometry of $M$ and our initial data, so that the application of the good pants constructions (Theorems 2.3 and 2.4) is valid. Note that, overall, we have only invoked Theorems 2.3 and 2.4 for a definite finite number of times, which is determined by the triangulation of $N$ .