Proposition Theorem 4.3.

We shall prove (2). Set $C_{X,\rho}=(\tau,L)\in\mathsf{T}\times\mathsf{ML}$ be
the ${\mathbb{C}{\rm P}}^{1}$ structure on $X$ with holonomy
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}\setminus
K$ in Thurston coordinates, and let $\kappa\colon C_{X,\rho}\to\tau$ be the
collapsing map. Since sufficiently away from the zero, the developing map is
well-approximated by the exponential map (Lemma 3.3).

If $K$ is sufficiently large, then for every vertical edge $v$ of $P$, the
restriction $\operatorname{Ep}_{X,\rho}|v$ is a $(1+\epsilon)$-bilipschitz
embedding on the Epstein surface. Therefore, by the closeness of
$\operatorname{Ep}_{X,\rho}$ and $\hat{\beta}_{X,\rho}$, $\kappa(v)$ is
$\epsilon$-close to a geodesic segment $s_{v}$ of length
$\sqrt{2}\operatorname{length}v$. By (1b), if $K$ is large enough,
$L(s_{v})<\epsilon$.

Every horizontal edge $h$ of $P$ is very short on the Epstein surface (Lemma
3.1). As the developing map is approximated by the Exponential map and ${\rm
Area}\tau=2\pi\rvert{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(S)\lvert$,
it follows that, if $\kappa(h)$ has length less than $\epsilon$ on $\tau$.

Therefore the image of $P$ on the hyperbolic surface is $\epsilon$-close to a
trancated ideal polygon.

\begin{overpic}[scale={.05}]{Figure/EuclideanPolygonToHyperbolicPolygon}
\end{overpic} Figure 16.

7.2

###  7.1. Equivariant circle systems

For
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}$,
we shall pick a system of a $\rho$-equivalent round circles on
${\mathbb{C}{\rm P}}^{1}$, which will be used to construct a circular train-
track structure of $C_{X,\rho}$. Let $\tilde{\mathbf{T}}_{X,\rho}$ be the
$\pi_{1}(S)$-invariant train-track structure on $\tilde{E}_{X,\rho}$ obtained
by lifting the train-track structure $\mathbf{T}_{X,\rho}$ on $E_{X,\rho}$.
Let $\operatorname{Ep}^{\ast}_{X,\rho}\colon T\tilde{E}_{X,\rho}\to
T\mathbb{H}^{3}$ be the differential of
$\operatorname{Ep}_{X,\rho}\colon\tilde{E}_{X,\rho}\to\mathbb{H}^{3}$.

######  Lemma 7.3.

For every $\epsilon>0$, there is a bounded subset $K_{\epsilon}$ of $\chi$
such that, if $\rho\colon\pi_{1}(S)\to{\rm PSL}(2,\mathbb{C})$ belongs to
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\setminus
K_{\epsilon}$, then, we can assign a round circle $c_{h}$ to every minimal
horizontal edge $h$ of $\tilde{\mathbf{T}}_{X,\rho}$ with the following
properties:

  1. (1)

The assignment $h\mapsto c_{h}$ is $\rho$-equivariant.

  2. (2)

The hyperbolic plane bounded by $c_{h}$ is $\epsilon$-almost orthogonal to the
$\operatorname{Ep}^{\ast}_{X,\rho}$-images of the vertical tangent vectors
along $h$.

  3. (3)

If $h_{1},h_{2}$ are horizontal edges of $\tilde{\mathbf{T}}_{X,\rho}$
connected by a vertical edge $v$ of length at least $\epsilon$, then the round
circles $c_{h_{1}}$ and $c_{h_{2}}$ are disjoint.

  4. (4)

If $h_{1},h_{2},h_{3}$ are “vertically consecutive” horizontal edges, such
that

     * •

$h_{1}$ and $h_{2}$ are connected by a vertical edge $v_{1}$ of
$E_{X,\rho}$-length at least $\epsilon$;

     * •

$h_{2}$ and $h_{3}$ are connected by a vertical edge $v_{3}$ of length at
least $\epsilon$;

     * •

$h_{1}$ and $h_{3}$ are on the different sides of $h_{2}$, i.e. the normal
vectors of $h_{2}$ in the direction of $v_{1}$ and $v_{3}$ are opposite,

then $c_{h_{1}}$ and $c_{h_{3}}$ are disjoint, and they bounds a round
cylinder whose interior contains $c_{h_{2}}$.

###### Proof.

Without loss of generality, we can assume that $\epsilon>0$ is sufficiently
small. For each minimal horizontal edge $h$ of $\tilde{\mathbf{T}}_{Y,\rho}$,
pick a round circle $c_{h}$, such that the assignment of $c_{h}$ is holonomy
equivariant and that the images of vertical tangent vectors along $h$ are
$\epsilon^{2}$-orthogonal to the hyperbolic plane bounded by $c_{h}$.

Then, if $v$ is a vertical edge sharing an endpoint with $h$, then
$\operatorname{Ep}_{X,\rho}(v)$ is $\epsilon^{2}$-almost orthogonal to the
hyperbolic plane bounded by $c_{h}$. For every sufficiently small
$\epsilon>0$, if $K>0$ is sufficiently large, then the geodesic segment of
length at least $\epsilon$ connects the hyperbolic planes bounded by
$c_{h_{1}}$ and $c_{h_{2}}$, and the geodesic segment is $\epsilon^{2}$-almost
orthogonal to both hyperbolic planes. Therefore, if $\epsilon>0$ is
sufficiently small, then, by elementary hyperbolic geometry, the hyperbolic
planes are disjoint, and (3) holds. By a similar argument, (4) also holds. ∎

The the circle system in Lemma 7.3 is not unique, but unique up to an
appropriate isotopy.

######  Proposition 7.4.

For every $\epsilon_{1}>0$, there is $\epsilon_{2}>0$, such that, for every
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\setminus
K_{\epsilon_{2}}$ given two systems of round circles $\\{c_{h}\\}$ and
$\\{c_{h}^{\prime}\\}$ realizing Lemma 7.3 for $\epsilon_{2}>0$, there is a
one-parameter family of equivalent circles systems
$\\{c_{t,h}\\}\leavevmode\nobreak\ (t\in[0,1])$ realizing Lemma 7.3 for
$\epsilon_{1}>0$ which continuously connects $\\{c_{h}\\}$ to
$\\{c_{h}^{\prime}\\}$.

###### Proof.

The proof is left for the reader. ∎

###  7.2. Pleated surfaces are close

The following gives a measure theoretic notion of almost parallel measured
laminations.

######  Definition 7.5 (Quasi-parallel).

Let $L_{1},L_{2}$ be two measured geodesic laminations on a hyperbolic surface
$\tau$. Then, $L_{1}$ and $L_{2}$ are $\epsilon$-quasi parallel, if a leaf
$\ell_{1}$ of $L_{1}$ and a leaf $\ell_{2}$ of $L_{2}$ intersect at a point
$p$ and $\angle_{p}(\ell_{1},\ell_{2})>\epsilon$, then letting $s_{1}$ and
$s_{2}$ be the unite length segments in $\ell_{1}$ and $\ell_{2}$ centered at
$p$,

$\min(L_{1}(s_{2}),L_{2}(s_{1}))<\epsilon.$

######  Proposition 7.6.

For every $\epsilon>0$, if a bounded subset
$K\subset{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
is sufficiently large, then $L_{Y,\rho}$ is $\epsilon$-quasi-parallel to
$L_{X,\rho}$ on $\tau_{X,\rho}$ away from the non-transversal graph $G_{Y}$.

###### Proof.

If $K$ is sufficiently large,
$\angle_{E_{X,\rho}}(H_{X,\rho},V_{Y,\rho}^{\prime})$ is uniformly bounded
from below by a positive number by Theorem 6.4. Then the assertion follows
from Lemma 3.1 and Proposition 7.2. ∎

In this section, we show that the pleated surfaces for $C_{X,\rho}$ and
$C_{Y,\rho}$ are close away from the non-transversal graph. Recall that
$\hat{\beta}_{X,\rho}\colon\tilde{C}_{X,\rho}\to\mathbb{H}^{3}$ denotes the
composition of the collapsing map and the bending map for $C_{X,\rho}$, and
similarly $\hat{\beta}_{Y,\rho}\colon\tilde{C}_{Y,\rho}\to\mathbb{H}^{3}$
denotes the composition of the collapsing map and the bending map for
$C_{Y,\rho}$.

######  Theorem 7.7.

Let $X,Y\in\mathsf{T}\sqcup\mathsf{T}^{\ast}$ with $X\neq Y$. For every
$\epsilon>0$, there is a bounded subset $K_{\epsilon}$ in
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}$
such that, for every
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K_{\epsilon}$, there are a homotopy equivalence map $\phi\colon E_{Y,\rho}\to
E_{Y,\rho}$ and a semi-diffeomorphism
$\psi\colon\mathbf{T}_{X,\rho}\to\mathbf{T}_{Y,\rho}^{\prime}$ (Proposition
6.29), such that

  1. (1)

$d_{E^{1}_{Y,\rho}}(\phi(z),z)<\epsilon$;

  2. (2)

the restriction of $\phi$ to $E_{X,\rho}\setminus N^{1}_{\epsilon}Z_{X,\rho}$
can be transformed to the identify by a homotopy along vertical leaves of
$E_{X,\rho}$;

  3. (3)

$\hat{\beta}_{X,\rho}(z)$ is $\epsilon$-close to
$\hat{\beta}_{Y,\rho}\tilde{\psi}(z)$ in $\mathbb{H}^{3}$ for every point
$z\in\tilde{E}_{X,\rho}$ which are not in the interior of the non-transversal
branches of $\mathbf{T}_{X,\rho}$.

Using Lemma 3.1, one can prove the following.

######  Lemma 7.8.

Let $\epsilon>0$ and let $X\in\mathsf{T}\sqcup\mathsf{T}^{\ast}$. Then there
is a compact subset $K$ of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}$
such that, for every
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\setminus
K$, if $\alpha$ is a monotone staircase closed curve in $E_{X,\rho}$, such
that

  * •

the total vertical length of $\alpha$ is more than $\epsilon$ times the total
horizontal length of $\alpha$, and

  * •

$\alpha$ is disjoint from the $\epsilon$-neighborhood of the singular set in
the normalized metric $E^{1}_{X,\rho}$,

then $\operatorname{Ep}_{X,\rho}\tilde{\alpha}$ is a
$(1+\epsilon,\epsilon)$-quasi-geodesic with respect to the vertical length.

######  Lemma 7.9.

Let $\alpha_{X}$ be staircase curve carried by $t_{X,\rho}$ satisfying the
conditions in Lemma 7.8. Then there is a staircase geodesic closed curve
$\alpha_{Y}$ carried by $\mathbf{T}_{Y,\rho}^{\prime}$ satisfying the
conditions in Lemma 7.8, such that the image of $\alpha$ by the semi-
diffeomorphism $\mathbf{T}_{X,\rho}\to\mathbf{T}_{Y,\rho}^{\prime}$ is
homotopic to $\alpha_{Y}$ in the train-track $\mathbf{T}_{Y,\rho}^{\prime}$.

###### Proof.

The proof is left for the reader. ∎

Let $W_{Y}$ be a realization of $[V_{Y}]_{X}$ on $\mathbf{T}_{X,\rho}$ (§6.3).
Let $x$ be a point of the intersection of the realization $W_{Y}$ and a
horizontal edge of $h_{X}$ of $\mathbf{T}_{X,\rho}$. Let $y$ be a
corresponding point of $V_{Y,\rho}$ (on $E_{Y,\rho}$). Recall that $r$ is the
train-track parameter, so that, in particular, horizontal edges are distance,
at least, $r$ away from the singular set in the normalized Euclidean metric.
Let $v_{x}$ be a vertical segment of length $r/2$ on $E^{1}_{X,\rho}$ such
that $x$ is the middle point of $v_{x}$. Similarly, let $v_{y}$ be the
vertical segment of length $r/2$ on $E^{1}_{Y,\rho}$ such that $y$ is the
middle point of $v_{y}$. We normalize the Epstein surfaces for $C_{X,\rho}$
and $C_{Y,\rho}$ so that they are $\rho$-equivariant for a fixed
representation $\rho\colon\pi_{1}(S)\to{\rm PSL}(2,\mathbb{C})$ (not a
conjugacy class).

######  Proposition 7.10 (Corresponding vertical edges are close in
$\mathbb{H}^{3}$).

For every $\epsilon>0$, there is a compact subset $K$ in $\textstyle\chi$,
such that, for every
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K$, if $v_{X}$ and $v_{Y}$ are vertical segments of $E_{X,\rho}$ and of
$E_{Y,\rho}$, respectively, as above, then there is a (bi-infinite) geodesic
$\ell$ in $\mathbb{H}^{3}$ satisfying the following:

  * •

$\operatorname{Ep}_{X,\rho}v_{X}$ is $\epsilon$-close to a geodesic segment
$\alpha_{X}$ of $\ell$ in $C^{1}$-metric;

  * •

$\operatorname{Ep}_{Y,\rho}v_{Y}$ is $\epsilon$-close to a geodesic segment
$\alpha_{Y}$ of $\ell$ in $C^{1}$-metric;

  * •

if $p_{X}$ and $p_{Y}$ are corresponding endpoints of $\alpha_{X}$ and
$\alpha_{Y}$, then the distance between $\operatorname{Ep}_{X,\rho}p_{X}$ and
$\operatorname{Ep}_{Y,\rho}p_{Y}$ is at most $\epsilon$ times the diameters of
$E_{X,\rho}$ and $E_{Y,\rho}$.

###### Proof of 7.10.

Then, pick a geodesic staircase closed curves $\ell_{X,1},\ell_{X,2}$ on
$E_{X,\rho}$ containing $v_{x}$ such that, for $i=1,2$, by taking appropriate
lift $\tilde{\ell}_{X,1}$ and $\tilde{\ell}_{X,2}$ to $\tilde{E}_{X,\rho}$,

  1. (1)

$\ell_{X,i}$ is carried by $\mathbf{T}_{X,\rho}$;

  2. (2)

$\tilde{\ell}_{X,1}\cap\tilde{\ell}_{X,2}$ is a single staircase curve
connecting singular points of $\tilde{E}_{X,\rho}$, and the projection of
$\tilde{\ell}_{X,1}\cap\tilde{\ell}_{X,2}$ to $E_{X,\rho}$ does not meet a
branch of $\mathbf{T}_{X,\rho}$ more than twice times;

  3. (3)

if a branch $B$ of $\tilde{T}_{X,H_{X}}$ intersects both $\tilde{\ell}_{X,1}$
and $\tilde{\ell}_{X,2}$, then $B$ intersects
$\tilde{\ell}_{X,1}\cap\tilde{\ell}_{X,2}$;

  4. (4)

$\tilde{\ell}_{X,1}$ and $\tilde{\ell}_{X,2}$ intersect, in the normalized
metric of $\tilde{E}^{1}_{X,\rho}$, the $\epsilon$-neighborhood of the
singular set on the near the endpoints of
$\tilde{\ell}_{X,1}\cap\tilde{\ell}_{X,2}.$

Then there is a homotopy of $\ell_{X,1},\ell_{X,2}$ to a staircase vertically-
geodesic curves $\ell^{\prime}_{X,1},\ell^{\prime}_{X,2}$ carried by
$\mathbf{T}_{X,\rho}$ such that the supported in the $2\epsilon$-neighborhood
of the singular set of $E^{1}_{X,\rho}$ so that the loops are disjoint from
the $\epsilon$-neighborhood of the zero set. Then
$\operatorname{Ep}_{X,\rho}\tilde{\ell}_{X,1}^{\prime}$ and
$\operatorname{Ep}_{X,\rho}\tilde{\ell}_{X,2}^{\prime}$ are
$(1+\epsilon,\epsilon)$-quasi-geodesics which are close only near the segment
corresponding to $\tilde{\ell}_{X,1}\cap\tilde{\ell}_{X,2}$.

Pick closed geodesic staircase-curves $\ell_{Y,1},\ell_{Y,2}$ on $E_{Y,\rho}$,
such that

  * •

$\ell_{Y,i}$ contains $v_{y}$;

  * •

the semi-diffeomorphism $\mathbf{T}_{X,\rho}\to\mathbf{T}_{Y,\rho}$ takes
$\ell_{X,i}^{\prime}$ to a curve homotopic to $\ell_{Y,i}$ on
$\mathbf{T}_{Y,\rho}$;

  * •

$\ell_{Y,i}$ is carried by $\mathbf{T}_{Y,\rho}$;

  * •

$\ell_{Y,i}$ is disjoint from $N_{\epsilon}^{1}Z_{X,\rho}$.

Let $\alpha$ be the geodesic such that a bounded neighborhood of $\alpha$
contains the quasi-geodesic
$\operatorname{Ep}_{X,\rho}\tilde{\ell}_{X,i}^{\prime}$. Let
$\tilde{\ell}_{Y,i}$ be a lift of $\ell_{Y,i}$ to $\tilde{E}_{Y,\rho}$
corresponding to $\tilde{\ell}_{X,i}^{\prime}$ (connecting the same pair of
points in the ideal boundary of $\tilde{S}$).

######  Lemma 7.11.

For every $\epsilon>0$, if a compact subset $K$ of $\textstyle\chi$ is
sufficiently large and $\upsilon>0$ is sufficiently small, then, for all
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K$, $\operatorname{Ep}_{Y,\rho}\tilde{\ell}_{Y,i}$ is
$(1+\epsilon,\epsilon)$-quasi-isoemtric with respect to the vertical length
for both $i=1,2$.

Then
$\operatorname{Ep}_{X,\rho}\tilde{\ell}_{X,1}^{\prime}\cup\tilde{\ell}_{X,2}^{\prime}$
and $\operatorname{Ep}_{Y,\rho}\tilde{\ell}_{Y,1}\cup\tilde{\ell}_{Y,2}$ are
both $\epsilon$-close in the Hausdorff metric of $\mathbb{H}^{3}$. Therefore,
corresponding endpoints of
$\operatorname{Ep}_{Y,\rho}\tilde{\ell}_{X,1}^{\prime}\cap\tilde{\ell}_{X,2}^{\prime}$
and
$\operatorname{Ep}_{Y,\rho}\tilde{\ell}_{Y,1}^{\prime}\cap\tilde{\ell}_{Y,2}^{\prime}$
have distance at most $\epsilon$ times the diameters of $E_{X,\rho}$ and
$E_{Y,\rho}$. By (2), the length of
$\tilde{\ell}_{X,1}^{\prime}\cap\tilde{\ell}_{X,2}^{\prime}$ can not be too
long relative to the diameter of $E_{X,\rho}$. Letting $\ell$ be the geodesic
in $\mathbb{H}^{3}$ fellow-traveling with
$\operatorname{Ep}_{X,\rho}\tilde{\ell}_{X,1}^{\prime}$ (or
$\operatorname{Ep}_{X,\rho}\tilde{\ell}_{X,2}^{\prime}$), the vertical segment
$v_{x}$ and $v_{y}$ have the desired property.  7.10

######  Proposition 7.12.

Suppose that a branch $B_{Y}^{\prime}$ of the train track
$\mathbf{T}_{Y,\rho}^{\prime}$ corresponds transversally to a branch $B_{X}$
of $\mathbf{T}_{X,\rho}$.

Then, there is an $\epsilon$-small isotopy of $B_{Y}^{\prime}$ in the
normalized surface $E_{Y,\rho}^{1}$ such that

  * •

in the complement of the $\frac{r}{2}$-neighborhood of the zero set, every
point of $B_{Y}^{\prime}$ moves along the vertical foliation $V_{Y,\rho}$,

  * •

after the homotopy $\hat{\beta}_{X,\rho}|B_{X}$ and
$\hat{\beta}_{Y,\rho}|B_{Y}^{\prime}$ are $\epsilon$-close pointwise by a
diffeomorphism $\psi\colon B_{Y}^{\prime}\to B_{X}$.

###### Proof.

By Proposition 7.10, we fine the map $\phi$ so that (3) holds along the one-
skeleton.

Then, if the branches are transversal, Theorem 4.3, then, since $B_{X}$ is a
transversal branch, if $K$ is sufficiently large, then the restriction of
$L_{X,\rho}$ and $L_{Y,\rho}$ are $\epsilon$-quasi parallel (Proposition 7.6).
Therefore we can extend to the interior of the branch by taking appropriate
$\psi$. ∎

##  8\. Compatible circular train-tracks

In §6, for every $\rho$ in
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
outside a large compact $K$, we constructed have semi-diffeomorphic train-
track structures $\mathbf{T}_{X,\rho}$ and $\mathbf{T}_{Y,\rho}^{\prime}$ of
the flat surfaces $E_{X,\rho}$ and $E_{Y,\rho}$, respectively. In this
section, as $E_{X,\rho}$ and $E_{Y,\rho}$ are the flat structures on
$C_{X,\rho}$ and $C_{Y,\rho}$, using Theorem 7.7, we homotope
$\mathbf{T}_{X,\rho}$ and $\mathbf{T}_{Y,\rho}^{\prime}$ to make them circular
in a compatible manner.

###  8.1. Circular rectangles

A round cylinder is a cylinder on ${\mathbb{C}{\rm P}}^{1}$ bounded by two
disjoin round circles. Given a round cylinder $A$, the boundary components of
$A$ bound unique (totally geodesic) hyperbolic planes in $\mathbb{H}^{3}$, and
there is a unique geodesic $\ell$ orthogonal to both hyperbolic planes.
Moreover $A$ is foliated by round circles which, in $\mathbb{H}^{3}$, bound
hyperbolic planes orthogonal to $\ell$ — we call this foliation the horizontal
foliation. In addition, $A$ is also foliated by circular arcs which are
contained in round circles bounding hyperbolic planes, in $\mathbb{H}^{3}$,
containing $\ell$ — we call this foliation the vertical foliation. Clearly,
the horizontal foliation is orthogonal to the vertical foliations of $A$.

######  Definition 8.1.

Let $\mathcal{R}$ be a ${\mathbb{C}{\rm P}}^{1}$-structure on a marked
rectangle $R$, and let $f\colon R\to{\mathbb{C}{\rm P}}^{1}$ be its developing
map. Then $\mathcal{R}$ is circular if there is a round cylinder $A$ on
${\mathbb{C}{\rm P}}^{1}$ such that

  * •

the image of $f$ is contained in $A$;

  * •

the horizontal edges of $R$ are immersed into different boundary circles of
$A$;

  * •

for each vertical edge $v$ of $R$, its development $f(v)$ is a simple arc on
$A$ transversal to the horizontal foliation.

Given a circular rectangle $\mathcal{R}$, the support of $\mathcal{R}$
consists of the round cylinder $A$ and the arcs on $A$ which are the
developments of the vertical edges of $\mathcal{R}$ in Definition 8.1. We
denote the support by $\operatorname{Supp}\mathcal{R}$. We can pull-back the
horizontal foliation on $A$ to a horizontal foliation on $\mathcal{R}$ by the
developing map.

Given projective structures $\mathcal{R}$ and $\mathcal{Q}$ on a marked
rectangle $R$, we say that $\mathcal{P}$ and $\mathcal{Q}$ are compatible if
$\operatorname{Supp}\mathcal{R}=\operatorname{Supp}\mathcal{Q}$.

Let $\mathcal{R}$ be a circular rectangle, such that the both vertical edges
are supported on the same arc $\alpha$ on a circular cylinder. Then, we say
that $\mathcal{R}$ is semi-compatible with $\alpha$.

####  8.1.1. Grafting a circular rectangle

([Bab10]) Let $\mathcal{R}$ be a circular ${\mathbb{C}{\rm P}}^{1}$-structure
on a marked rectangle $R$. Let $A$ be the round cylinder in ${\mathbb{C}{\rm
P}}^{1}$ which supports $\mathcal{R}$. Pick an . $\alpha$ on $\mathcal{R}$,
such that $\alpha$ connects the horizontals edges and it is transversal to the
horizontal foliation of $\mathcal{R}$. Then $\alpha$ is embedded into $A$ by
$\operatorname{dev}\mathcal{R}$ — we call such an arc $\alpha$ an admissible
arc. By cutting and gluing $A$ and $\mathcal{R}$ along $\alpha$ in an
alternating manner, we obtain a new circular ${\mathbb{C}{\rm
P}}^{1}$-structure on $R$ whose support still is
$\operatorname{Supp}\mathcal{R}$. This operation is the grafting of
$\mathcal{R}$ along $\alpha$, and the resulting structure on $R$ is denoted by
$\operatorname{Gr}_{\alpha}\mathcal{R}$.

One can easily show that $\operatorname{Gr}_{\alpha}\mathcal{R}$ is
independent on the choice of $\alpha$, since an isotopy of $\alpha$ preserving
its initial conditions do not change $\operatorname{Gr}_{\alpha}\mathcal{R}$.

###  8.2. Circular staircase loops

Let $C=(f,\rho)$ be a ${\mathbb{C}{\rm P}}^{1}$-structure on $S$. A
topological staircase curve is a piecewise smooth curve, such that

  * •

its smooth segments are labeled by “horizontal” or “vertical” alternatively,
and

  * •

at every singular point, the horizontal and vertical tangent directions are
linearly independent.

Then a topological staircase curve $s$ on $C$ is circular, if the following
conditions are satisfied: Letting $\tilde{s}$ be a lift of $s$ to $\tilde{S}$,

  * •

every horizontal segment $h$ of $\tilde{s}$ is immersed into a round circle in
${\mathbb{C}{\rm P}}^{1}$ by $f$, and

  * •

for every vertical segment $v$ of $\tilde{s}$, letting $h_{1},h_{2}$ be the
horizontal edges starting from the endpoints of $v$,

    * –

the round circles $c_{1},c_{2}$ containing $f(h_{1})$ and $f(h_{2})$ are
disjoint, and

    * –

$f|v$ is contained in the round cylinder bounded by $c_{1},c_{2}$ and, it is
transverse to the horizontal foliation of the round cylinder.

###  8.3. Circular polygons read once more 3.22.21

Let $P$ be a marked polygon with even number of edges. Then let
$e_{1},e_{2},\dots,e_{2n}$ denote its edges in the cyclic order so that the
edges with odd indices are vertical edges and with even indices horizontal
edges. Suppose that $c_{2},c_{4}\dots c_{2n}$ are round circles in
${\mathbb{C}{\rm P}}^{1}$ such that, for every $i\in\mathbb{Z}/n\mathbb{Z}$,

  * •

$c_{2i}$ and $c_{2(i+1)}$ are disjoint, and

  * •

$c_{2(i-1)}$ and $c_{2(i+1)}$ are contained in the same component of
${\mathbb{C}{\rm P}}^{1}\setminus c_{2i}$.

Let $\mathcal{A}_{i}$ denote the round cylinder bounded by $c_{2i}$ and
$c_{2(i+1)}$. A circular ${\mathbb{C}{\rm P}}^{1}$-structure $\mathcal{P}$ on
$P$ is supported on $\\{c_{2i}\\}_{i=1}^{n}$ if

  * •

$e_{2i}$ is immersed into the round circles of $c_{2i}$ by
$\operatorname{dev}\mathcal{P}$ for every $i=1,\dots n$, and

  * •

$e_{2i+1}$ is immersed into $\mathcal{A}_{i}$ and its image is transversal to
the horizontal foliation of $\mathcal{A}_{i}$ (Figure 17) for every
$i=0,1,\dots,n-1$.

\begin{overpic}[scale={.07}]{Figure/CircularPolygon}
\put(40.0,20.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\vector(1,0){15.0}}
\put(42.0,22.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$dev\mathcal{P}$}}
\put(65.0,33.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}${\mathbb{C}{\rm
P}}^{1}$}} \put(18.0,18.0){$P$} \end{overpic} Figure 17. A developing map of a
projective polygon supported on round circles (in general, the developing map
is not embedding).

Let $\mathcal{P}$ be a circular ${\mathbb{C}{\rm P}}^{1}$-structure on a
polygon $P$ supported on a circle system $\\{c_{2i}\\}_{i}^{n}$. For
$\epsilon>0$, $\mathcal{P}$ is $\epsilon$-circular, if

  * •

for every vertical $v_{i}$ edge is $\epsilon$-parallel to the vertical
foliation $\mathcal{V}$ of the support cylinder $\mathcal{A}_{i}$, and

  * •

the total transversal measure of $v$ given by the vertical foliation
$\mathcal{V}$ is less than $\epsilon$.

(Here, by the “total” transversal measure, we mean that if $v$ intersects a
leaf of $\mathcal{V}$ more than once, and the measure is counted with
multiplicity.)

Let $\mathcal{P}_{1},\mathcal{P}_{2}$ be circular ${\mathbb{C}{\rm
P}}^{1}$-structures on a $2n$-gon $P$. Then $\mathcal{P}_{1}$ and
$\mathcal{P}_{2}$ are compatible if, for each $i=1,\dots,n$,
$\operatorname{dev}\mathcal{P}_{1}$ and $\operatorname{dev}\mathcal{P}_{2}$
take $e_{2i}$ to the same round circle and the arcs $f_{1}(v_{2i-1})$ and
$f_{2}(v_{2i-1})$ are the same.

Let $A$ be a flat cylinder with geodesic boundary; then its universal cover
$\tilde{A}$ is an infinite Euclidean strip. A projective structure $(f,\rho)$
on $A$ is circular, if the developing map $f\colon\tilde{A}\to{\mathbb{C}{\rm
P}}^{1}$ is a covering map onto a round cylinder in ${\mathbb{C}{\rm P}}^{1}$.

Next let $A$ be a spiral cylinder. Then each boundary component $b$ of $A$ is
a monotone staircase loop. Let $\tilde{b}$ be the lift of $b$ to the universal
cover universal cover $\tilde{A}$. Let $\\{e_{i}\\}_{i\in\mathbb{Z}}$ be the
segments of $\tilde{b}$ linearly indexed so that $e_{i}$ with an odd index is
a vertical edge and with an even index is a horizontal edge; clearly
$\tilde{b}=\cup_{i\in\mathbb{Z}}e_{i}$. Then, a ${\mathbb{C}{\rm
P}}^{1}$-structure $(f,\rho)$ on $A$ is circular, if, for each boundary
staircase loop $b$ of $A$ and each $i\in\mathbb{Z}$,

  * •

the horizontal edge $e_{2i}$ is immersed into a round circle $c_{i}$ on
${\mathbb{C}{\rm P}}^{1}$;

  * •

$c_{i-1}$, $c_{i}$ and $c_{i+1}$ are disjoint, and the round annulus bounded
by $c_{i-1}$ and $c_{i+1}$ contains $c_{i}$ in its interior;

  * •

$f$ embeds $v_{i}$ in the round cylinder $\mathcal{A}_{i}$ bounded by $c_{i}$
and $c_{i+1}$, and $f(v_{i})$ is transverse to the circular foliation of
$\mathcal{A}_{i}$.

Two circular ${\mathbb{C}{\rm P}}^{1}$-structures
$\mathcal{A}_{1}=(f_{1},\rho_{1}),\mathcal{A}_{2}=(f_{2},\rho_{2})$ on a
spiral cylinder $A$ are compatible

  * •

$\rho_{1}$ is equal to $\rho_{2}$ up to conjugation by an element of ${\rm
PSL}(2,\mathbb{C})$ (thus we can assume $\rho_{1}=\rho_{2}$);

  * •

for each boundary component $h$ of $\tilde{A}$, $f_{1}$ and $f_{2}$ take $h$
to the same round circle;

  * •

for each vertical edge $v$ of $\tilde{A}$, $f_{1}|v=f_{2}|v$.

More generally, let $\mathcal{F}=(f_{1},\rho_{1})$ and
$\mathcal{F}^{\prime}=(f_{2},\rho_{2})$ be two circular ${\mathbb{C}{\rm
P}}^{1}$-structures on stair-case surfaces $F$ and $F^{\prime}$. First suppose
that there is a diffeomorphism $\phi\colon F\to F^{\prime}$, which takes the
vertices of $F$ bijectively to those of $F^{\prime}$. Then $\mathcal{F}$ is
compatible with $\mathcal{F}^{\prime}$ if

  * •

$\rho_{1}$ is conjugate to $\rho_{2}$ (thus we can assume that
$\rho_{1}=\rho_{2}$);

  * •

for every vertex $p_{1}$ of $\mathcal{F}_{1}$, the development of $p_{1}$
coincides with the development of $\phi(p_{2})$;

  * •

for every a horizontal edge $h$ of $\mathcal{F}_{1}$, letting $h^{\prime}$ be
its corresponding horizontal edge of $\mathcal{F}^{\prime}$, then the
developments of $h$ and $h^{\prime}$ are contained in the same round circle;

  * •

for every vertical edge $v$ of $\mathcal{F}$, letting $v^{\prime}$ be its
corresponding edge $v^{\prime}$ of $\mathcal{F}^{\prime}$, then the
developments of $v^{\prime}$ and $v$ coincides.

Next, in stead of a diffeomorphism, we suppose that there is a semi-
diffeomorphism $\phi\colon F\to F^{\prime}$. Then $\mathcal{F}$ is semi-
compatible with $\mathcal{F}^{\prime}$ if

  * •

$\rho_{1}$ is conjugate to $\rho_{2}$ (thus we can assume that
$\rho_{1}=\rho_{2}$);

  * •

for every vertex $p_{1}$ of $\mathcal{F}_{1}$, the development of $p_{1}$
coincides with the development of $\phi(p_{2})$;

  * •

if a horizontal edge $h$ of $\mathcal{F}$ corresponds to a horizontal edge
$h^{\prime}$ of $\mathcal{F}^{\prime}$, then $h$ and $h^{\prime}$ are
supported on the same round circle on ${\mathbb{C}{\rm P}}^{1}$;

  * •

for every vertical edge $v$ of $\mathcal{F}$, letting $v^{\prime}$ be its
corresponding vertical edge (segment) of $\mathcal{F}^{\prime}$, then the
developments of $v^{\prime}$ and $v$ coincide.

###  8.4. Construction of circular train tracks $\mathcal{T}_{Y,\rho}$

In this section, if $\rho$ is in
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
minus a large compact subset, we construct a circular train-track structure of
$C_{Y,\rho}$ related to the polygonal train-track decomposition
$\mathbf{T}^{\prime}_{Y,\rho}$.

Two train-track structures $T_{1},T_{2}$ on a flat surface $E$ is
$(p,q)$-quasi-isometric for $p>1$ and $q>0$ if there is a continuous
$(p,q)$-quasi-isometry $\phi\colon E\to E$ homotopic to the identity such that
$\phi(T_{1})=T_{2}$ and the restriction of $\phi$ to $T_{1}$ is a homotopy
equivalence between $T_{1}$ and $T_{2}$.

######  Theorem 8.2.

For every $\epsilon>0$, there is a bounded subset $K=K_{\epsilon}$ in
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$,
such that, for every
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K$, there is an $\epsilon$-circular surface train track decomposition
$\mathcal{T}_{Y,\rho}$ of $C_{Y,\rho}$ with the following properties:

  1. (1)

$\mathcal{T}_{Y,\rho}$ is diffeomorprhic to $\mathbf{T}_{Y,\rho}^{\prime}$,
and it is $(1+\epsilon,\epsilon)$-quasi-isometric to both
$\mathbf{T}_{Y,\rho}$ and $\mathbf{T}_{Y,\rho}^{\prime}$ in the normalized
metric $E^{1}_{Y,\rho}$.

  2. (2)

For every vertical edge $v$ of $\mathbf{T}_{Y,\rho}^{\prime}$, then its
corresponding edge of $\mathcal{T}_{Y,\rho}$ is contained in the leaf of the
vertical foliation $V_{Y,\rho}$.

  3. (3)

For a branch $B_{X}$ of $\mathbf{T}_{X,\rho}$, letting $B_{Y}$ be its
corresponding branch of $\mathbf{T}_{Y,\rho}^{\prime}$, then letting
$\mathcal{B}_{Y}$ be the branch of $\mathcal{T}_{Y,\rho}$ corresponding to
$B_{Y}$, then the restriction of $\hat{\beta}_{X,\rho}$ to
$\partial\tilde{B}_{X}$ is $\epsilon$-close to the restriction of
$\hat{\beta}_{Y,\rho}$ to $\partial\tilde{\mathcal{B}}_{Y}$ pointwise;
moreover, if $B_{X}$ is a transversal branch, then
$\hat{\beta}_{X,\rho}|\tilde{B}_{X}$ is $\epsilon$-close to
$\hat{\beta}_{Y,\rho}|\tilde{\mathcal{B}}_{Y}$ pointwise.

We fix a metric on the unit tangent bundle of $\mathbb{H}^{3}$ which is left-
invariant under ${\rm PSL}(2,\mathbb{C})$.

######  Proposition 8.3.

For every $\epsilon>0$, if a bounded subset $K_{\epsilon}$ of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}$
is sufficiently large, then, for every
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\setminus
K_{\epsilon}$ and every horizontal edge $h$ of $\mathbf{T}_{X,\rho}$, the
$\operatorname{Ep}^{\ast}_{X,\rho}$-images of the vertical unite tangent
vectors of along $h$ are $\epsilon$-close.

###### Proof.

The assertion immediately follows from Theorem 3.10 (2). ∎

Recall that we have constructed a system of equivariant circles for horizontal
edges of $\tilde{\mathbf{T}}_{X,\rho}$ in Lemma 7.3. Let $h=[u,w]$ denote the
horizontal edge of $\mathbf{T}_{X,\rho}$ where $u,w$ are the endpoints. We
shall perturb the endpoints of each horizontal edge of $\mathbf{T}_{Y,\rho}$
so that the endpoints map to the corresponding round circle.

######  Proposition 8.4.

For every $\epsilon>0$, there are sufficiently small $\delta>0$ and a (large)
bounded subset $K_{\epsilon}$ of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
satisfying the following: For every
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K_{\epsilon}$, if ${\bf c}=\\{c_{h}\\}$ is a circle system for horizontals
edges $h$ of $\mathbf{T}_{X,\rho}$ given by Lemma 7.3 for $\delta$, then, for
every horizontal edge $h=[u,w]$ of $\tilde{\mathbf{T}}_{Y,\rho}$, there are,
with respect to the normalized metric $E_{Y,\rho}^{1}$, $\epsilon$-small
perturbations $u^{\prime}$ and $w^{\prime}$ of $u$ and $w$ along $V_{Y,\rho}$,
respectively, such that $f_{Y,\rho}(u^{\prime})$ and $f_{Y,\rho}(w^{\prime})$
are contained in the round circle $c_{h}$.

###### Proof.

This follows from Theorem 7.7 and Lemma 7.3 (2). ∎

###### Proof of Theorem 8.2.

By Proposition 8.4, for each horizontal edge $h=[u,w]$ of
$\mathbf{T}_{Y,\rho}^{\prime}$, there is an $\epsilon$-homotopy of $h$ to the
circular segment $h^{\prime}$ the perturbations $u^{\prime},w^{\prime}$ such
that, letting $\tilde{h}$ be a lift of $h$ to $\tilde{E}_{Y,\rho}$, the
corresponding lift $\tilde{h}^{\prime}$ of $h^{\prime}$ is immersed into the
round circle $c_{\tilde{h}}$. For each vertical edge $v$ of
$\mathbf{T}_{Y,\rho}^{\prime}$, at each endpoint of $v$, there is a horizontal
edge of $\mathbf{T}_{Y,\rho}^{\prime}$ starting from the point; then the round
circles corresponding to the horizontal edges bound a round cylinder.

Note that a vertex $u$ of $\mathbf{T}_{Y,\rho}^{\prime}$ often be an endpoint
of different horizontal edges $h_{1}$ and $h_{2}$. Thus, if the perturbations
$u_{1}^{\prime}$ and $u_{2}^{\prime}$ of $u$ are different for $h_{1}$ and
$h_{2}$, then $\mathcal{T}_{Y,\rho}$ has a new short vertical edge connecting
$u_{1}^{\prime}$ and $u_{2}^{\prime}$, and $\mathcal{T}_{Y,\rho}$ is often
non-diffeomorphic to $\mathbf{T}_{Y,\rho}^{\prime}$.

Recall that the $\delta/4$-neighborhood of the singular points of
$E_{Y,\rho}^{1}$ is disjoint from the one-skeleton of $\mathbf{T}_{Y,\rho}$ by
Proposition 6.23. Thus every vertical edge $v$ of
$\mathbf{T}_{Y,\rho}^{\prime}$ is $\epsilon$-circular with respect to the
round cylinder by Corollary 3.8. Thus we have (2). Thus we obtained an
$\epsilon$-circular train-track decomposition $\mathcal{T}_{Y,\rho}$ of
$E_{Y,\rho}$.

As the applies homotopied are $\epsilon$-small, $\mathcal{T}_{Y,\rho}$ are
$\epsilon$-close to $\mathbf{T}_{Y,\rho}^{\prime}$ (1). Thus we may in
addition assume that $\mathcal{T}_{Y,\rho}$ are $\epsilon$-close to
$\mathbf{T}_{Y,\rho}^{\prime}$ by Proposition 6.23. Moreover, Theorem 7.7 give
(3).  8.2

###  8.5. Construction of $\mathcal{T}_{X,\rho}$

Given a train-track structure on a surface, the union of the edges of its
branches is a locally finite graph embedded on the surface. An edge of a
train-track decomposition is an edge of the graph, which contains no vertex in
its interior (whereas an edge of a branch many contain a vertex).

######  Definition 8.5.

Let $C$, $C^{\prime}$ be ${\mathbb{C}{\rm P}}^{1}$-structures on $S$ with the
same holonomy $\rho\colon\pi_{1}(S)\to{\rm PSL}(2,\mathbb{C})$, so that
$\operatorname{dev}C$ and $\operatorname{dev}C^{\prime}$ are
$\rho$-equivariant. A circular train-track decomposition
$\mathcal{T}=\cup_{i}\mathcal{P}_{i}$ of $C$ is semi-compatible with a
circular train-track decomposition
$\mathcal{T}^{\prime}=\cup\mathcal{P}_{j}^{\prime}$ of $C^{\prime}$ if there
is a marking-preserving continuous map $\Theta\colon C\to C^{\prime}$ such
that for each branch $\mathcal{B}$ of $\mathcal{T}$, $\Theta$ takes
$\mathcal{B}$ to a branch of $\mathcal{B}^{\prime}$ of $\mathcal{T}^{\prime}$
and $\mathcal{B}$ and $\mathcal{B}^{\prime}$ are compatible by $\Theta$.

######  Theorem 8.6.

For every $\epsilon>0$, if a bounded subset $K_{\epsilon}$ in
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
is sufficiently large, then, for every
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K_{\epsilon}$, there is an $\epsilon$-circular train track decomposition
$\mathcal{T}_{X,\rho}$ of $C_{X,\rho}$, such that

  1. (1)

$\mathcal{T}_{X,\rho}$ is semi-compatible with $\mathcal{T}_{Y,\rho}$, and

  2. (2)

$\mathcal{T}_{X,\rho}$ additively $2\pi$-hausdorff-close to
$\mathbf{T}_{X,\rho}$ with respect to the (unnormalized) Euclidean metric
$E_{X,\rho}$: More precisely, in the vertical direction,
$\mathcal{T}_{X,\rho}$ is $\epsilon$-close to $\mathbf{T}_{X,\rho}$, and in
the horizontal direction, $2\pi$-close in the Euclidean metric of $E_{X,\rho}$
for all
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\setminus
K_{\epsilon}$.

###### Proof.

First, we transform $\mathbf{T}_{X,\rho}$ by perturbing horizontal edges so
that horizontal edges are circular. Recall that, the branches of
$\mathcal{T}_{Y,\rho}$ are circular with respect to a fixed system ${\bf c}$
of equivariant circles given by Lemma 7.3. Thus the $\beta_{X,\rho}$-images of
vertical tangent vectors along $h$ are $\epsilon$-close to a vector orthogonal
to the hyperbolic plane bounded by $c_{h}$. Therefore, similarly to Theorem
8.2, we can modify the train-track structure $\mathbf{T}_{X,\rho}$ so that
horizontal edges are circular and $\epsilon$-Hausdorff close to the original
train-track structure in the Euclidean metric of $E_{X,\rho}$ (this process
may create new short vertical edges). Thus we obtained an $\epsilon$-circular
train-track $\mathbf{T}_{X,\rho}^{\prime}$ whose horizontal edges map to their
corresponding round circles of ${\bf c}$.

Next, we make the vertical edges compatible with $\mathcal{T}_{Y,\rho}$.
Recall that $\mathbf{T}_{X,H_{X}}$ has no rectangles with short vertical edges
(Lemma 5.6). Therefore, we have the following.

######  Lemma 8.7.

For every $R>0$, if the bounded subset $K$ of $\textstyle\chi$ is sufficiently
large, then, for each vertical edge of $\mathbf{T}_{X,\rho}^{\prime}$, the
horizontal distance to adjacent vertical edges is at least $R$.

Thus by Lemma 8.7, there is enough room to move vertical edges, less than
$2\pi$, so that the train-track is compatible with $\mathcal{T}_{Y,\rho}$
along vertical edges as well.

Since $\mathbf{T}_{X,\rho}$ is semi-diffeomorphic to $\mathbf{T}_{Y,\rho}$
(Proposition 6.29 (3)), $\mathcal{T}_{X,\rho}$ is semi-compatible with
$\mathcal{T}_{Y,\rho}$.  8.6

##  9\. Grafting cocycles and intersection of holonomy varieties

In this section, given certain pairs ${\mathbb{C}{\rm P}}^{1}$-structures on
$S$ with the same holonomy, we shall construct an $\mathbb{Z}$-valued cocycle.

###  9.1. Relative degree of rectangular ${\mathbb{C}{\rm P}}^{1}$-structures

Let $a<b$ be real numbers. Let $f,g\colon[a,b]\to\mathbb{S}^{1}$ be
orientation preserving immersions or constant maps, such that $f(a)=g(a)$ and
$f(b)=g(b)$.

######  Definition-Lemma 9.1.

$\sharp f^{-1}(x)-\sharp g^{-1}(x)$ is independent on
$x\in\mathbb{S}^{1}\setminus\\{f(a),f(b)\\}$, where $\sharp$ denotes the
cardinality. We call this integer the degree of $f$ relative to $g$, or
simply, relative degree, and denote it by $\deg(f,g)$.

Clearly it is not important that $f$ and $g$ are defined on the same interval
as long as corresponding endpoints map to the same point on $\mathbb{S}^{1}$.
Then, the degree is additive in the following sense.

######  Lemma 9.2 (Subdivision of relative degree).

Suppose in addition that $f(c)=g(c)$ for some $c\in(a,b)$. Then

$\deg(f,g)=\deg(f|_{[a,c]},g|_{[a,c]})+\deg(f|_{[a,c]},g|_{[a,c]}).$

The proofs of the lemmas above are elementary. Let $R,Q$ be circular
projective structures on a marked rectangle, and suppose that $R$ and $Q$ are
compatible. Then, by their developing maps, corresponding horizontal edges of
$R$ and $Q$ are immersed into the same round circle on ${\mathbb{C}{\rm
P}}^{1}$ and the corresponding vertices map to the same point.

Then, the degree of $R$ relative to $Q$ is the degree of a horizontal edge of
$R$ relative to its corresponding horizontal edge of $Q$ — we similarly denote
the degree by $\deg(R,Q)\in\mathbb{Z}$. Although $R$ has two horizontal edges,
this degree is well-defined:

######  Lemma 9.3 (c.f. Lemma 6.2 in [Bab15]).

The degree $\deg(R,Q)$ is independent on the choice of the horizontal edge.

###### Proof.

Let $A$ be the round cylinder on ${{\mathbb{C}{\rm P}}}^{1}$ supporting both
$R$ and $Q$. Then, the horizontal foliation $\mathcal{F}_{A}$ of $A$ by round
circles $c$ yields a foliation $\mathcal{F}_{R}$ and $\mathcal{F}_{Q}$ on $R$
and $Q$, respectively. Then, for each leaf $c$ of $\mathcal{F}_{A}$, the
corresponding leaves $\ell_{R}$ and $\ell_{Q}$ of $\mathcal{F}_{R}$ and
$\mathcal{F}_{Q}$, respectively, are immersed into $c$, and the endpoints of
$\ell_{R}$ and $\ell_{Q}$ on the corresponding vertical edges of $R$ and $Q$
map to the same point on $c$. The degree of $\ell_{R}$ relative to $\ell_{Q}$
is integer, and it changes continuously in the leaves $c$ of
$\mathcal{F}_{A}$. Thus the assertion follows immediately. ∎

######  Lemma 9.4 (cf. Lemma 6.2 in [Bab15]).

Let $R,Q$ be ${\mathbb{C}{\rm P}}^{1}$-structures on a rectangle with
$\operatorname{Supp}R=\operatorname{Supp}Q$. Then

  * •

if $\deg(R,Q)>0$, then $R$ is obtained by grafting $Q$ along an admissible arc
$\deg(R,Q)$ times;

  * •

if $\deg(R,Q)<0$, then $Q$ is obtained by grafting $R$ along an admissible arc
$-\deg(R,Q)$ times;

  * •

if $\deg(R,Q)=0$, then $R$ is isomorphic to $Q$ (as ${\mathbb{C}{\rm
P}}^{1}$-structures).

By Lemma 9.4, the “difference” of ${\mathbb{C}{\rm P}}^{1}$-rectangles $R$ and
$Q$ is represented by an arc $\alpha$ with weight $\deg(R,Q)$ such that
$\alpha$ sits on the base rectangle connecting the horizontal edges.

######  Lemma 9.5.

Let $R$ and $Q$ be circular projective structures on a marked rectangle with
$\operatorname{Supp}R=\operatorname{Supp}Q$. Let $A$ be the round cylinder on
${\mathbb{C}{\rm P}}^{1}$ supporting $R$ and $Q$. Suppose that there are
admissible arcs $\alpha_{R}$ on $R$ and $\alpha_{Q}$ on $Q$ which develop onto
the same arc on $A$ transversal to the horizontal foliation, so that the arcs
decompose $R$ and $Q$ into two circular rectangles $R_{1},R_{2}$ and
$Q_{1},Q_{2}$, respectively and
$\operatorname{Supp}R_{1}=\operatorname{Supp}Q_{1}$ and
$\operatorname{Supp}R_{2}=\operatorname{Supp}Q_{2}$. Then

$\deg(R,Q)=\deg(R_{1},Q_{1})+\deg(R_{2},Q_{2}).$

###### Proof.

This follows from Lemma 9.2. ∎

###  9.2. Train-track graphs for planar polygons

Let $P$ be a $L^{\infty}$-convex staircase polygon in $\mathbb{E}^{2}$, which
contains no singular points. We can decompose $P$ into finitely many
rectangles $P_{1},P_{2},\dots,P_{n}$ by cutting $P$ along $n-1$ horizontal
arcs each connecting a vertex and a point on a vertical edge. Let
$\mathcal{P},\mathcal{Q}$ be compatible circular projective structures on $P$
such that the round circles supporting horizontal edges are all disjoint. The
decomposition $P$ into $P_{1},P_{2},\dots P_{n}$ gives decompositions of
$\mathcal{P}$ into $\mathcal{P}_{1},\mathcal{P}_{2},\dots,\mathcal{P}_{n}$ and
$\mathcal{Q}$ into $\mathcal{Q}_{1},\mathcal{Q}_{2},\dots,\mathcal{Q}_{n}$
such that
$\operatorname{Supp}\mathcal{P}_{i}=\operatorname{Supp}\mathcal{Q}_{i}$ for
$i=1,2,\dots,n$. As in §9.1, for each $i$, we obtain an arc $\alpha_{i}$
connecting horizontal edges of $P_{i}$ with weight
$\deg(\mathcal{P}_{i},\mathcal{Q}_{i})$. Then, by splitting and combining
$\alpha_{1},\alpha_{2},\dots,\alpha_{n}$ appropriately, we obtain a
$\mathbb{Z}$-valued train-track graph $\Gamma(\mathcal{P},\mathcal{Q})$ on $P$
transversal to the decomposition (Figure 18).

\begin{overpic}[scale={.15}]{Figure/TraintrackGraphOnPolygon}
\put(11.0,30.0){\small$P_{1}$} \put(50.0,45.0){\small$P_{2}$}
\put(80.0,10.0){\small$P_{3}$}
\put(11.0,53.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\small$1$}}
\put(40.0,62.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\small$2$}}
\put(55.0,31.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\small$-1$}}
\put(38.0,12.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\small$-2$}}
\put(80.0,30.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\small$-3$}}
\end{overpic} Figure 18. An example of weighted train track on a convex
polygon in $\mathbb{E}^{2}$.

###  9.3. Train-track graphs for cylinders

Let $A_{X}$ be a cylindrical branch of $\mathbf{T}_{X,\rho}$, and let $A_{Y}$
be the corresponding cylindrical branch of $\mathbf{T}_{Y,\rho}$.

Pick a monotone staircase curve $\alpha$ on $A_{X}$, such that

  1. (1)

$\alpha$ connects different boundary components of $A_{X}$, and its endpoints
are on horizontal edges (Figure 19),

  2. (2)

the restriction of $[V_{Y,\rho}]_{X}$ to $A_{X}$ has a leaf disjoint from
$A_{X}$.

\begin{overpic}[scale={.15}]{Figure/RectangleToSpiralCyliner}
\put(22.0,25.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$\alpha$}}
\put(71.0,16.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\small$[V_{Y}]_{X}$}}
\end{overpic} Figure 19.

Then $[V_{Y,\rho}]_{X}$ is essentially carried by $A_{X}$. Then one can easily
show that the choice of $\alpha$ is unique though an isotopy preserving the
properties.

######  Lemma 9.6.

Suppose there are two staircase curves $\alpha_{1},\alpha_{2}$ on $A_{X}$
satisfying Conditions (1) and (2). Then, $\alpha_{1}$ and $\alpha_{2}$ are
isotopic through staircase curves $\alpha_{t}$ satisfying Conditions (1) and
(2).

In Proposition 6.7, pick a realization of $[V_{Y}]_{X}$ on the decomposition
$(A_{X},\alpha_{X})$ by a homotopy of $[V_{Y}]_{X}$ sweeping out triangles.
This induces an $\epsilon$-almost staircase curve $\alpha_{Y}$. Similarly to
Lemma 7.3, pick a system of round circles ${\bf c}=\\{c_{h}\\}$ corresponding
to horizontal edges $h$ of $\alpha_{X}$ so that the
$\operatorname{Ep}_{X,\rho}$-images of vertical tangent vectors along $h$ are
$\epsilon$-close to a single vector orthogonal to the hyperbolic plane bounded
by $c_{h}$.

Then, as in §8.4 we can accordingly isotope the curve $\alpha_{Y}$ so that the
horizontal edges are supported on their corresponding circle of ${\bf c}$ and
vertical edges remains vertical— let $\alpha_{Y}^{\bf c}$ denote the curve
after this isotopy. Then $\mathcal{A}_{Y}\setminus\alpha_{Y}^{\bf c}$ is a
circular projective structure on a staircase polygon in $\mathbb{E}^{2}$.

Then, (similarly to Theorem 8.6), we can isotope $\alpha_{X}$ to an
$\epsilon$-almost circular staircase curve $\alpha^{\bf c}_{X}$ so that

  * •

$\alpha_{X}$ is $2\pi$-hausdorff close to $\alpha_{X}^{\bf c}$;

  * •

the horizontal edge $h$ of $\alpha_{X}^{\bf c}$ is supported on $c_{h}$;

  * •

$\mathcal{A}_{X}\setminus\alpha_{X}^{\alpha}$ is a $\epsilon$-almost circular
staircase polygon compatible with $\mathcal{A}_{Y}\setminus\alpha_{Y}^{\bf
c}$.

As in §9.2, $\mathcal{A}_{X}\setminus\alpha_{X}^{\bf c}$ and
$\mathcal{A}_{Y}\setminus\alpha_{Y}^{\bf c}$ yield a $\mathbb{Z}$-valued
weighted train track $\Gamma_{A\setminus\alpha_{X}}$ on the polygon
$A\setminus\alpha_{X}$ such that $\Gamma_{A_{X}\setminus\alpha_{X}}$ is
transversal to horizontal foliation. Up to an homotopy preserving endpoints on
the horizontal edges, the endpoints of $\Gamma_{A_{X}\setminus\alpha_{X}}$
match up along $\alpha_{X}$ as $\mathbb{Z}$-weighted arcs. Thus we obtain a
weighted train-track graph $\Gamma_{A_{X}}$ on $A_{X}$.

Consider the subset of the boundary of the circular cylinder $\mathcal{A}_{X}$
which is the union of vertical boundary edges and the vertices of
$\mathcal{T}_{X,\rho}$ contained in $\partial\mathcal{A}_{X}$. Let
$\mathsf{A}$ be the homotopy class of arcs in $\mathcal{A}_{X}$ connecting
different points in this subset. Then
$[\Gamma_{A_{X}}]\colon\mathsf{A}\to\mathbb{Z}$ be the map which takes an arc
to its total signed intersection number with $\Gamma_{A_{X}}$.

Then Lemma 9.6 give a uniqueness of $[\Gamma_{A_{X}}]$:

######  Proposition 9.7.

$[\Gamma_{A_{X}}]\colon\mathsf{A}\to\mathbb{Z}$ is independent on the choice
of the staircase curve $\alpha$ and the realization of $[V_{Y}]_{X}$ on
$(\mathcal{A}_{X},\alpha_{X})$.

###  9.4. Weighted train tracks and ${\mathbb{C}{\rm P}}^{1}$-structures with
the same holonomy

In this section we suppose that Riemann surfaces $X,Y$ have the same
orientation. Let $\mathsf{C}$ be the set of the homotopy classes of closed
curves on $S$ (which are not necessarily simple). Given a weighted train-track
graph immersed on $S$, it gives an cocycle taking $\gamma\in\mathsf{C}$ to its
weighted intersection number with the graph.

######  Theorem 9.8.

For all distinct $X,Y\in\mathsf{T}$, there is a bounded subset $K$ in
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$,
such that

  1. (1)

for each
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K$, the semi-compatible train track decompositions $\mathcal{T}_{X,\rho}$ of
$C_{X,\rho}$ and $\mathcal{T}_{Y,\rho}$ of $C_{Y,\rho}$ in Theorem 8.6 yield a
$\mathbb{Z}$-weighted train track graph $\Gamma_{\rho}$ carried by
$\mathcal{T}_{X,\rho}$ (immersed in $S$);

  2. (2)

the intersection cocycle $[\Gamma_{\rho}]\colon\mathsf{C}\to\mathbb{Z}$ is
independent on the choices for the construction of $\mathcal{T}_{X,\rho}$ and
$\mathcal{T}_{Y,\rho}$;

  3. (3)

$[\Gamma_{\rho}]\colon\mathsf{C}\to\mathbb{Z}$ is continuous in
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K$.

Since $[\Gamma_{\rho}]$ takes values in $\mathbb{Z}$, the continuity
immediately implies the following.

######  Corollary 9.9.

For sufficiently large boundary subset $K$ of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$,
$[\Gamma_{\rho}]$ is well-defined and constant on each connected component of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K$.

We first construct a weighted train-track in (1). Let $h_{X,1}\dots h_{X,n}$
be the horizontal edges of branches of $\mathcal{T}_{X,\rho}$.

Since $\mathcal{T}_{X,\rho}$ and $\mathcal{T}_{Y,\rho}$ are semi-compatible
(Proposition 8.6), for each $i=1,2,\dots,n$, letting $h_{Y,i}$ be its
corresponding edge of a branch of $\mathcal{T}_{Y,\rho}$ or a vertex of
$\mathcal{T}_{Y,\rho}$. Then, $h_{X,i}$ and $h_{Y,i}$ develop into the same
round circle on ${\mathbb{C}{\rm P}}^{1}$ and corresponding endpoints map to
the same point by the semi-compatibility. Thus we have the degree of $h_{X,i}$
relative to $h_{Y,i}$ taking value in $\mathbb{Z}$ (Definition 9.1) for each
horizontal edge:

$\gamma_{\rho}\colon\\{h_{X,1},\dots,h_{X,n}\\}\to\mathbb{Z}.$

We will construct a $\mathbb{Z}$-weighted train track $\Gamma_{\rho}$ carried
by $\mathcal{T}_{X,\rho}$ so that the intersection number with $h_{X,i}$ is
$\gamma_{\rho}(h_{X,i})$. The train-track graph $\Gamma_{\rho}$ will be
constructed on each branch of $\mathcal{T}_{X,\rho}$:

  * •

For each rectangular branch $R$ of $\mathcal{T}_{X,\rho}$, we will construct a
$\mathbb{Z}$-train track graph embedded in $R$ (Proposition 9.10).

  * •

For each cylinder $A$ of $\mathcal{T}_{X,\rho}$, we have obtained a
$\mathbb{Z}$-weighted train track graph embedded in $A$ (§9.3).

  * •

For each transversal branch, we will construct a $\mathbb{Z}$-weighted train
track graph embedded in the branch (Proposition 9.12).

  * •

For each non-transversal branch of $\mathcal{T}_{X,\rho}$, we will construct a
$\mathbb{Z}$-weighted train track immersed in the branch (Lemma 9.15).

####  9.4.1. Train tracks for rectangular branches

Given an ordered pair of compatible ${\mathbb{C}{\rm P}}^{1}$-structures on a
rectangle, Lemma 9.4 gives a $\mathbb{Z}$-weighted arc connecting the
horizontal edges of the rectangle. Since the horizontal edges of a rectangular
branch of $\mathcal{T}_{X,\rho}$ may contain a vertex, we transform the
weighted arc to a weighted train-track graph so that it matches with
$\gamma_{\rho}$.

######  Proposition 9.10.

For every $\epsilon>0$, there is a bounded subset $K$ of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$,
such that, for every
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K$, for each rectangular branch $\mathcal{R}_{X}$ of $\mathcal{T}_{X,\rho}$,
there is a $\mathbb{Z}$-weighted train track graph
$\Gamma_{\rho,\mathcal{R}_{X}}$ embedded in $\mathcal{R}_{X}$ satisfying the
following:

  * •

$\Gamma_{\mathcal{R}_{X}}$ induces $\gamma_{\rho}$;

  * •

$\Gamma_{\mathcal{R}_{X}}$ is transversal to the horizontal foliation on
${\mathcal{R}_{X}}$;

  * •

each horizontal edge of $\mathcal{T}_{X,\rho}$ in $\partial\mathcal{R}_{X}$
contains at most one endpoint of $\Gamma_{\rho,\mathcal{R}_{X}}$;

  * •

As cocycles, $\Gamma_{R}$ is $(1+\epsilon,\epsilon)$-quasi-isometric to
$(V_{X,\rho}|R_{X})-(V_{Y,\rho}|R_{Y})$, where $R_{X}$ is a branch of
$\mathbf{T}_{X,\rho}$ corresponding to $\mathcal{R}_{X}$ and $R_{Y}$ is the
branch of $\mathbf{T}_{Y,\rho}^{\prime}$ corresponding to $R_{X}$ .

\begin{overpic}[scale={.05}]{Figure/TraintrackOnRectangle}
\put(33.0,12.0){$n_{4}$} \put(33.0,33.0){$n_{5}$} \put(62.0,35.0){$n_{3}$}
\put(62.0,21.0){$n_{2}$} \put(62.0,8.0){$n_{1}$} \put(45.0,21.0){$n$}
\end{overpic} Figure 20. A $\mathbb{Z}$-weight train-track graph for a
rectangle. Here $n=n_{1}+n_{2}+n_{3}=n_{4}+n_{5}.$ The black dots are vertices
of $\mathcal{T}_{X,\rho}$.

###### Proof.

Since $\mathcal{R}_{X}$ and $\mathcal{R}_{Y}$ share their support, let
$n=\deg(\mathcal{R}_{X},\mathcal{R}_{Y})$ as seem in Lemma 9.4. Let $h_{X}$
and $h_{Y}$ be corresponding horizontal edges of $\mathcal{R}_{X}$ and
$\mathcal{R}_{Y}$. Then $h_{X}=h_{X,1}\cup\dots\cup h_{X,m}$ be the
decomposition of $h_{X}$ into horizontal edges of $\mathcal{T}_{X}$, and let
$h_{Y}=h_{Y,1}\cup\dots\cup h_{Y,m}$ be the corresponding decomposition into
horizontal edges and vertices of $\mathcal{T}_{X}$ compatible with the semi-
diffeomorphism $\mathcal{T}_{X,\rho}\to\mathcal{T}_{Y,\rho}$. Let
$n_{i}\in\mathbb{Z}$ be $\deg(h_{X,i},h_{Y,i})$. Then, by Lemma 9.2,
$n=n_{1}+\dots+n_{m}$. Then it is easy to construct a desired
$\mathbb{Z}$-weighted train track realizing such decomposition for both pairs
of corresponding horizontal edges (see Figure 20).

The last assertion follows from Lemma 4.1 and Theorem 7.7. ∎

####  9.4.2. Train tracks for cylinders

For each cylindrical branch $\mathcal{A}_{X}$ of $\mathcal{T}_{X,\rho}$, in
§9.3, we have constructed a train-track graph $\Gamma_{\rho,\mathcal{A}_{X}}$
on $\mathcal{A}_{X}$, representing the difference between $\mathcal{A}_{X}$
and its corresponding cylindrical branch $\mathcal{A}_{Y}$ of
$\mathcal{T}_{Y,\rho}$.

######  Proposition 9.11.

For every $\epsilon>0$, if a bounded subset $K$ of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
is sufficiently large, then, for each cylindrical branch $\mathcal{A}_{X}$ of
$\mathcal{T}_{X,\rho}$, the induced cocycle
$[\Gamma_{\mathcal{A}_{X}}]\colon\mathsf{A}\to\mathbb{Z}$ times $2\pi$ is
$(1+\epsilon,\epsilon)$-quasi-isometric to
$V_{Y,\rho}|A_{X}-V_{X,\rho}|A_{Y}$, where $A_{X}$ and $A_{Y}$ are the
corresponding cylindrical branches of $\mathbf{T}_{X,\rho}$ and
$\mathbf{T}_{Y,\rho}^{\prime}$.

###### Proof.

Recall that $\Gamma_{\mathcal{A}_{X}}$ is obtained from $\mathbb{Z}$-weighted
train-track graphs on the rectangles. There is a uniform upper bound for the
number of the rectangles used to define $[\Gamma_{\mathcal{A}_{X}}]$ (which
depends only on $S$), since the decomposition was along horizontal arcs
starting from singular points. Then, on each rectangle, the weighted graph is
$(1+\epsilon,\epsilon)$-quasi-isometric to the difference of $V_{X,\rho}$ and
$V_{Y,\rho}$ by Proposition 9.10. Thus $[\Gamma_{\mathcal{A}_{X}}]$ is also
$(1+\epsilon,\epsilon)$-quasi-isometric to the difference of $V_{X,\rho}$ and
$V_{Y,\rho}$ if $K$ is sufficiently large. ∎

####  9.4.3. Train-track graphs for transversal polygonal branches

Recall that all transversal branches are polygonal or cylindrical (Lemma
6.18), i.e. their Euler characteristics are non-negative.

######  Proposition 9.12.

For every $\epsilon>0$, there is a bounded subset $K$ in
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
such that, for each transversal polygonal branch $\mathcal{P}_{X}$ of
$\mathcal{T}_{X,\rho}$, there is a $\mathbb{Z}$-weighted train track
$\Gamma_{\rho,\mathcal{P}_{X}}$ embedded in $\mathcal{P}_{X}$, letting $P_{X}$
and $P_{Y}$ be the branch of $\mathbf{T}_{X,\rho}$ and $\mathbf{T}_{Y,\rho}$
corresponding to $\mathcal{P}_{X}$, respectively, such that

  1. (1)

each horizontal edge $h$ of $\mathcal{P}_{X}$ contains, at most, one endpoint
of $\Gamma_{\rho,P}$;

  2. (2)

$[\Gamma_{\mathcal{P}_{X}}]$ agrees with $\gamma_{\rho}$ on the horizontal
edges of $\mathcal{T}_{X,\rho}$ contained in $\partial\mathcal{P}_{X}$;

  3. (3)

$\Gamma_{P}$ is transversal to horizontal foliation $H_{X,\rho}$ on $P_{X}$;

  4. (4)

$2\pi[\Gamma_{P}]$ is $(1+\epsilon,\epsilon)$-quasiisometric to
$(V_{X,\rho}|P_{X}-V_{Y,\rho}|P_{Y})$.

For every $\epsilon>0$, if $K$ is sufficiently large, then, by Theorem 7.2
(2), let $\hat{\mathcal{Q}}_{X}$ be an ideal circular polygon whose truncation
${\mathcal{Q}}_{X}$ is $\epsilon$-close to $\mathcal{P}_{X}$ in $C_{X,\rho}$.
Similarly, $\hat{\mathcal{Q}}_{Y}$ be an ideal circular polygon whose
truncation $\mathcal{Q}_{Y}$ is $\epsilon$-close to $\mathcal{P}_{Y}$. Since
$\operatorname{Supp}\mathcal{P}_{X}=\operatorname{Supp}\mathcal{P}_{Y}$ as
circular polygons, we may in addition assume that
$\operatorname{Supp}\mathcal{Q}_{X}=\operatorname{Supp}\mathcal{Q}_{Y}$ as
truncated idea polygons.

Let $\overline{\mathcal{Q}}_{X}$ be the canonical polynomial ${\mathbb{C}{\rm
P}}^{1}$-structure on $\mathbb{C}$ which contains $\hat{\mathcal{Q}}_{X}$
(§7.0.1). Let $\hat{\mathcal{L}}_{X}$ be the restriction of the Thurston
lamination of $\overline{\mathcal{Q}}_{X}$ to $\hat{\mathcal{Q}}_{X}$.
Similarly, let $\overline{\mathcal{Q}}_{Y}$ be the canonical polynomial
${\mathbb{C}{\rm P}}^{1}$-structure on $\mathbb{C}$ which contains
$\mathcal{Q}_{Y}$. Let $\hat{\mathcal{L}}_{Y}$ be the restriction of the
Thurston lamination of $\overline{\mathcal{Q}}_{Y}$ to
$\hat{\mathcal{Q}}_{Y}$. As
$\operatorname{Supp}\mathcal{Q}_{X}=\operatorname{Supp}\mathcal{Q}_{Y}$, thus
$\overline{\mathcal{Q}}_{X}$ and $\overline{\mathcal{Q}}_{Y}$ share their
ideal vertices. Then $\mathcal{Q}_{X}$ and $\mathcal{Q}_{Y}$ are
$\epsilon$-close to $\mathcal{P}_{X}$ and $\mathcal{P}_{Y}$, respectively.
Thus, since $\gamma_{\rho}$ takes values in $\mathbb{Z}$,
$\hat{\mathcal{L}}_{X}-\hat{\mathcal{L}}_{Y}$ satisfies (2).

Theorem 7.2 (1) implies taht $\hat{\mathcal{L}}_{X}$ is
$(1+\epsilon,\epsilon)$-quasi-isometric to $V_{X}|P_{X}$ and
$\hat{\mathcal{L}}_{Y}$ is $(1+\epsilon,\epsilon)$-quasi-isometric to
$V_{Y}|P_{Y}$, and therefore $\hat{\mathcal{L}}_{X}-\hat{\mathcal{L}}_{Y}$
satisfies (4).

(1) is easy to be realized by homotopy combining the edges of the train-track
graph ending on the same horizontal edge. We show that there is a
$\mathbb{Z}$-weighted train track $\Gamma$ which is $\epsilon$-close to
$\hat{\mathcal{L}}_{X}-\hat{\mathcal{L}}_{Y}$, satisfying (3).

Let $\Gamma_{X}^{P}$ be a weighted train track graph on $P$ which represents
$\hat{\mathcal{L}}_{X}$. Let $\Gamma_{Y}^{P}$ be the weighted train-track
graph which represents $\hat{\mathcal{L}}_{Y}$. By Theorem 7.7, the pleated
surface of $\mathcal{P}_{X}$ is $\epsilon$-close to the plated surface of
$\mathcal{P}_{Y}$, because of the quasi-parallelism in Proposition 7.6. Let
$\check{\Gamma}_{X}^{P}$ be the subgraph of $|\Gamma_{X}^{P}|$ obtained by
eliminating the edges of weights less than a sufficiently small $\epsilon$.
Similarly, let $\check{\Gamma}_{Y}^{P}$ be the subgraph of $\Gamma_{Y}^{P}$
obtained by eliminating the edges of weight less than $\epsilon$.

Then there is a minimal train-track graph $\Gamma^{P}$ containing both
$\check{\Gamma}_{Y}^{P}$ and $\check{\Gamma}_{X}^{P}$ and satisfying (1).
Since the pleated surfaces are sufficiently close, by approximating the
weights of $\Gamma_{X}^{P}-\Gamma_{Y}^{P}$ by integers, we obtain a desired
$\mathbb{Z}$-weighted train track graph supported on $\Gamma^{P}$.

Since $\mathcal{P}_{X}$ is a transversal branch, $\Gamma_{X}^{P}$ and
$\Gamma_{Y}^{P}$ are both transversal to $H_{X,\rho}|P$, thus $\Gamma^{P}$ is
transversal to $H_{X,\rho}|P$ (3).  9.12

####  9.4.4. Train-track graphs for non-transversal branches

Let $P_{X}$ and $P_{Y}$ be the corresponding branches of $\mathbf{T}_{X,\rho}$
and $\mathbf{T}_{Y,\rho}$, respectively, which are non-transversal. Let
$\mathcal{P}_{X}$ and $\mathcal{P}_{Y}$ be the branches of
$\mathcal{T}_{X,\rho}$ and $\mathcal{T}_{Y,\rho}$ corresponding to $P_{X}$ and
$P_{Y}$, respectively. Then, by Theorem 8.2,
$\hat{\beta}_{X,\rho}|\partial\mathcal{P}_{X}$ is $\epsilon$-close to
$\hat{\beta}_{Y,\rho}|\partial\mathcal{P}_{Y}$ in the $C^{0}$-metric and
$C^{1}$-close along the vertical edges. Let $\sigma_{X}$ be a pleated surface
with crown-shaped boundary whose truncation approximates
$\hat{\beta}_{X}|\mathcal{P}_{X}$, and let $\sigma_{Y}$ be the pleated
surfaced boundary with crown-shaped boundary whose truncation approximates
$\hat{\beta}_{Y}|\mathcal{P}_{Y}$, such that $\sigma_{X}$ and $\sigma_{Y}$
share their boundary (in $\mathbb{H}^{3}$). In particular, the base hyperbolic
surfaces for $\sigma_{X}$ and $\sigma_{Y}$ are diffeomorphic preserving
marking and spikes.

Let $\nu_{X}$ and $\nu_{Y}$ be the bending measured laminations for
$\sigma_{X}$ and $\sigma_{Y}$, respectively; then $\nu_{X}$ and $\nu_{Y}$
contain only finite many leaves whose leaves connect ideal points.

In Thurston coordinates, the developing map and the pleated surface of a
${\mathbb{C}{\rm P}}^{1}$-structure are related by the nearest point
projection to the supporting planes of the pleated surface ([KP94, Bab20]).
For small $\upsilon>0$, let $F_{X}$, $F_{Y}$ be the surfaces in
$\mathbb{H}^{3}$ which are at distance $\upsilon$ from $\sigma_{X}$ and
$\sigma_{Y}$ in the direction of the nearest point projections of $P_{X}$ and
$P_{Y}$ ([EM87, Chapter II.2] ).

Thus, if necessarily, refining $\nu_{X}$ to $\nu_{Y}$ to ideal triangulations
of the hyperbolic surfaces appropriately, pick an (irreducible) sequence of
flips $w_{i}$ which connects $\nu_{X}$ to $\nu_{Y}$. Clearly the sequence
$w_{i}$ corresponds to a sequence of triangulations.

######  Lemma 9.13.

If a bounded subset
$K\subset{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}$
is sufficiently large, for every
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K$ and all non-transversal branches $P_{X}$ and $P_{Y}$, there is a uniform
upper bound on the length of the flip sequence which depend only on
$X,Y\in\mathsf{T}$, or appropriate refinements into triangulations.

###### Proof.

This follows from the length bound in Lemma 6.2. ∎

A triangulation in the sequence given by $w_{i}$ is realizable, if there is an
equivariant pleated surface homotopic to $\sigma_{X}$ (and $\sigma_{Y}$)
relative to the boundary such that the pleating locus agrees with the
triangulation. In general, a triangulation in the sequence is not be
realizable when the endpoints of edges develop to the same point on
${\mathbb{C}{\rm P}}^{1}$. However a generic perturbation makes the
triangulation realizable:

######  Lemma 9.14.

For almost every perturbation of the holonomy of $P_{X}$ and holonomy
equivariant perturbation of the (ideal) vertices of $\tilde{\sigma}_{X}$ (and
$\tilde{\sigma}_{Y}$) in ${\mathbb{C}{\rm P}}^{1}$, all triangulations in the
flip sequence $w_{i}$ are realizable. Moreover, the set of realizable
perturbation is connected.

###### Proof.

If $\sigma_{X}$ is an ideal polygon, the holonomy is trivial. Then, since
there are only finitely many vertices and ${\mathbb{C}{\rm P}}^{1}$ has real
dimension two, almost every perturbation are realizable.

If $\sigma_{X}$ is not a polygon, an edge of a triangulation forms a loop if
the endpoints are at the same spike of $\tau_{X}$. For each loop $\ell$ of
$\tau_{X}$, the condition that the holonomy of $\ell$ is the identity is a
complex codimension at least one in the character variety (and also in the
representation variety). Since the flip sequence is finite, for almost all
perturbations of the holonomy, if an edge of a triangulation in the sequence
forms a loop, then its holonomy is non-trivial. Clearly such a perturbation is
connected. Then, for every such perturbation of the holonomy, it is easy to
see that, for almost all equivariant perturbations of the ideal points, the
triangulations in the sequence is realizable. ∎

For every perturbation of the holonomy and the ideal vertices given by Lemma
9.14, the flip sequence $w_{i}$ gives the sequence of pleated surfaces
$\sigma_{X}=\sigma_{1}\xrightarrow{w_{1}}\sigma_{2}\xrightarrow{w_{2}}\dots\xrightarrow{w_{n-1}}\sigma_{n}=\sigma_{Y}$
in $\mathbb{H}^{3}$ connecting $\sigma_{Y}$ to $\sigma_{X}$, such that
$\sigma_{i}^{\prime}s$ share their boundary geodesics and ideal vertices.

. \begin{overpic}[scale={.1}]{Figure/HomotopyBetweenPleatedSurfaces}
\end{overpic}

Figure 21. The blue lines are the pleating lamination interpolating the
pleated surfaces which differ by a flip.

For each flip $w_{i}$, the pairs of triangles of the adjacent pleated surfaces
$\sigma_{i}$ and $\sigma_{i+1}$ bounds a tetrahedron in $\mathbb{H}^{3}$. To
be precise, i9f the four vertices are contained in plane, we have the
tetrahedron is collapsed into a quadrangle, but it does not affect the
following argument. The edges exchanged by the flip correspond to the opposite
edges of the tetrahedron. Then pick a geodesic segment connecting those
opposite edges. Then there is a path $\sigma_{t}\leavevmode\nobreak\ (i\leq
t\leq i+1)$ of pleated surfaces with a single cone point of angle more than
$2\pi$ such that

  * •

$\sigma_{t}$ connects $\sigma_{i}$ to $\sigma_{i+1}$;

  * •

$\sigma_{t}$ supported on the ideal quadrangle supporting the flip $w_{i}$;

  * •

by the homotopy the quadrangle sweeps out the tetrahedron;

  * •

the cone point on the geodesic segment (see Figure 21).

In this manner, this sequence of pleated surfaces $\sigma_{i}$ continuously
extends to a homotopy of the pleated surfaces with at most one singular point
of cone angle greater than $2\pi$. This interpolation also connects a bending
cocycle on $\sigma_{i}$ to a bending cocycle on $\sigma_{i+1}$ continuous,
although the induced cocycle on $\sigma_{i+1}$ may corresponds to a measured
lamination only immersed on the surface, since the edges of the triangulations
transversally intersect. Thus, $\nu_{X}$ induces a sequence of the bending
(immersed) measured laminations $\nu_{i}$ of $\sigma_{i}$ supported a union of
the pleating loci of $\sigma_{1},\dots,\sigma_{i}$.

For each $i$, the difference $\nu_{i+1}-\nu_{i}$ of the transverse cocycles is
supported on the geodesics corresponding to the edges of the tetrahedron, so
that, on the surface, the edges form an ideal rectangle with both diagonals.
Let $\mu_{i}$ be the difference cocycle $\nu_{i+1}-\nu_{i}$. From each vertex
of the ideal rectangle there are three leaves of $\nu_{i+1}-\nu_{i}$ starting,
and the sum of their weights is zero by Euclidean geometry (Figure 22).

\begin{overpic}[scale={.2}]{Figure/FlipAndBendingAngles}
\put(45.0,6.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$\sigma_{i}$}}
\put(35.0,16.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$\sigma_{i+1}$}}
\end{overpic} Figure 22. The link of a vertex of the ideal rectangle —– the
sum of the indicated singed angles is zero.

Note that $P_{X}$ can be identified with $\sigma_{X}$ by collapsing each
horizontal edge of $P_{X}$ to a point. Hence, for every $i$, if $\alpha$ is a
closed curve on $P_{X}$ or an arc connecting vertical edges of $P_{X}$, then
$\mu_{i}(\alpha)=0$. By regarding $\nu_{j}$ is a geodesic lamination on
$\sigma_{X}$, their union $\cup_{j=1}^{i}\nu_{j}$ is a graph on $\sigma_{X}$
whose vertices are the transversal intersection points of the triangulations.
A small regular neighborhood $N$ of $\cup_{j=1}^{i}\nu_{j}$ is decomposed into
a small regular neighborhood $N_{0}$ of the vertices and small regular
neighborhood of the edges minus $N_{0}$ in $N\setminus N_{0}$.

Since, after the whitehead moves, the pleated surface $\sigma_{X}$ is
transformed to a pleated surface $\sigma_{Y}$. Thus $\nu_{n}\setminus\nu_{Y}$
gives an integral transversal cocycle.

By the construction of the regular homotopy, we have the following.

######  Proposition 9.15 (Train-tracks for irregular branches).

For every non-transversally compatible branches $\mathcal{P}_{X}$ of
$\mathcal{T}_{X,\rho}$ and $\mathcal{P}_{Y}$ of $\mathcal{T}_{Y,\rho}$, there
is a $\mathbb{Z}$-weighted immersed train-track graph $\Gamma_{P_{X}}$
representing the transversal cocycle supported on $\cup_{j=1}^{i}\nu_{j}$.
Moreover, the train-track cocycle is independent on the choice of the flip
sequence $w_{i}$.

###### Proof.

Given two flip sequences $(w_{i}),(w_{j}^{\prime})$ connecting the
triangulations of $\sigma_{X}$ to $\sigma_{Y}$, there are connected by a
sequence of sequences $(v_{i}^{k})$ of triangulations connecting $\sigma_{X}$
to $\sigma_{Y}$, such that $(v_{i}^{k})$ and $(v_{i}^{k+1})$ differ by either
an involutivity, a commutativity or a pentagon relation ([Pen12, Chapter 5,
Corollary 1.2]). Clearly, the difference by an involutivity and a
commutativity do not affect the resulting cocycle. Also by the pentagon
relation, the pleated surface does not change including the bending measure
since each flip preserves the total bending along the vertices. Therefore
$(v_{i}^{k})$ and $(v_{i}^{k+1})$ give the same train-track cocycle. ∎

Therefore we obtain $\mathsf{A}\to\mathbb{Z}$. By continuity and the
connectedness in Lemma 9.14. we have the following.

######  Corollary 9.16.

$[\Gamma_{P}]$ is independent on the choice of the perturbation in Lemma 9.14.

There are only finitely many combinatorial types of the train-tracks
$\mathbf{T}_{X,\rho}$. We consider a branch $B_{X}$ of $\mathbf{T}_{X,\rho}$
and $B_{X}^{\prime}$ of $\mathbf{T}_{X,\rho^{\prime}}$ are equivalent if they
are diffeomorphic and isotopic on $S$. Then there are only finitely many
isotopy classes of branches of $\mathbf{T}_{X,\rho}$ for all
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\setminus
K$. Let $\alpha$ be an arc $\alpha$ on a branch $B_{X}$ such that each
endpoint of $\alpha$ is at either on a vertical edge or a vertex of
$\mathbf{T}_{X,\rho}$.

######  Proposition 9.17.

Let $X,Y\in\mathsf{T}$. For every $\epsilon>0$, there a compact subset $K$ in
$\textstyle\chi$ with the following property: For every pair $(B_{X},\alpha)$
of an isotopy class of a branch $B_{X}$ and an arc $\alpha$ as above, there is
a constant $k_{\alpha}>0$ such that, if $B_{X}$ is a (non-transversal) branch
of $\mathbf{T}_{X,\rho}$ for some
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K$, then, $[\Gamma_{P_{X}}](\alpha)$ is $(1+\epsilon,k_{\alpha})$-quasi-
isometric to $(V_{X}|P_{X}-V_{Y}|P_{Y})(\alpha)$.

###### Proof.

Since the length of the flip sequence is bounded from above, the difference
between $\nu_{X}$ and $\nu_{n}$ is uniformly bounded in the space of cocycles
on $S$. Then the assertion follows. ∎

####  9.4.5. Independency of the transverse cocycle

For the train-track decompositions $\mathcal{T}_{X,\rho}$ and
$\mathcal{T}_{Y,\rho}$, we have constructed a weighted train-track graph
$\Gamma_{\rho}$ (Theorem 9.8 (1)). Next we show its cocycle is independent on
the train-track decompositions $\mathcal{T}_{X,\rho}$ and
$\mathcal{T}_{Y,\rho}$ ( Theorem 9.8(2)).

Recall that the train-track decompositions $\mathcal{T}_{X,\rho}$ and
$\mathcal{T}_{Y,\rho}$ are determined by

  1. (1)

the holonomy equivariant circle system ${\bf c}=\\{c_{h}\\}$ indexed by
horizontal edges $h$ of $\tilde{\mathbf{T}}_{X,\rho}$ (given by Lemma 7.3),

  2. (2)

the realization $W_{Y}$ of $[V_{Y,\rho}]_{X,\rho}$ on $\mathbf{T}_{X,\rho}$
(Lemma 6.7), and

  3. (3)

the choice of vertical edges of $\mathcal{T}_{X,\rho}$ (Theorem 8.6 (2)).

######  Proposition 9.18.

The cocycle $[\Gamma_{\rho}]\colon\mathsf{C}\to\mathbb{Z}$ constructed above
is independent of the construction for $\mathcal{T}_{X,\rho}$ and
$\mathcal{T}_{Y,\rho}$, i.e. (1), (2), (3).

###### Proof.

(1) By Proposition 7.4, given two appropriate circle systems $\\{c_{h}\\}$ and
$\\{c^{\prime}_{h}\\}$, there is a equivariant isotopy of circles systems
$\\{c_{t,h}\\}$ connecting $\\{c_{h}\\}$ to $\\{c^{\prime}_{h}\\}$. Then
accordingly, we obtain a continuous family of cocycles
$[\Gamma_{t,\rho}]\colon\mathsf{C}\to\mathbb{Z}$. As it takes discrete values,
$[\Gamma_{t,\rho}]$ must remain the same.

By the different choices for (2) and (3), the $\mathbb{Z}$-weights on $\Gamma$
shift across bigon regions corresponding vertical edges of $T_{Y,\rho}$ by
integer values. These weight shifts clearly preserve the cocycle
$[\Gamma_{\rho}]\colon\mathsf{C}\to\mathbb{Z}$. ∎

####  9.4.6. Continuity of the transversal cocycle

Next we prove the continuity of $[\Gamma_{\rho}]$ in $\rho$ claimed in (3).

######  Definition 9.19 (Convergence and semi-convergence of train tracks).

Suppose that $C_{i}\in\mathsf{P}$ converges to $C\in\mathsf{P}$. In addition,
suppose, for each $i$, there are a train-track structure $\mathcal{T}_{i}$ of
$C_{i}$ and a train-track structure $\mathcal{T}$ for $C$. Then,

  * •

$\mathcal{T}_{i}$ converges to $\mathcal{T}$ if

    * –

for every branch $\mathcal{P}$ of $\mathcal{T}$, there is a sequence of
branches $\mathcal{P}_{i}$ of $\mathcal{T}_{i}$ converging to $\mathcal{P}$,
and

    * –

for every sequence of branches $\mathcal{P}_{i}$ of $\mathcal{T}_{i}$, up to a
subsequence, converges to either a branch of $\mathcal{T}$ or an edge of a
branch of $\mathcal{T}$.

  * •

$\mathcal{T}_{i}$ semi-converges to $\mathcal{T}$ if there is a subdivision of
$\mathcal{T}$ into another circular train-track structure
$\mathcal{T}^{\prime}$ so that $\mathcal{T}_{i}$ converges to
$\mathcal{T}^{\prime}$.

######  Lemma 9.20.

Let $\rho_{i}$ be a sequence in
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
converging to
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K$, where $K$ is a sufficiently large compact (as in Theorem 8.6). Pick an
equivariant circle system ${\bf c}_{i}$ for $\mathbf{T}_{X,\rho_{i}}$ by Lemma
7.3 which converges to a circle system ${\bf c}$ for $\mathbf{T}_{X,\rho}$.
Then, up to a subsequence,

  * •

the circular train track $\mathcal{T}_{X,\rho_{i}}$ of $C_{X,\rho_{i}}$ semi-
converges to a circular train track $\mathcal{T}_{X,\rho}$ of $C_{X,\rho}$;

  * •

the circular train track $\mathcal{T}_{Y,\rho_{i}}$ of $C_{Y,\rho_{i}}$ semi-
converges to a circular train track $\mathcal{T}_{Y,\rho}$ of $C_{Y,\rho}$;

  * •

$\mathcal{T}_{X,\rho}$ is semi-compatible with $\mathcal{T}_{Y,\rho}$.

###### Proof.

By Theorem 6.12 (1) By Lemma 6.19, $\mathbf{T}_{X,\rho_{i}}$ semi-converges to
$\mathbf{T}_{X,\rho}$. Therefore $\mathbf{T}_{X,\rho_{i}}$ converges to a
subdivision $\mathbf{T}_{X,\rho}^{\prime}$ of $\mathbf{T}_{X,\rho}$ as
$i\to\infty$. Then, if $\mathbf{T}_{X,\rho}\neq\mathbf{T}_{X,\rho}^{\prime}$,
then $\mathbf{T}_{X,\rho}$ is obtained from $\mathbf{T}_{X,\rho}^{\prime}$ by
gluing non-rectangular branches with rectangular branches of small width or
replacing long rectangles into spiral cylinders (as in §5.1 and §6.7).

Recall that the realization of $[V_{Y,\rho_{i}}]_{X,\rho_{i}}$ in the train
track $\mathbf{T}_{X,\rho_{i}}$ is unique up to shifting across vertical edges
of non-rectangular branches (Proposition 6.12 (2)). Therefore, up to a
subsequence, the realization of $[V_{Y,\rho_{i}}]_{X,\rho_{i}}$ on
$\mathbf{T}_{X,\rho_{i}}$ converges to a realization of $[V_{Y,\rho}]_{X}$ on
$\mathbf{T}_{X,\rho}^{\prime}$, Since $\mathbf{T}_{X,\rho}^{\prime}$ is a
subdivision of $\mathbf{T}_{X,\rho}$, the limit can be regarded as also a
realization on $\mathbf{T}_{X,\rho}$. Since the realization determines the
train-track structure of $E_{Y,\rho}$, up to a subsequence,
$\mathbf{T}_{Y,\rho_{i}}$ converges to a bounded train-track
$\mathbf{T}_{Y,\rho}^{\prime}$. Then $\mathbf{T}_{Y,\rho}$ is transformed to
$\mathbf{T}_{Y,\rho}^{\prime}$ by possibly sliding vertical edges and
subdividing spiral cylinders to wide rectangles. Moreover, by Theorem 8.2 (1),
$\mathcal{T}_{Y,\rho}$ is $(1+\epsilon,\epsilon)$-quasiisometric to
$\mathbf{T}_{Y,\rho}$. Therefore, up to a subsequence,
$\mathcal{T}_{Y,\rho_{i}}$ converges to a circular train-track structure
$\mathcal{T}_{Y}^{\prime}$ of $C_{Y,\rho}$. If $\mathcal{T}_{Y,\rho}$ is
different from $\mathcal{T}_{Y,\rho}^{\prime}$, then $\mathcal{T}_{Y,\rho}$
can be transformed to $\mathcal{T}_{Y,\rho}^{\prime}$ by sliding vertical
edges and subdividing spiral cylinders into rectangles.

By Theorem 8.6, $\mathcal{T}_{X,\rho_{i}}$ is additively $2\pi$-close to
$\mathbf{T}_{X,\rho_{i}}$ in the Hausdorff metric of $E^{1}_{X,\rho}$.
Therefore, up to a subsequence $\mathcal{T}_{X,\rho_{i}}$ converges to a
circular train track decomposition $\mathcal{T}_{X,\rho}^{\prime}$ semi-
diffeomorphic to $\mathcal{T}_{Y,\rho}$. Moreover $\mathcal{T}_{X,\rho}$ can
be transformed to $\mathcal{T}_{X,\rho}^{\prime}$ possibly by subdividing and
sliding by $2\pi$ or $4\pi$.

We have already shown that $\mathcal{T}_{X,\rho}$ is semi-diffeomorphic to
$\mathcal{T}_{Y,\rho}$ (Theorem 8.6).  9.20

Finally we have the continuity (3).

######  Corollary 9.21.

$[\Gamma_{\rho_{i}}]\colon\mathsf{C}\to\mathbb{Z}$ converges to
$[\Gamma_{\rho}]\colon\mathsf{C}\to\mathbb{Z}$ as $i\to\infty$.

###### Proof.

Since $\mathcal{T}_{X,\rho_{i}}$ semi-converges to $\mathcal{T}_{X,\rho}$, up
to taking a subsequence, $\mathcal{T}_{X,i}$ converges to a subdivision
$\mathcal{T}^{\prime}_{X,\rho}$ of $\mathcal{T}_{X,\rho}$. Accordingly, there
is a subdivision $\mathcal{T}^{\prime}_{Y,\rho}$ of $\mathcal{T}_{Y,\rho}$ ,
such that, to up a subsequence, $\mathcal{T}_{Y,i}$ converges to
$\mathcal{T}^{\prime}_{Y,\rho}$ and that $\mathcal{T}^{\prime}_{X,\rho}$ is
semi-diffeomorphic to $\mathcal{T}^{\prime}_{Y,\rho}$.

Let $\Gamma_{\rho_{i}}$ be the $\mathbb{Z}$-weighted train-track given by
$\mathcal{T}_{X,\rho_{i}}$ and $\mathcal{T}_{Y,\rho_{i}}$. Let
$\Gamma_{\rho}^{\prime}$ be the $\mathbb{Z}$-weighted train track given
$\mathcal{T}^{\prime}_{X,\rho}$ and $\mathcal{T}^{\prime}_{Y,\rho}$. Then, by
the convergence of the train tracks, $\Gamma_{\rho_{i}}$ converges to
$\Gamma_{\rho}^{\prime}$ as $i\to\infty$. Since
$\mathcal{T}^{\prime}_{X,\rho}$ and $\mathcal{T}^{\prime}_{Y,\rho}$ are
obtained by sliding and subdividing $\mathcal{T}_{X,\rho}$ and
$\mathcal{T}_{Y,\rho}$ respectively, thus $\Gamma_{\rho}^{\prime}$ and
$\Gamma_{\rho}$ yield the same cocycle $\mathsf{C}\to\mathbb{Z}$. ∎

###  9.5. Approximation of the grafting cocycle $[\Gamma_{\rho}]$ by vertical
foliations

Suppose that $X,Y$ be distinct marked Riemann surfaces homeomorphic to $S$
such that $X$ and $Y$ have the same orientation. For a branch $B_{X}$ of
$\mathbf{T}_{X,\rho}$, let $\mathsf{A}(B_{X})$ be the homotopy class of arcs
$\alpha$ on $\mathcal{R}_{X}$ such that every endpoint of $\alpha$ is either
on a vertical edge or a vertex of $\mathbf{T}_{X,\rho}.$

######  Theorem 9.22.

Let $c_{1},\dots,c_{n}$ be essential closed curves on $S$. Then, for every
$\epsilon>0$, there is a bounded subset $K_{\epsilon}$ of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
such that, for every
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K$, the intersection cocycle $[\Gamma_{\rho}]$ is $(1+\epsilon,q)$-quasi-
isometric to $V_{X,\rho}-V_{Y,\rho}$ along $c_{1},\dots,c_{n}$. That is,

(2)
$(1-\epsilon)\Gamma_{\rho}(c_{i})-q<V_{X,\rho}(c_{i})-V_{Y,\rho}(c_{i})<(1+\epsilon)\Gamma_{\rho}(c_{i})+q$

for all $i=1,2,\dots,n$.

###### Proof.

Let $H\in\mathsf{PML}$. Recall that $E^{1}_{X,H}$ is the flat surface
conformal to $X$, such that the horizontal foliation is $H$ and ${\rm
Area}E^{1}(X,H)=1$. Let $\mathbf{T}_{X,H}$ be the bounded train-track
structure of $E_{X,H}$.

Then, every closed curve $c$ can be isotoped to a closed curve $c^{\prime}$ so
that, for each branch of $B$ of $\mathbf{T}_{X,\rho}$, $c^{\prime}|B$ is an
arc connecting different vertices. Let $c^{\prime}_{1},\dots,c^{\prime}_{m}$
be the decomposition into sub-arcs. By the finiteness of possible train-
tracks, the number $m$ of the subarcs is bounded from above for all $\rho$.
Then, if $B$ is regular, then $\Gamma_{\rho}|c^{\prime}_{j}$ is
$(1+\epsilon,\epsilon)$-quasi-isometric to
$(V_{X,\rho}|B-V_{Y,\rho}|B)c_{k}^{\prime}$ if $B$ is a transversal branch by
Proposition 9.12(4), Proposition 9.11, Proposition 9.10, and
$(1+\epsilon,q)$-quasi-isometric if non-transversal by Proposition 9.17. As
the number of subarcs is bounded, the assertion follows. ∎

##  10\. The discreteness

###  10.1. The discreteness of the intersection of holonomy varities

######  Theorem 10.1.

Suppose that $X,Y$ are marked Riemann surfaces structures on $S$ with the same
orientation. Then every (connected) component of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
is bounded.

###### Proof.

Let $K$ be a component of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$.
Suppose, to the contrary, that $V$ is unbounded in $\textstyle\chi$. Then,
there is a path $\rho_{t}$ in
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
which leaves every compact subset. Then, by Theorem 4.9, by taking a diverging
sequence $t_{1}<t_{2}<\dots$, there are
$k_{i},k_{i}^{\prime}\in\mathbb{R}_{>0}$ such that
$\frac{k_{i}}{k_{i}^{\prime}}\to 1$ as $i\to\infty$ and

$\lim_{i\to\infty}k_{i}H_{X,\rho_{t_{i}}}=\lim_{i\to\infty}k^{\prime}_{i}H_{Y,\rho_{t_{i}}}\in\mathsf{ML}.$

By taking a subsequence, we may in addition assume that their vertical
foliations $[V_{X,\rho_{t_{i}}}]$ and $[V_{Y,\rho_{t_{i}}}]$ converge in
$\mathsf{PML}$. Thus let $[V_{X,\infty}]$ and $[V_{Y,\infty}]$ be their
respective limits in $\mathsf{PML}$. Since $X\neq Y$, $V_{X,\infty}$ and
$V_{Y,\infty}$ can not be asymptotically the same, in comparison to their
horizontal foliations. Then $V_{X,\rho_{t_{i}}}-V_{Y,\rho_{t_{i}}}$ “diverges
to $\infty$”. That is, there is a closed curve $\alpha$ on $S$, such that

$|V_{X,\rho_{t_{i}}}(\alpha)-V_{Y,\rho_{t_{i}}}(\alpha)|\to\infty$

as $i\to\infty$.

Let $[\Gamma_{\rho_{t}}]\colon\mathsf{C}\to\mathbb{Z}$ be the function given
by Theorem 9.8. As $[\Gamma_{\rho_{t}}]$ is continuous (Theorem 9.8 (3)),
$[\Gamma_{\rho_{t}}]\colon\mathsf{C}\to\mathbb{Z}$ is a constant function (for
$t\gg 0$). On the other hand, by Theorem 9.22, there is $q>0$ such that
$(V_{X,\rho_{t_{i}}}-V_{Y,\rho_{t_{i}}})(\alpha)$ is
$(1+\epsilon_{i},q)$-quasi-isometrically close to
$[\Gamma_{\rho_{t_{i}}}](\alpha)$, and $\epsilon_{i}\to 0$ as $i\to\infty$. We
thus obtain a contradiction.  10.1

Since
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}$
and
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}$
are complex analytic, thus their intersection is also a complex analytic set
(Theorem 5.4 in [FG02]). Therefore, since every bounded connect analytic set
is a singleton (Proposition 2.6), Theorem 10.1 implies the following.

######  Corollary 10.2.

${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
is a discrete set.

We will, moreover, show that this intersection is non-empty in §12 (Corollary
12.3).

###  10.2. A weak simultaneous uniformization theorem

In this section, using Corollary 10.2, we prove a weak version of a
simultaneous uniformization theorem for general representations. Let
$\rho\colon\pi_{1}(S)\to{\rm PSL}(2,\mathbb{C})$ be any non-elementary
representation which lifts to ${\rm SL}(2,\mathbb{C})$. Let $C,D$ be
${\mathbb{C}{\rm P}}^{1}$-structures on $S^{+}$ with L the holonomy $\rho$.
Then, if a neighborhood $U_{\rho}$ of $\rho$ in $\textstyle\chi$ is
sufficiently small, then there are (unique) neighborhoods $V_{C}$ and $W_{D}$
of $C$ and $D$ in $\mathsf{P}$, respectively, which are biholomorphic to
$U_{\rho}$ by
$\operatorname{Hol}\colon\mathsf{P}\to{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}$.
Then, for every $\eta\in U_{\rho}$, there are unique ${\mathbb{C}{\rm
P}}^{1}$-structures $C_{\eta}$ in $V_{C}$ and $D_{\eta}$ in $W_{D}$ with
holonomy $\eta$. Let $\Phi_{\rho,U}=\Phi\colon
U_{\rho}\to\mathsf{T}\times\mathsf{T}$ be the map which takes $\eta\in
U_{\rho}$ to the pair of the marked Riemann surface structures of $C_{\eta}$
and $D_{\eta}$.

######  Theorem 10.3.

$\Phi_{\rho,U}$ is a finite-to-one open mapping.

###### Proof.

By Corollary 10.2, the fiber of $\Phi$ is discrete. In addition, $\Phi$ is
holomorphic and $\dim U_{\rho}=2\dim\mathsf{T}$. Therefore, by Theorem 2.7,
$\Phi$ is an open map. ∎

##  11\. Opposite orientations

In this section, when the orientations of the Riemann surfaces are opposite,
we show the discreteness of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
corresponding to Theorem 11.1) and the local uniformization theorem
corresponding to Theorem 10.3.

######  Theorem 11.1.

Fix $X\in\mathsf{T}$ and $Y\in\mathsf{T}^{\ast}$ . Then,
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
is a non-empty discrete set.

Since the proof is similar to the case when the orientations coincide, we
simply outline the proof, but explaining the parts which must be modified. We
leave the details to the reader.

Recall that we have constructed compatible train track decomposition
regardless of the orientation (§8.4, §8.5). In summary, we have the following
(in the case of opposite orientataions):

######  Proposition 11.2.

Fix $X\in\mathsf{T}$ and $Y\in\mathsf{T}^{\ast}$. For every $\epsilon>0$,
there is a bounded subset $K_{\epsilon}$ in
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$,
such that, if
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K)_{\epsilon}$, then there are circular polygonal train-track decompositions
$\mathcal{T}_{X,\rho}$ of $C_{X,\rho}$ and $\mathcal{T}_{Y,\rho}$ of
$C_{Y,\rho}$, such that

  * •

$\mathcal{T}_{X,\rho}$ and $\mathcal{T}_{Y,\rho}$ are semi-diffeomorphic, and

  * •

$\mathcal{T}_{X,\rho}$ and $\mathcal{T}_{Y,\rho}$ are
$(1+\epsilon,\epsilon)$-quasi-isometric to the train track decompositions
$\mathbf{T}_{X,\rho}$ of the flat surface $E_{X,\rho}^{1}$ and
$\mathbf{T}_{Y,\rho}$ of the flat surface $E_{Y,\rho}^{1}$, respectively, with
respect to the normalized metrics.

In the case when the orientation of $X$ and $Y$ are the same, in Theorem 9.8,
we constructed a $\mathbb{Z}$-weighted train-track graph representing the
“difference” of projective structures on $X$ and $Y$ with the same holonomy.
As the orientations of $X$ and $Y$ are different, we shall construct a
$\mathbb{Z}$-weighted train track graph representing, in this case, the “sum”
of the ${\mathbb{C}{\rm P}}^{1}$-structures on $X$ and $Y$ with the same
holonomy.

Let $\mathcal{T}_{X,\rho}$ and $\mathcal{T}_{Y,\rho}$ be circular train track
decompositions of $C_{X,\rho}$ and $C_{Y,\rho}$ given by Proposition. 11.2 Let
$\\{h_{X,1},h_{X_{2}},\dots,h_{X_{n}}\\}$ be the horizontal edges of
$\mathcal{T}_{X,\rho}$. Similarly to §9.4, we first define the
$\mathbb{Z}$-valued function on the set of horizontal edges. For each
$i=1,\dots,n$, let $c_{i}$ be the round circle on ${\mathbb{C}{\rm P}}^{1}$
supporting the development of $h_{X,i}$. First suppose that $h_{X,i}$
corresponds to an edges $h_{Y,i}$ of $\mathcal{T}_{Y,\rho}$ by the collapsing
map $\mathcal{T}_{X,\rho}\to\mathcal{T}_{Y,\rho}$. Since
$\mathcal{T}_{X,\rho}$ and $\mathcal{T}_{Y,\rho}$ are compatible the
corresponding endpoints of $h_{X,i}$ and $h_{Y,i}$ map to the same point on
$c_{i}$ by their developing maps. Thus, by identifying the endpoints, we
obtain a covering map from a circle $h_{X,i}\cup h_{Y,i}$ onto $c_{i}$. Then,
define $\gamma_{X,i}(h_{X,i})\in\mathbb{Z}_{>0}$ to be the covering degree.

Next, suppose that $h_{X,i}$ corresponds to a vertex of $\mathcal{T}_{Y,i}$.
Then the endpoints of $h_{X,i}$ develop to the same point on $c_{i}$. The
circle obtained by identifying endpoints of $h_{X,i}$ covers $c_{i}$. Thus,
let $\gamma_{X,i}(h_{X,i})\in\mathbb{Z}_{>0}$ be the covering degree.

Similarly to §9.4, we shall construct a $\mathbb{Z}$-weighted train-track
graph $\Gamma_{\rho}$ immersed in $\mathcal{T}_{X,\rho}$. On each branch
$\mathcal{P}_{X}$ of $\mathcal{T}_{X,\rho}$, we construct a
$\mathbb{Z}$-weighted train-track graph $\Gamma_{\mathcal{P}_{X}}$ on
$\mathcal{P}_{X}$ such that, for every $\epsilon>0$, there is a compact subset
$K$ of $\textstyle\chi$ satisfying the following:

  * •

Endpoints of $\Gamma_{\mathcal{P}_{X}}$ are on horizontal edges of
$\mathcal{P}_{X}$.

  * •

They agree with $\gamma_{X,\rho}$ along the horizontal edges.

  * •

If $\mathcal{P}_{X}$ is a transversal branch, then, for
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K$, then $\Gamma_{\mathcal{P}_{X}}(\alpha)$ is $(1+\epsilon,\epsilon)$-quasi-
isometric to $(V_{X,\rho}|P_{X})(\alpha)+(V_{Y,\rho}|P_{Y})(\alpha)$ for all
$\alpha\in\mathcal{A}(\mathcal{P}_{X})$.

  * •

For every smooth isotopy class of a staircase surface $B$ on $S$ and every arc
$\alpha$ on $B$ connecting points on horizontal edges or vertices, there is a
positive constant $q(B,\alpha)$ such that, if $\mathcal{T}_{X,\rho}$ contains
a non-transversal branch $\mathcal{B}_{X}$ smoothly isotopic to $B$ on $S$,
then $\Gamma_{\mathcal{B}_{X}}(\alpha)$ is $(1+\epsilon,q(B,\alpha))$-quasi-
isometric to $V_{X,\rho}|B_{X}(\alpha)+V_{Y,\rho}|B_{Y}(\alpha)$.

######  Theorem 11.3.

Let $X\in\mathsf{T}$ and $Y\in\mathsf{T}^{\ast}$. For every $\epsilon>0$,
there is a bounded subset
$K_{\epsilon}\subset{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
such that, for every
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K_{\epsilon}$, there is a $\mathbb{Z}$-weighted graph $\Gamma_{\rho}$ carried
in $\mathcal{T}_{X,\rho}$ such that

  1. (1)

the induced cocycle $[\Gamma_{\rho}]\colon\mathsf{C}\to\mathbb{Z}$ changes
continuously in
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K$,

  2. (2)

for every loop $\alpha$ on $S$, there is $q_{\alpha}>0$, such that
$\Gamma_{\rho}(\alpha)$ is $(1+\epsilon,q_{\alpha})$-quasi-isometric to
$V_{X,\rho}(\alpha)+V_{Y,\rho}(\alpha)$ for all
$\rho\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\setminus
K$.

###### Proof.

The proof is similar to Theorem 9.8 (3) and Theorem 9.22.∎

Then, Theorem 11.3 implies, similarly to Theorem 10.1, the following:

######  Theorem 11.4.

Each connected component of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
is bounded.

##  12\. The completeness

In this section, we prove the completeness in A. Let $Q$ be a connected
component of the Bers’ space $\mathsf{B}$; then $Q$ is a complex submanifold
of
$(\mathsf{P}\sqcup\mathsf{P}^{\ast})\times(\mathsf{P}\sqcup\mathsf{P}^{\ast})$,
and $\dim_{\mathbb{C}}Q=6g-6$. We call that
$\psi\colon\mathsf{P}\sqcup\mathsf{P}^{\ast}\to\mathsf{T}\sqcup\mathsf{T}^{\ast}$
is the uniformization map and $\Psi\colon
Q\to(\mathsf{T}\sqcup\mathsf{T}^{\ast})^{2}$ is defined by
$\Psi(C,D)=(\psi(C),\psi(D))$. Then, by Theorem 10.3, $\Psi$ is an open
holomorphic map. In this section, we prove the completeness of $\Psi$.

######  Lemma 12.1.

The open map $\Psi\colon Q\to(\mathsf{T}\sqcup\mathsf{T}^{\ast})^{2}$ has a
local path lifting property. That is, for every $z\in Q$, there is a
neighborhood $W$ of $\Psi(z)$ such that if path $\alpha_{t},\,0\leq t\leq 1$
in $W$ satisfies $\zeta(z)=\alpha_{0}$, then there is a lift
$\tilde{\alpha}_{t}$ of $\alpha_{t}$ to $Q$ with $\tilde{\alpha}_{0}=z$.

###### Proof.

Since $\Psi$ is an open map and $\dim
Q=\dim(\mathsf{T}\sqcup\mathsf{T}^{\ast})^{2}$, $\Psi$ is a locally branched
covering map. Then, for every $z\in Q$, there is an open neighborhood $V$ of
$z$ in $Q$ and a finite group $G_{z}$ biholomorphically acting $V$, such that
$\Psi$ is $G_{z}$-invariant, and $\Psi$ induces the biholomorphic map
$V/G_{z}\to\Psi(V)$.

For $g\in G_{z}\setminus\\{id\\}$, let $F_{g}\subset W$ be the (pointwise)
fixed point set of $g$. Clearly $F_{g}$ is a proper analytic subset, and thus
$F\coloneqq\cup_{g\in G_{z}\setminus\\{id\\}}F_{g}$ is an analytic subset
strictly contained in $W$.

For every path $\alpha\colon[0,1]\to V$ with $\alpha(0)=\Psi(z)$, we can take
a one-parameter family of paths $\alpha_{t}\,(t\in[0,1])$ in $W$ with
$\alpha_{t}(0)=\Psi(z)$ such that $\alpha_{1}=\alpha$ and, for $t<1$,
$\alpha_{t}$ is disjoint from $\Psi(F)$ (since $\Psi(F)$ has complex
codimension at least one.)

Then, for $t<1$, $\alpha_{t}$ continuously lifts a path
$\tilde{\alpha}_{t}\colon[0,1]\to Q$, and $\alpha_{t}$ converges to a desired
lift of $\alpha_{1}$ as $t\to 1$. ∎

Now we are ready to prove the completeness.

######  Theorem 12.2.

$\Psi\colon Q\to(\mathsf{T}\sqcup\mathsf{T}^{\ast})^{2}\setminus\Delta$ is a
complete map, where $\Delta$ is the diagonal set.

The completeness of Theorem 12.2 immediately implies the following:

######  Corollary 12.3.

$\Psi\colon Q\to(\mathsf{T}\sqcup\mathsf{T}^{\ast})^{2}\setminus\Delta$ is
surjective onto a connected component of
$(\mathsf{T}\sqcup\mathsf{T}^{\ast})^{2}\setminus\Delta$.

###### Proof of Theorem 12.2.

By Lemma 12.1, $\Psi$ has a local path lifting property. Thus, suppose that
$(X_{t},Y_{t})\colon[0,1]\to(\mathsf{T}\sqcup\mathsf{T}^{\ast})^{2}\setminus\Delta\leavevmode\nobreak\
(0\leq t\leq 1)$ be a path and there is a (partial) lift
$(C_{t},D_{t})\colon[0,1)\to Q$ of the path $(X_{t},Y_{t})$. For each
$t\in[0,1]$, let
$\rho_{t}\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}$
denote the common holonomy of $C_{t}$ and $D_{t}$. By the continuity the
orientations of Riemann surfaces in the pairs obviously remain the same along
$\rho_{t}$ for all $t>0$.

First we, in addition, suppose that there is an increasing sequence $0\leq
t_{1}<t_{2}<\dots$ converging to $1$, such that $\rho_{t_{i}}$ converges to a
representation in $\textstyle\chi$. By Corollary 10.2 or, in the case of
opposite orientations, Theorem 11.1,
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X_{1}}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y_{1}}$
is a discrete subset of $\textstyle\chi$. Then, since $(X_{t},Y_{t})$
converges to a point $(X_{1},Y_{1})$ in $\mathsf{T}\times\mathsf{T}$, and
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X_{t}}$
and
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y_{t}}$
change continuously in $t\in[0,1]$, every neighborhood
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X_{1}}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y_{1}}$
contains $\rho_{t_{i}}$ for all sufficiently large $i$. Thus the sequence
$\rho_{t_{i}}$ converges to a point in
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X_{1}}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y_{1}}$.
Since
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X_{t}}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y_{t}}$
is a discrete set in $\textstyle\chi$ which continuously changes in
$t\in[0,1]$, we indeed have a genuine convergence.

######  Lemma 12.4.

$\rho_{t}$ converges to $\rho_{1}$ as $t\to 1$.

By this lemma, $(C_{t},D_{t})$ converges in
$(\mathsf{P}\sqcup\mathsf{P}^{\ast})\times(\mathsf{P}\sqcup\mathsf{P}^{\ast})$
as $t\to 1$, so that the partial lift $(C_{t},D_{t})$ extends to $t=1$.

Thus it is suffices to show the addition assumption always holds

######  Proposition 12.5.

There is a compact subset $K$ in $\textstyle\chi$, such that, for every $t>0$,
there is $t^{\prime}>0$ such that $\rho_{t^{\prime}}\in K$.

###### Proof.

The proof is essentially the same as the proof of Theorem 10.1 or Theorem
11.1, which states that each component of
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{X}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}$
is a bounded subset of $\textstyle\chi$.

For $0\leq t<1$, let $V_{C_{t}}$ and $V_{D_{t}}$ be the vertical measured
foliations of $C_{t}$ and $D_{t}$, respectively. Suppose, to the contrary,
that $\rho_{t}$ leaves every compact subset of $\textstyle\chi$. As
$(X_{t},Y_{t})$ converges to
$(X,Y)\in(\mathsf{T}\sqcup\mathsf{T}^{\ast})^{2}\setminus\Delta$ and
$\operatorname{Hol}(C_{t})=\operatorname{Hol}(D_{t})$, similarly to Theorem
9.8 or Theorem 11.3, for $t$ close to $1$, we can construct a
$\mathbb{Z}$-weighed train track $\Gamma_{t}$ on $S$, such that

  * •

the intersection function $[\Gamma_{t}]\colon\mathsf{C}\to\mathbb{Z}$ is
continuous in $t$, and

  * •

for every closed curve $\alpha$ on $S$, there is a constant $q_{\alpha}>0$ and
a function $\epsilon_{t}>0$ converging to $0$, such that, for all sufficiently
large $t>0$, $[\Gamma_{t}](\alpha)$ is $(1+\epsilon_{t},q_{\alpha})$-quasi-
isometric to $V_{C_{t}}(\alpha)-V_{D_{t}}(\alpha)$ if the orientation of $X$
and $Y$ are the same and to $V_{C_{t}}(\alpha)+V_{D_{t}}(\alpha)$ if the
orientation of $X$ and $Y$ are different.

The first condition implies that the intersection number is constant in $t$,
whereas the second condition implies that the intersection number with some
closed curve $\alpha$ diverges to infinity as $t\to 1$. This is a
contradiction. ∎

12.2

Last we remark the behavior of $\operatorname{Hol}$ near the diagonal
$\Delta$.

######  Proposition 12.6.

Let $(C_{t},D_{t}),0\leq t<1$ be a path in $\mathsf{B}$, such that
$\Psi(C_{t},D_{t})$ converges to a diagonal point $(X,X)$ of
$(\mathsf{T}\sqcup\mathsf{T}^{\ast})^{2}$. Then
$\operatorname{Hol}C_{t}=\operatorname{Hol}D_{t}$ leaves every compact in
$\textstyle\chi$ as $t\to 1$.

###### Proof.

Recall that
$\operatorname{Hol}\colon\overline{\mathsf{P}}\to{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}$
is a locally biholomorphic map. Therefore, for every compact subset $K$ in
$\textstyle\chi$, if a neighborhood $U$ of $X$ in $\overline{\mathsf{T}}$ is
sufficiently small, for every $Y,Z\in U$,
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Y}\cap{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{Z}$
is disjoint from $K$. Then the assertion is immediate. ∎

##  13\. A proof of the simultaneous uniformization theorem

In this section, using Theorem A, we give a new proof simultaneous
uniformization theorem without using the measurable Riemann mapping theorem.
Recall $\mathsf{QF}$ is the quasi-Fuchsian space, and it is embedded in
$\mathsf{B}$.

Given $(C,D)$ in $\mathsf{QF}$, the universal covers $\tilde{C},\tilde{D}$ are
the connected components of ${\mathbb{C}{\rm P}}^{1}$ minus its equivariant
Jordan curve equivariant via $\operatorname{Hol}C=\operatorname{Hol}D$.

######  Lemma 13.1.

$\mathsf{QF}$ is a union of connected components of $\mathsf{B}$.

###### Proof.

As being a quasi-isometric embedding is an open condition, $\mathsf{QF}$ is an
open subset of $\mathsf{B}$. Thus, it suffices to show that $\mathsf{QF}$ is
closed.

Let $(C_{i},D_{i})\in\mathsf{P}\times\mathsf{P}^{\ast}$ be a sequence in
$\mathsf{QF}$ which converges to $(C,D)\in\mathsf{P}\times\mathsf{P}^{\ast}$.
Let $\rho_{i}\colon\pi_{1}(S)\to{\rm PSL}(2,\mathbb{C})$ be the quasi-Fuchsian
representation of $C_{i}$ and $D_{i}$. We show that the holonomy
$\rho\colon\pi_{1}(S)\to{\rm PSL}(2,\mathbb{C})$ of the limits $C$ and $D$ is
also quasi-Fuchsian. Let $\tilde{C}_{i}$ and $\tilde{D}_{i}$ be the universal
covers of $C_{i}$ and $D_{i}$, respectively. Then $\tilde{C}_{i}$ and
$\tilde{D}_{i}$ are the components of ${\mathbb{C}{\rm P}}^{1}$ minus the
$\rho_{i}$-equivariant Jourdan curve. Let
$f_{i}\colon\tilde{C_{i}}\cup\mathbb{S}^{1}\to{\mathbb{C}{\rm P}}^{1}$ and
$g_{i}\colon\tilde{D_{i}}\cup\mathbb{S}^{1}\to{\mathbb{C}{\rm P}}^{1}$ be the
extension of the embedding to its boundary circle by a theorem of
Carathéodory. Let $h_{i}$ be the homeomorphism
$\tilde{C}_{i}\cup\mathbb{S}^{1}\cup\tilde{D}_{i}\to{\mathbb{C}{\rm P}}^{1}$.

Since embeddings $\operatorname{dev}C_{i}$ converge to $\operatorname{dev}C$
uniformly on compact as $i\to\infty$, the limit $\operatorname{dev}C$ is also
an embedding. Then, by the convergence of corresponding convex pleated
surfaces in $\mathbb{H}^{3}$, the equivariant property implies that
$\operatorname{dev}C$ extends to the boundary circle continuously and
equivariantly Similarly, since the embedding $\operatorname{dev}D_{i}$
converging to $\operatorname{dev}D$, then $\operatorname{dev}D$ is also an
embedding, and $\operatorname{dev}D$ extends to the boundary circle
continuously and equivariantly. Therefore $h_{i}$ converges to a continuous
map

$h\colon\mathbb{S}^{2}\cong\tilde{C}\cup\mathbb{S}^{1}\cup\tilde{D}\to{\mathbb{C}{\rm
P}}^{1}$

such that the restriction of $h$ to $\tilde{C}\sqcup\tilde{D}$ is an
embedding.

The domain and the target of $h$ are both homeomorphic to $\mathbb{S}^{2}$.
Therefore, if $h|\mathbb{S}^{1}$ is not a Jordan curve on ${\mathbb{C}{\rm
P}}^{1}$, then there is a point $z\in{\mathbb{C}{\rm P}}^{1}$, such that
$h^{-1}(z)$ is a single segment of $\mathbb{S}^{1}$. By the equivariant
property, $h^{-1}(z)=\mathbb{S}^{1}$, and ${\rm Im}\,h$ is a wedge of two
copies of $\mathbb{S}^{2}$, which is a contradiction. ∎

The following asserts that the diagonal of $\mathsf{T}\times\mathsf{T}^{\ast}$
corresponds to the Fuchsian representations.

######  Lemma 13.2.

Let $X\in\mathsf{T}$. Let $\eta\colon\pi_{1}(S)\to{\rm PSL}(2,\mathbb{C})$ be
a quasi-Fuchsian representation, such that the ideal boundary of
$\mathbb{H}^{3}/{\rm Im}\rho$ realizes the marked Riemann surface $X$ and its
complex conjugate $X^{\ast}\in\mathsf{T}^{\ast}$. Then $\eta$ is the Fuchsian
representation $\pi_{1}(S)\to{\rm PSL}(2,\mathbb{R})$ such that
$X=\mathbb{H}^{2}/{\rm Im}\eta$.

###### Proof.

By the Riemann uniformization theorem, the universal covers $\tilde{X}$ and
$\tilde{X}^{\ast}$ are the upper hand the lower half planes. Then by
identifying their ideal boundaries equivariantly, we obtain ${\mathbb{C}{\rm
P}}^{1}$ so that the universal covers $\tilde{X}$ and $\tilde{X}^{\ast}$ are
round open disks. Let $(C,D)\in\mathsf{P}\times\mathsf{P}^{\ast}$ be the pair
corresponding to $\eta$, such that the complex structure of $C$ is $X$ and the
complex structure of $D$ is on $X^{\ast}$.

On the other hand, the universal covers $\tilde{C}$ and $\tilde{D}$ are
connected component of ${\mathbb{C}{\rm P}}^{1}\setminus\Lambda(\eta)$, where
$\Lambda(\eta)$ is the $\eta$-equivariant Jordan curve in ${\mathbb{C}{\rm
P}}^{1}$. Thus, there is a $\eta$-equivariant homeomorphism
$\phi\colon{\mathbb{C}{\rm P}}^{1}\to{\mathbb{C}{\rm P}}^{1}$, such that
$\phi$ restricts to a biholomorphism from ${\mathbb{C}{\rm
P}}^{1}\setminus{\mathbb{R}\cup\\{\infty\\}}\to{\mathbb{C}{\rm
P}}^{1}\setminus\Lambda(\eta)$.

Then, by Morera’s theorem for the line integral along triangles (see [SS03]),
$\phi$ is a genuine biholomorphic map ${\mathbb{C}{\rm
P}}^{1}\to{\mathbb{C}{\rm P}}^{1}$. Therefore $\phi$ is a Möbius
transformation, and therefore $\eta$ is conformally conjugate to the Fuchsian
representation uniformizing $X$. ∎

######  Proposition 13.3.

$\mathsf{QF}$ is a single connected component of $\mathsf{B}$.

###### Proof.

By Lemma 13.1, $\mathsf{QF}$ is the union of some connected components of
$\mathsf{B}$. By Theorem A, for every component $Q$ of $\mathsf{QF}$, the
image $\Psi(Q)$ contains the diagonal $\\{(X,X^{\ast})\\}$ of
$\mathsf{T}\times\mathsf{T}^{\ast}$. Then, by Lemma 13.2, every diagonal pair
$(X,X^{\ast})\in\mathsf{T}\times\mathsf{T}^{\ast}$ corresponds to a unique
point in $\mathsf{QF}$. Therefore $\mathsf{QF}$ is connected. ∎

Last we reprove the simultaneous uniformization theorem.

######  Theorem 13.4.

$\mathsf{QF}$ is biholomorphic to $\mathsf{T}\times\mathsf{T}^{\ast}$ by
$\Psi$.

###### Proof.

By Theorem A, $\Psi$ is a complete local branched covering map. Since $\Psi$
is surjective, by Lemma 13.2, $\Psi$ is a degree-one over the diagonal
$\\{(X,\overline{X})\mid X\in\mathsf{T}\\}$, and the diagonal corresponds to
the Fuchsian space.

The set of ramification points of $\Psi$ is an analytic set. The Fuchsian
space is a totally real subspace of dimension $6g-6$. Therefore, if the
ramification locus contains the Fuchsian space, then the locus must have the
complex dimension $6g-6$, the full dimension. This is a contradiction as
$\Psi$ is a locally branched covering map. Therefore,
$\mathsf{QF}\to\mathsf{T}\times\mathsf{T}^{\ast}$ has a degree one, and thus
it is biholomorphic. ∎

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