\put(28.0,11.0){{\color[rgb]{1,0.39,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.39,0.13}\pgfsys@color@cmyk@stroke{0}{0.61}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.61}{0.87}{0}$e_{2,t}$}}
\put(7.0,7.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$R_{t}$}}
\put(45.0,40.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$\tilde{\ell}_{t}$}}
\end{overpic} Figure 10. Left, $F_{t}\subset R_{t}\subset\mathbb{C}$. Right,
the $\exp$-image of $F_{t}$.
\begin{overpic}[scale={.05}]{Figure/ParabolicAlmostLimit} \end{overpic} Figure
11.

In addition, by the convergence of $\gamma_{t}$ to the parabolic element
$\gamma_{\infty}$, we may in addition assume that, for $t\gg 0$, $\exp
e_{1,t}$ and $\exp e_{2,t}$ are contained in a compact subset of
$\mathbb{C}{\rm P}^{1}\setminus\operatorname{Fix}(\gamma_{\infty})$. where
$\operatorname{Fix}\gamma_{\infty}$ is the parabolic fixed point of
$\gamma_{\infty}$.

Then $d_{t}|T_{1,t}$ and $d_{t}|T_{2,t}$ each converge to a fundamental domain
of a hyperbolic cusp neighborhood. Then, since $\exp(P_{t})$ is contained in
the $\epsilon$-part of $A_{t}$, for every neighborhood of
$\operatorname{Fix}\gamma_{\infty}$, if $t\geq 0$ is sufficiently large,
$d_{t}(P_{t})$ is contained in the neighborhood. ∎

Let $\ell_{t}$ be the leaf of $A_{t}$ dividing $A_{t}$ into two isomorphic
(projective) cylinders. Note that $\ell_{t}$ is an admissible loop on $A_{t}$.
Let $B_{t}$ be the $2\pi n$-grafting $\operatorname{Gr}_{\ell_{t}}^{n}(A_{t})$
of $A_{t}$ along $\ell_{t}$, which remains to be a projective structure on a
cylinder $A$ whose holonomy is the isomorphism onto
$\langle\gamma_{t}\rangle$. Let $\phi\colon\tilde{A}\to A$ be the universal
covering map.

######  Proposition 9.8.

There is a multiloop $M$ on the interior of $A$ consisting $n$-parallel copies
of $m$, such that, up to an isotopy,
$\operatorname{dev}(B_{t})\colon\tilde{A}_{t}\to\mathbb{C}{\rm P}^{1}$
converges to a $\langle\gamma_{\infty}\rangle$-equivariant continuous map
$d_{\infty}\colon\tilde{A}\to\mathbb{C}{\rm P}^{1}$ such that

  * •

$d_{\infty}$ is a local homeomorphism in the complement of $\phi^{-1}(M)$, and

  * •

$d_{\infty}$ yields a $\mathbb{C}{\rm P}^{1}$-structure $A_{\infty}$ on
$A\setminus M$, and for each component $F$ of $A\setminus M$, if $F$ is
bounded by two loops of $M$, $A_{\infty}|F$ is a biinfinite Euclidean
cylinder, and if $F$ is bounded by a single loop of $M$, $A_{\infty}|F$ has a
single hyperbolic cusp neighborhood.

###### Proof.

We describe how the fundamental domain $F_{t}$ of $A_{t}$ in $\mathbb{C}$
changes by the grafting. Notice that
$\phi_{t}^{-1}(\ell_{t})\eqqcolon\tilde{\ell}_{t}$ divides the infinite strip
$R_{t}$ into two isometric infinite stripes. Then $\tilde{\ell}_{t}$
intersects $F_{t}$ in a segment $s$ in the parallelogram $P_{t}$. Then the
fundamental domain $F_{t}^{n}$ for $\operatorname{Gr}_{\ell_{t}}^{n}(A_{t})$
is obtained by inserting, along $\tilde{\ell}_{t}$ a parallelogram $P^{n}$
bounded by vertical translates of $s_{t}$ and vertical edges of length $2\pi
n$ (see Figure 12 for $P^{1}$).

The exponential map $\exp\colon\mathbb{C}\to\mathbb{C}\setminus 0$ conformally
identifies $R_{t}$ with $\tilde{A}_{t}\subset\mathbb{C}{\rm P}^{1}$,
$\exp^{-1}(\omega_{t}ti\ell_{t})$ is a set of parallel lines in $\mathbb{C}$
transitive by the vertical translations by $2\pi$-multiples. Then the
universal cover of $\tilde{B}_{t}$ is identified with the domain between two
lines, so that the developing map corresponds to $\omega_{t}^{-1}\exp.$ Then
let $M_{t}$ be the multiloop on $B_{t}$ obtained from the inverse-image of
$\ell_{t}$ by $\operatorname{dev}B_{t}$. Then, the inverse image of $\ell_{t}$
decomposes the inserted parallelogram $P^{n}$ into $n$ isometric
parallelograms $P^{\prime}_{1},\dots,P^{\prime}_{n}$, whose vertical edges
have length $2\pi$.

Then, for each $i=1,\dots,n$, $P_{i}^{\prime}$ converges to the fundamental
domain of the Euclidean cylinder $(\mathbb{C}{\rm P}^{1}\setminus
Fix(\rho_{\infty}))/\langle\gamma_{\infty}\rangle$ such that the boundary
components of $P_{i}$ converges to the constant map to the parabolic fixed
point of $\gamma_{\infty}$. ∎

\begin{overpic}[scale={.03}]{Figure/GrafttedFundamentalDomain}
\put(52.0,43.0){{\color[rgb]{1,0.39,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.39,0.13}\pgfsys@color@cmyk@stroke{0}{0.61}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>\end{overpic} Figure 12. $F_{t}^{1}$

###### Proof of Theorem 9.6.

By Theorem 9.3, we already know the convergence of the developing map in the
every thick part. The convergence of the thin part is given by Proposition 9.8
and Lemma 9.7. ∎

###  9.1. Convergence in holomorphic quadratic differential in the case of
parabolic cusps

By Theorem 6.1, Corollary 6.2, Theorem 5.7, we have

######  Theorem 9.9.

Suppose that $\rho_{\infty}(m)$ is parabolic. Then $X_{t}$ converges to a
nodal Riemann surface $X_{\infty}$ such that $X_{\infty}$ minus the node is
homeomorphic to $S\setminus m$ and $q_{t}$ converges a regular quadratic
differential $q_{\infty}$ on $X_{\infty}$.

##  10\. $\rho_{\infty}(m)$ can not be hyperbolic

In this section, we show that $\rho_{\infty}(m)$ can not be a hyperbolic
element.

######  Claim 10.1.

Suppose that $X_{t}$ is pinched along a loop $m$ and, to the contrary,
$\rho_{\infty}(m)$ is hyperbolic. Then

  1. (1)

$C_{t}$ converges to a $\mathbb{C}{\rm P}^{1}$-structure $C_{\infty}$ on a
compact surface with two punctures, which is homeomorphic to $S\setminus m$,
in the sense that, for every $\epsilon>0$, the $\epsilon$-thick part of
$C_{t}$ converges to the $\epsilon$-thick part of $C_{\infty}$ uniformly, and

  2. (2)

$\rho_{\infty}(F)$ is non-elementary for each component $F$ of $S\setminus m$.

###### Proof.

(1) is an immediate corollary of Proposition 8.4.

(2) Let $F_{\infty}$ be the component of $C_{\infty}$ corresponding to $F$.
Then let $(\sigma,\nu)$ be the Thurston parametrizaition of $F_{\infty}$. Note
that $\sigma$ is a hyperbolic surface with geodesic boundary, such that the
lengths of the boundary components are the translation length of
$\rho_{\infty}(m)$ (see the proof of Lemma 5.4). Let
$(\tilde{\sigma},\tilde{\nu})$ be the universal cover of $(\sigma,\nu)$ so
that $\tilde{\sigma}$ is a convex subset of $\mathbb{H}^{2}$ bounded by
geodesics and that $\tilde{\nu}$ is a $\pi_{1}(\sigma)$-invariant lamination
on $\tilde{\sigma}$. Let $\alpha\colon\tilde{\sigma}\to\mathbb{H}^{3}$ be its
pleated surface equivariant by the holonomy of $F_{\infty}$.

Let $\ell$ be a boundary geodesic of $\tilde{\sigma}$. Then the endpoints of
$\alpha(\ell)$ are in the limit set $\Lambda$ of
$\operatorname{Hol}F_{\infty}$, as $\alpha(\ell)$ is the axis of the
hyperbolic $\rho_{\infty}(m)$. Every component $R$ of
$\tilde{\sigma}\setminus\tilde{\nu}$ has at least three ideal points. Then the
ideal points of $\alpha(R)$ are in $\Lambda$ (see [Bab]). Thus
$\rho_{\infty}|F$ is non-elementary. ∎

######  Lemma 10.2.

For each cusp $p$ of $C_{\infty}$, there is a neighborhood of $p$ foliated by
isomorphic admissible loops which develop to simple curves on $\mathbb{C}{\rm
P}^{1}$ invariant under a one-parameter subgroup in ${\rm PSL}(2,\mathbb{C})$
containing $\rho_{t}(m)$.

###### Proof.

The developing map near a cusp neighborhood is the restriction of the
exponential map $\exp\colon\mathbb{C}\to\mathbb{C}^{\ast}$; moreover, by
taking an appropriate neighborhood, one can assume that the restriction is to
a half plane bounded by a straight line in $\mathbb{C}$ invariant under the
deck transformation corresponding to $\rho_{t}(m)$.

The half plane is foliated by straight lines parallel to the boundary, and
this foliation descends to a desired foliation of the cusp neighborhoods by
admissible loops. ∎

######  Claim 10.3.

If $\epsilon>0$ is sufficiently small, then, for every sufficiently large
$t>0$, there is a cylinder $A_{t}$ in $C_{t}$ homotopy equivalent to $m$ such
that

  * •

$A_{t}$ changes continuously in $t\gg 0$;

  * •

$A_{t}$ is foliated by admissible loops whose developments are invariant under
a one-parameter subgroup $G_{t}$ in ${\rm PSL}(2,\mathbb{C})$ containing
$\rho_{t}(m)$;

  * •

$A_{t}$ contains the conformally $\epsilon$-thin part of $C_{t}$;

  * •

$C_{t}\setminus A_{t}$ converges to a $\mathbb{C}{\rm P}^{1}$-structure on
$S\setminus m$ whose boundary components are admissible loops.

###### Proof.

Consider the cusp neighborhoods of $C_{\infty}$ foliated by admissible loops
by Lemma 10.2. By the convergence of Lemma 10.1 and the stability of the
admissible loops, for $t\gg 0$, there is a cylinder $A_{t}$ foliated
admissible loops whose developments are invariant under $G_{t}$. Then it is
easy to realized other desired properties. ∎

By Claim 10.1 (2), the developing map of $C_{t}\setminus A_{t}$ converges
uniformly on compacts. The developing map of the cylinder $A_{t}$ is the
immersion into the region bounded by the developments of the two boundary
components of $C_{t}\setminus A_{t}$. Then, because of the convergence of
$C_{t}\setminus A_{t}$, $A_{t}$ must converge. Thus $C_{t}$ converges to a
$\mathbb{C}{\rm P}^{1}$-structure on $S$— this contradicts against the
divergence of $C_{t}$ in the deformation space. Hence $\rho_{\infty}(m)$ can
not be hyperbolic.

##  11\. $\rho_{\infty}(m)$ can not be elliptic

In this section, we show that $\rho_{\infty}(m)$ cannot be elliptic. To show
this, we assume, to the contrary, that $\rho_{\infty}(m)$ is elliptic and
reduce a contradiction against the convergence of $\rho_{t}$ as $t\to\infty$.
By Proposition 8.4, we have

######  Proposition 11.1.

Suppose that $\rho_{\infty}(m)$ is elliptic. Then $C_{t}$ converges to a
$\mathbb{C}{\rm P}^{1}$-structure $C_{\infty}$ on a compact surface minus two
points homeomorphic to $S\setminus m$, in the sense that, for every
$\epsilon>0$, the $\epsilon$-thick part of $C_{t}$ converges to the
$\epsilon$-thick part of $C_{\infty}$.

######  Lemma 11.2.

For each component $F_{\infty}$ of $C_{\infty}$, the stabilizer of
$\rho_{\infty}(F_{\infty})$ by conjugation is a discrete subgroup in ${\rm
PSL}(2,\mathbb{C})$.

###### Proof.

Let $F_{\infty}$ be a component of $C_{\infty}$. Then let $(\sigma,\nu)$ be
the Thurston parametrization of $F_{\infty}$, and let
$(\tilde{\sigma},\tilde{\nu})$ be the universal cover of $(\sigma,\nu)$. Then
the rotation angle of the elliptic element $\rho_{\infty}(m)$ is, modulo
$2\pi$, equal to the total weight, give by $\nu$, of the leaves ending at a
puncture (Proposition 5.2). Let
$\beta_{\infty}\colon\tilde{\sigma}\to\mathbb{H}^{3}$ be the equivariant
pleated surface. Pick an isolated leaf $\ell$ of $\nu$ whose endpoints are at
cusps of $\nu$. Let $\tilde{\ell}$ be a leaf of $\tilde{\nu}$ which is a lift
of $\ell$. Then its image $\beta_{\infty}(\ell)$ is a geodesic in
$\mathbb{H}^{3}$, and its endpoints are fixed points some elliptic elements in
$\rho_{\infty}(F_{\infty})$.

Moreover, the geodesic $\ell$ bounds a component $P$ of
$\tilde{\sigma}\setminus\tilde{\nu}$, and $P$ has at least three ideal points.
Then for each ideal point $p$ of $P$, let $\gamma\in\pi_{1}(F_{\infty})$ be
such that $\gamma$ fixes $p$. Then $\beta_{\infty}(p)$ is fixed by the
elliptic element $\rho_{\infty}(\gamma)$. Therefore, the stabilizer of
$\rho_{\infty}(F_{\infty})$ is a discrete subgroup of ${\rm
PSL}(2,\mathbb{C})$. ∎

Lemma 11.1 and Lemma 11.2 imply:

######  Proposition 11.3.

If $\epsilon>0$ is sufficiently small, then for every sufficiently large
$t>0$, there is a cylinder $A_{t}$ in $C_{t}$ homotopy equivalent to $m$ such
that

  * •

$A_{t}$ changes continuously in $t\gg 0$;

  * •

$A_{t}$ is foliated by loops whose developments are invariant under the one-
dimensional subgroup $G_{t}$ of ${\rm PSL}(2,\mathbb{C})$ containing
$\rho_{t}(m)$, and $G_{t}$ converges to a one-dimensional subgroup
$G_{\infty}$ of ${\rm PSL}(2,\mathbb{C})$ containing $\rho_{\infty}(m)$;

  * •

$A_{t}$ contains the conformally $\epsilon$-thin part of $C_{t}$ homotopic to
$m$;

  * •

$C_{t}\setminus A_{t}$ converges to a $\mathbb{C}{\rm P}^{1}$-structure on
$S\setminus m$ such that the boundary components cover round circles on
$\mathbb{C}{\rm P}^{1}$.

######  Proposition 11.4.

Suppose that $\rho_{\infty}(m)$ is parabolic. Then $C_{t}$ converges to a
$\mathbb{C}{\rm P}^{1}$-structure on $S$, which is a desired contradiction.

###### Proof.

Fix sufficiently small $\epsilon>0$, and let $A_{t}$ be a cylinder given by
Proposition 11.3. Let $\phi_{t}\colon\tilde{C}_{t}\to C_{t}$ be the universal
covering map. Then the developing map of
$\tilde{C_{t}}\setminus\phi_{t}^{-1}(A_{t})$ converges uniformly on compacts.
Let $\tilde{A}_{t}$ be a component of $\phi^{-1}(A_{t})$ which changes
continuously in $t$. Then $f_{t}$ converges on the boundary components of
$\tilde{A}_{t}$ and thus $A_{t}$ must converge (up to isotopy). Therefore
$C_{t}$ converges to a $\mathbb{C}{\rm P}^{1}$-structure on $S$, which is a
contradiction. ∎

##  12\. Limit when $\rho_{\infty}(m)=I$.

Let $A$ be a regular neighborhood of $m$ on $S$. Let $(\tau_{t},L_{t})$ be
Thurston parameters of $C_{t}$. Let
$\beta_{t}\colon\mathbb{H}^{2}\to\mathbb{H}^{3}$ be its $\rho_{t}$-equivariant
pleated surface. Let $\kappa_{t}\colon C\to\tau$ be the collapsing map, and
$\tilde{\kappa}_{t}\colon\tilde{C}\to\mathbb{H}^{2}$ denote the lift of
$\kappa$ to the map between their universal covers.

######  Theorem 12.1.

Suppose that $\rho_{\infty}(m)=I$. Then the following holds:

  1. (1)

$\rho_{t}(m)\neq I$ for sufficiently large $t>0$.

  2. (2)

The Fenchel-Nielsen twisting parameter (in $\mathbb{R}$) of $X_{t}$ along $m$
diverges to either $\infty$ or to $-\infty$.

  3. (3)

For every diverging sequence $0<t_{1}<t_{2}<\dots$, there is a subsequence
such that

    1. (a)

the axis $a_{t_{i}}$ converges to a point on $\mathbb{C}{\rm P}^{1}$ or a
geodesic in $\mathbb{H}^{3}$, denoted by $a_{\infty}$;

    2. (b)

there is a $\mathbb{C}{\rm P}^{1}$-structure in $\mathcal{P}(S\setminus m)$
such that, for every $\epsilon>0$, the $\epsilon$-thick part of $C_{t_{i}}$
converges to the $\epsilon$-thick part of $C_{\infty}$ uniformly;

    3. (c)

up to an isotopy of $S$ in $t$, the restriction of $f_{t_{i}}$ to
$\tilde{S}\setminus\phi^{-1}(A)$ converges to a $\rho_{\infty}$-equivariant
continuous map
$f_{\infty}\colon\tilde{S}\setminus\phi^{-1}(A)\to\mathbb{C}{\rm P}^{1}$ as
$t_{i}\to\infty$ such that, for each component $\tilde{A}$ of $\phi^{-1}(A)$,
its boundary components map onto the ideal points of $a_{\infty}$.

  4. (4)

the pleated surface
$\tilde{\kappa}_{t_{i}}\circ\beta_{t_{i}}\colon\tilde{S}\to\mathbb{H}^{3}$
converges to a $\rho_{\infty}$-equivariant continuous map
$\tilde{S}\to\mathbb{H}^{3}\cup\mathbb{C}{\rm P}^{1}$, up to an isotopy of
$S$.

Notice that, in (3c), if $a_{\infty}$ is a geodesic, then the different
boundary components of $\tilde{A}$ map to the different endpoints of
$a_{\infty}$.

We will prove (4) in the next section (§12.0.1). In this section, we will
prove the other assertions: (1) will be proved in Lemma 12.7; (2) will be
proved in Lemma 12.8; (3c) will be proved in Proposition 12.11. The proof of
(3b) is similar to the proof of Proposition 8.4.

Let $C_{\infty}\cong(\sigma_{\infty},\nu_{\infty})$ denote the Thurston
parameterization, where $\sigma_{\infty}$ be a hyperbolic structure in the
Teichmüller space $\mathsf{T}(S\setminus m)$ and $\nu_{\infty}$ be a measured
lamination on $\sigma_{\infty}$. Then $\sigma_{\infty}$ has two cusps, and, at
each cusp $c$ of $\sigma_{\infty}$, there are only finitely many leaves of
$\nu_{\infty}$ ending at $c$, and, since $\rho_{\infty}(m)=I$, the total
weight of those leaves is a positive $2\pi$-multiple.

######  Lemma 12.2.

If $\nu_{\infty}$ contains an irrational sublamination, then the holonomy of
$C_{\infty}$ is non-elementary.

###### Proof.

Suppose that $\nu_{\infty}$ contains an irrational sublamination. Then there
is a minimal irrational sublamination $N$ of $L$, so that every leaf of $N$ is
dense in $N$. Let $F$ be a (topologically) smallest subsurface of $S$
containing $N$, such that $F\subset N$ is a $\pi_{1}$-injective. Let $\ell$ be
a geodesic loop in $\sigma_{\infty}$ which is a good approximation of $N$. Let
$\beta_{\infty}\colon\mathbb{H}^{2}\to\mathbb{H}^{3}$ be the equivariant
pleated surface corresponding to $(\sigma_{\infty},\nu_{\infty})$. Then, for
each component $R$ of $F\setminus\ell$, the restriction of $\beta_{\infty}$ to
$R$ is a quasi-isometric embedding ([Bab10]). Thus $\rho_{\infty}|\pi_{1}R$ is
non-elementary, which immediately implies the lemma. ∎

Using the assumption that $C_{t}$ is pinched along a single loop (not a
multiloop), we prove the following:

######  Proposition 12.3.

For each component $F$ of $S\setminus m$, the restriction of $\rho_{\infty}$
to $\pi_{1}F$ is a non-trivial representation.

###### Proof.

If $\nu_{\infty}$ contains an irrational lamination, by Lemma 12.2,
$\rho_{\infty}$ is non-elementary. Then we can assume, without loss of
generality, that $\nu_{\infty}$ contains only isolated leaves, and
$\nu_{\infty}$ divides $\sigma_{\infty}$ into ideal polygons.

Since each component of $\sigma_{\infty}$ has one or two cusps, there is a
leaf $\ell$ of $\nu_{\infty}$ whose endpoints are at a single cusp $c$ of
$\sigma_{\infty}$. Let $D$ be a small horodisk quotient neighborhood of $c$.
Then $\ell\setminus D$ is a long geodesic segment, and by connecting its
endpoints by a horocyclic arcin $\partial D$, we obtained a simple loop
$\gamma$ (which is a good approximation of $\ell$).

Notice that $\beta_{\infty}(\ell)$ is a (biinfinite) geodesic in
$\mathbb{H}^{3}$. Then the holonomy along $\gamma$ takes one endpoint of
${\beta}_{\infty}(\ell)$ to a point close to the other endpoint, and thus
$\rho_{\infty}(\gamma)\neq I$. ∎

######  Lemma 12.4.

Let $G$ be a non-trivial subgroup of ${\rm PSL}(2,\mathbb{C})$. Consider the
(pointwise) stabilizer of $G$ by conjugation by ${\rm PSL}(2,\mathbb{C})$.
Suppose that the stabilizer is continuous. Then there is a set $\Lambda$ of
one or two points of $\mathbb{C}{\rm P}^{1}$ fixed pointwise by the action of
$G$.

###### Proof.

Suppose that $G$ has a continuous stabilizer. Then, clearly, $G$ is an
elementary subgroup of ${\rm PSL}(2,\mathbb{C})$. First suppose, in addition,
that $G$ contains a hyperbolic element $h$. Then no element in $G$ exchanges
the fixed points of $h$, as otherwise the stabilizer can not be continuous.
Therefore $\Lambda$ is the fixed point set of $h$, and all elements in
$G\setminus\\{I\\}$ must be hyperbolic or elliptic elements with the same
axis.

Next suppose that $G$ contains a parabolic element $p$. Then there is no
elliptic element or hyperbolic element in $G$, as otherwise the stabilizer can
not be continuous. Then $\Lambda$ must be the single fixed point of $p$, and
all $G\setminus\\{I\\}$ are all parabolic elements with the same fixed point.

Suppose that $G$ contains an elliptic element $e$ and contains no hyperbolic
element. Then, similarly, $\Lambda$ must the fixed point set of $e$, and $G$
contains no parabolic element. Moreover $G\setminus\\{I\\}$ are all elliptic
elements with a common axis. Then $\Lambda$ is the set of the two endpoints of
the axis. ∎

Given a $\mathbb{C}{\rm P}^{1}$-surface with a puncture such that the holonomy
around the puncture is trivial, its developing map continuously extends to the
puncture, so that it is a branched covering map near the puncture.

######  Lemma 12.5.

Let $F$ be a compact surface with finitely many punctures, such that the Euler
characteristic of $F$ is negative. Let $(f,\rho)$ be a developing pair of a
$\mathbb{C}{\rm P}^{1}$-structure $C$ on $F$ such that

  * •

$\rho\colon\pi_{1}(F)\to{\rm PSL}(2,\mathbb{C})$ is not the trivial
representation,

  * •

the holonomy around each puncture is trivial, and

  * •

the stabilizer of $\operatorname{Im}\,\rho$ in ${\rm PSL}(2,\mathbb{C})$ is
continuous; thus let $\Lambda\subset\mathbb{C}{\rm P}^{1}$ be the one- or two-
point set in Lemma 12.4.

Then, there is a puncture $p$ of $F$ such that $f(p)$ is not a point of
$\Lambda$.

###### Proof.

Notice that $\mathbb{C}{\rm P}^{1}$ minus $\Lambda$ admits a complete
Euclidean metric invariant under $\operatorname{Im}\rho$ (unique up to
scaling). Thus, if $f$ takes all punctures of $F$ into $\Lambda$, then $F$
minus finitely many points admits a complete Euclidean metric. This is a
contraction as the Euler characteristic of $F$ is negative. ∎

######  Proposition 12.6.

Let $F$ be a compact connected surface with two punctures, such that the Euler
characteristic of $F$ is negative. Let $C=(f,\rho)$ be a $\mathbb{C}{\rm
P}^{1}$-structure on $F$, such that

  * •

$\operatorname{Im}\,\rho$ has a continuous stabilizer in ${\rm
PSL}(2,\mathbb{C})$;

  * •

the holonomy around punctures are trivial;

  * •

the degrees around the two punctures are the same.

Then no puncture of $C$ maps to a point of $\Lambda$ by $f$, where $\Lambda$
is as in Lemma 12.4.

###### Proof.

By Lemma 12.5, at least one puncture $q$ must develop to a point in the
complement of $\Lambda$. Suppose, to the contrary, that there is a puncture
$p$ of $C$ which develops to a point in $\Lambda$.

Let $(\tau,L)$ be the Thurston parameters of $C$. As $\rho$ is elementary, by
Lemma 12.2, $L$ consists of isolated leaves. By the third condition, the total
weights of leaves of $L$ ending $p$ and $q$ are the same $2\pi$-multiple; let
$2\pi d$ denote the total weights, where $d$ is the degree. As
$\operatorname{Im}\rho$ has a continuous stabilizer, $f(p)$ must be preserved
by $\operatorname{Im}\rho$. Thus every leaf of $L$ starting from $p$ must end
at $q$. Thus, as the total weights agree, every leaf of $L$ must connects $p$
and $q$.

Let $R$ be a component of $\tau\setminus L$, and let $\tilde{R}$ be a lift of
$R$ to the universal cover $\tilde{\tau}\cong\mathbb{H}^{2}$. Then
$\beta\colon\mathbb{H}^{2}\to\mathbb{H}^{3}$ isometrically embeds $\tilde{R}$
into a totally geodesic hyperbolic plane, which has at least three ideal
vertices. As all lifts of $p$ to the ideal boundary of $\tilde{\tau}$ map to
the single point in $\Lambda$, there is exactly one point of $\tilde{R}$
corresponding to $p$. Then there must be an edge of $\tilde{R}$ connecting
different lifts of $q$. This is a contradiction.  12.6

Let $\tilde{m}$ be a lift of $m$ to $\tilde{S}$. Let $Q$ and $R$ be the
adjacent components of $\tilde{S}\setminus\phi^{-1}(m)$ along $\tilde{m}$. Let
$\operatorname{Stab}{Q}$ and $\operatorname{Stab}{R}$ denote the stabilizers,
in $\pi_{1}(S)$, of $Q$ and $R$. We first prove (1) in Theorem 12.1.

######  Lemma 12.7.

For sufficiently large $t>0$, $\rho_{t}(m)\neq I$.

###### Proof.

Suppose, to the contrary, that there is a diverging sequence $0\leq
t_{1}<t_{2}<\dots$ such that $\rho_{t_{i}}(m)=I$ for each $i$. We may, in
addition, assume that $C_{t_{i}}$ converges to $C_{\infty}$ as $i\to\infty$
uniformly on compacts as $i\to\infty$. Then, as $\rho_{t_{i}}(m)=I$ and
$C_{t}$ is pinched along $m$, for $i\gg 0$, there is a cylinder $A_{i}$ in
$C_{t_{i}}$ homotopic to $m$ such that

  * •

$A_{i}$ is bounded by round circles (i.e. the development of each boundary
component is a round circle on $\mathbb{C}{\rm P}^{1}$),

  * •

$\operatorname{Mod}A_{i}\to\infty$, and

  * •

$C_{i}\setminus A_{i}$ converges to $C_{\infty}$ minus cusp neighborhoods
bounded by round circles.

Then, by Proposition 12.6, Lemma 12.5, and Proposition 12.3, either
$\rho_{t_{i}}|\operatorname{Stab}R$ or $\rho_{t_{i}}|\operatorname{Stab}Q$
must diverge to $\infty$ in the ${\rm PSL}(2,\mathbb{C})$-character variety.
This is a contradiction against the convergence of $\rho_{t}$. ∎

######  Lemma 12.8.

The Fenchel-Nielsen twist coordinate along $m$ must diverge to $\infty$ or
$-\infty$ as $t\to\infty$.

###### Proof.

The proof is similar to that of Lemma 12.7. ∎

Then, for each $t>0$, let $\iota_{t}$ be some power of the Dehn twist of $S$
along $m$ such that the twist coordinated of $\iota_{t}C_{t}$ along $m$ is
bounded from above and below uniformly in $t>0$. Then, by Lemma 12.8, the
power must diverge to either $\infty$ or $-\infty$ as $t\to\infty$.

There is a sequence $0\leq t_{1}<t_{2}<\dots$ such that $C_{t_{i}}\to
C_{\infty}$ as $i\to\infty$ uniformly on compact. Let $F$ be a component of
$S\setminus m$. Let $\tilde{F}$ be the universal cover of $F$.

First suppose that $\rho_{\infty}|F$ has a discrete stabilizer (in ${\rm
PSL}(2,\mathbb{C})$). Let $F_{\infty}$ be a component of $C_{\infty}$ which
corresponds to $F$. Then $\operatorname{dev}F_{\infty}$ is the limit of
$f_{t_{i}}|\tilde{F}$, so that $\lim_{i\to\infty}f_{t_{i}}$ takes each
boundary components of $\tilde{F}$ to a single point corresponding to a cusp
of $C_{\infty}$.

Pick a fundamental domain $D_{i}$ in $\tilde{F}$ with an arc $s_{i}$ on
$\partial D_{i}\cap\partial\tilde{F}$ such that $s_{i}$ descends to a loop
$m_{i}$ isotopic to $m$, it is contained in the $\epsilon_{i}$-thin part of
$C_{t_{i}}$ with $\epsilon_{i}\searrow 0$ as $i\to\infty$, and the development
of $m_{i}$ is invariant under a one-dimensional subgroup $G_{i}$ of ${\rm
PSL}(2,\mathbb{C})$ containing $\rho_{i}(m)$. As $\rho_{t_{i}}(m)\to I$, the
image of $s_{i}$ becomes more and more like a round circle $c_{i}$ as
$i\to\infty$.

Next suppose that $\rho_{\infty}(F)$ has a continuous stabilizer. Then
$\rho_{\infty}(F)$ is elementary, and the restriction of $f_{t_{i}}$ to
$\tilde{F}$ may not converges to a local homeomorphism, even, up to a
subsequence. Nonetheless, as $C_{t}$ accumulates to a bounded subset of
$\mathsf{P}(S\setminus m)$, we can certainly normalize it in a nice way:

######  Lemma 12.9.

Suppose that there is no subsequence of $t_{i}$ such that
$f_{t_{i}}|\tilde{F}$ converges to a developing map of $F_{\infty}$. Then
there is a sequence $\gamma_{i}$ of ${\rm PSL}(2,\mathbb{C})$ such that, up to
a subsequence, $\gamma_{i}(f_{t_{i}}|\tilde{F},\rho_{t_{i}}|F)$ converges to a
developing pair $(h_{\infty},\zeta_{\infty})$ of $F_{\infty}$.

######  Proposition 12.10.

Suppose that there is no subsequence such that $f_{t_{i}}|\tilde{F}$ converges
to a developing map of $F_{\infty}$. Then $f_{t_{i}}|\tilde{F}$ converges to a
$\rho_{\infty}|F$-equivariant continuous map
$f_{F,\infty}\colon\tilde{F}\to\mathbb{C}{\rm P}^{1}$ uniformly on compacts,
such that each boundary component of $\tilde{F}$ maps to a single point.
Moreover, $f_{F,\infty}$ is either a constant map to a fixed point of
$\rho_{\infty}|F$ or a there are open disks $D_{1},\dots,D_{n}$ on $F$ such
that $f_{F,\infty}$ takes $\tilde{F}\setminus\phi^{-1}(D_{1}\sqcup\dots\sqcup
D_{n})$ to a fixed point $p$ of $\rho_{\infty}(F)$ and each lift
$\tilde{D}_{i}$ of $D_{i}$ to $\mathbb{C}{\rm P}^{1}\setminus\\{p\\}$ for all
$i=1,\dots,n$.

###### Proof.

Let $\gamma_{i}\in{\rm PSL}(2,\mathbb{C})$ be the sequence obtained by Lemma
12.9.

First suppose that $\rho_{\infty}(F)$ is loxodromic or elliptic. Then let
$\ell$ be the axis of $\rho_{\infty}(F)$. Then $\gamma_{i}$ will be a
hyperbolic element for sufficiently large $i$, and, as $i\to\infty$, the
translation length of $\gamma_{i}$ diverges to infinity and
$\operatorname{Axis}(\gamma_{i})$ converges to $\ell$. Let $p$ be the limit of
the repelling fixed point of $\gamma_{i}$, and let $q$ be the limit of the
attracting fixed point of $\gamma_{i}$, so that $\\{p,q\\}$ are the endpoints
of $\ell$.

Take a connected fundamental domain $Q$ in $\tilde{F}$ so that the
$h_{\infty}(\partial Q)\cap\\{p,q\\}=\emptyset$. Suppose that $h_{\infty}(Q)$
is disjoint from $q$. Then, letting $f_{i}=f_{t_{i}}$, the restriction
$f_{i}|Q$ converges to the constant map to $p$ uniformly, as $i\to\infty$, and
thus $f_{i}\colon\tilde{F}\to\mathbb{C}{\rm P}^{1}$ converges to the constant
map to $p$ uniformly on compacts. Suppose that
$h_{\infty}(Q)\cap\\{q\\}\neq\emptyset$. Then there are finitely many points
of $h_{\infty}^{-1}(q)$ in the interior of $Q$. Pick disjoint open disk
neighborhood of the points in $h_{\infty}^{-1}(q)$ in $Q$, and let
$D_{1},\dots,D_{n}$ be their disjoint images on $F$. Then, up to an isotopy of
$F$, the restriction of $f_{i}$ to
$\tilde{F}\setminus\phi^{-1}(D_{1}\sqcup\dots\sqcup D_{n})$ converges to the
constant map to $p$ uniformly on compacts and for each lift $\tilde{D}_{i}$ of
$D_{i}$ to $\tilde{F}$, $f_{i}|D_{i}$ converges to a homeomorphism to
$\mathbb{C}{\rm P}^{1}\setminus\\{p\\}$, as desired. By Lemma 8.5 and
Proposition 8.6, the boundary components of $\tilde{F}$ all map to $p$.

Next suppose that $\rho_{\infty}(F)$ is parabolic. Let $p$ be the parabolic
fixed point of $\rho_{\infty}(F)$. Then then $\operatorname{Axis}\gamma_{i}$
converges to $p$ as $i\to\infty$. By normalizing $\mathbb{C}{\rm P}^{1}$
appropriately, we may assume that $\gamma_{i}$ is asymptotically in the form
of $\begin{bmatrix}1&c_{i}\\\ 0&1\end{bmatrix}$ and $\|c_{i}\|\to\infty$. Then
up to a subsequence, we may in addition assume that $\arg c_{i}$ converges as
well. Then, similarly, pick a fundamental domain $Q$ in $\tilde{F}$ so that
$h_{\infty}(\partial Q)$ does not intersect $p$. Suppose that there is no
point of $Q$ maps to $p$ by $h_{\infty}$. Then $f_{i}|\tilde{F}$ converges to
the constant map to $p$ uniformly on compacts.

Otherwise there are finitely many points $z_{1},z_{2},\dots,z_{n}$ of the
interior of $Q$, which map to $p$. Normalize $\mathbb{C}{\rm
P}^{1}=\mathbb{C}\cup\\{\infty\\}$ so that $p=\infty$. Then consider a half
plane $H$ of $\mathbb{C}$ (bounded to by a straight line) such that $\partial
H$ is asymptotically orthogonal to $c_{i}$ (regarded as a vector) but not
containing $c_{i}$ (as a point) for sufficiently large $i$, so that the
translation of $H$ by $c_{i}$ will asymptotically be the entire complex plane
$\mathbb{C}$ as $i\to\infty$. Then, for each point $z_{j}$ in $Q$ mapping to
$p$, takes a sufficiently small round ball such that it boundary contain
$z_{j}$ and its $h_{\infty}$-image is the half plane $H$. Then, take disjoint
open disks $D_{1},\dots,D_{n}$ on $F$ which lift to those open balls in $Q$,
we can show the desired convergence.

Last suppose that $\rho_{\infty}(F)$ is the trivial representation. Then the
normalized holonomy $\zeta_{\infty}$ is a parabolic representation. Let $p$ be
the parabolic fixed point of $\zeta_{\infty}$. Pick a connected fundamental
domain $Q$ in $\tilde{F}$ so that $h^{-1}(p)$ is disjoint from $\partial Q$.
Suppose in addition that no point of $Q$ maps to $p$. Then, up to a
subsequence, $f_{i}|\tilde{F}$ converges to a constant map uniformly on
compacts. Suppose there are (finitely many) points of $Q$ which map to $p$.
Then, similarly to the case of hyperbolic and elliptic element, take disjoint
open ball neighborhoods of those points in $Q$, and let $D_{1},D_{2}\dots
D_{n}$ be disjoint disks on $F$ which lift to those open balls. Then the
desired convergence follows similarly. ∎

By Proposition 12.10, the restriction of $f_{i}$ to
$\tilde{S}\setminus\phi^{-1}(A)$ converges to a $\rho_{\infty}$-equivariant
map $f_{\infty}\colon\tilde{S}\setminus\phi^{-1}(A)\to\mathbb{C}{\rm P}^{1}$
(3c).

######  Proposition 12.11.

For each component $\tilde{A}$ of $\phi^{-1}(A)$, let $\gamma\in\pi_{1}(S)$ be
the representative of $m$ preserving $\tilde{A}$. Then, by taking a
subsequence so that $\operatorname{Axis}(\rho_{t_{i}}\gamma)\eqqcolon a_{i}$
converges to a subset $a_{\infty}\in\mathbb{H}^{3}\cup\mathbb{C}{\rm P}^{1}$,
which is either a point on $\mathbb{C}{\rm P}^{1}$ or a geodesic in
$\mathbb{H}^{3}$, then $f_{\infty}$ takes the boundary components of
$\tilde{A}$ onto the ideal points of $a_{\infty}$.

###### Proof.

By Lemma 12.7, $\rho_{t_{i}}(m)\neq I$ for sufficiently large
$i\in\mathbb{Z}_{>0}$. Then, by taking a subsequence, we may in addition
assume that $\rho_{t_{i}}(\gamma)$ converges to $I$ tangentially to a unite
tangent vector of ${\rm PSL}(2,\mathbb{C})$ at $I$. We let $G_{i}$ denote the
one-parameter subgroup of ${\rm PSL}(2,\mathbb{C})$ which contains
$\rho_{t_{i}}(\gamma)$, such that the cyclic group generated by
$\rho_{t_{i}}(\gamma)$ is asymptotically dense in $G_{i}$ with respect to the
intrinsic metric on $G_{i}$. Then the trajectories of $G_{i}$ yields a unique
foliation of $\mathbb{H}^{3}$ except that, if $\rho_{t_{i}}(\gamma)$ is
elliptic, only of $\mathbb{H}^{3}\setminus a_{i}$ (Figure 14). We have chosen
a subsequence $t_{i}$ so that $C_{t_{i}}\to C_{\infty}$ uniformly on every
thick part and the axis $a_{i}$ converges to a closed subset $a_{\infty}$ of
$\overline{\mathbb{H}^{3}}$. Let $P,Q$ be the components of
$\tilde{S}\setminus\phi^{-1}(N)$ adjacent across $\tilde{N}$.

######  Claim 12.12.

Let $\ell$ be the common boundary component of $P$ and $\tilde{N}$. Suppose,
to the contrary, that $\lim f_{t_{i}}(\ell)$ is not a point of the limit axis
$a_{\infty}$. Then $\rho_{t_{i}}|Q$ diverges to $\infty$ in $\chi$.

###### Proof.

Let $\iota_{i}$ be some power of the Dehn twist of $S$ along $m$ so that the
Fenchel-Nielsen twist parameter of $\iota_{i}X_{t_{i}}$ along $m$ is bounded
from above and below uniformly in $i$.

Let $r$ be the common boundary component of $\tilde{N}$ and $Q$. By
Proposition 12.3, there is $\gamma\in\pi_{1}S$ belonging to
$\operatorname{Stab}Q$ such that $\rho_{\infty}(\gamma)$ is not the identity
matrix. We may in addition assume the axis of $\rho_{i}(\iota_{i}\cdot\gamma)$
converges to the point $f_{F,\infty}(\ell)$ on $\mathbb{C}{\rm P}^{1}$ (if
$\rho_{\infty}(\operatorname{Stab}Q)$ is elementary, use Lemma 12.5 and
Proposition 12.6). By the tangential convergence of $\rho_{i}(m)\to I$, the
$G_{i}$-invariant foliation of $\mathbb{H}^{3}$ by $\mathcal{F}_{i}$ converges
to a foliation of $\mathcal{F}_{\infty}$ of $\mathbb{H}^{3}$. If
$f_{P,\infty}(\ell)$ is not the ideal point of $a_{\infty}$,
$\operatorname{Axis}(\rho_{i}(\iota\gamma))$ be eventually disjoint from every
compact subset in the space of the leaves of $\mathcal{F}_{\infty}$. Therefore
$\operatorname{Axis}(\rho_{i}(\gamma))$ leaves every compact subset of the
leaf space of $\mathcal{F}_{\infty}$, and thus $\rho_{i}(\gamma)$ must diverge
to $\infty$ in ${\rm PSL}(2,\mathbb{C})$, which is a contradiction. (Figure
13.) ∎

This claim completes the proof.  12.11

It remains only to prove the surjectivity in (3c):

######  Lemma 12.13.

Suppose that $a_{\infty}$ is a geodesic in $\mathbb{H}^{3}$. Then
$f_{\infty}(\ell)$ and $f_{\infty}(r)$ are the different endpoints of
$a_{\infty}$.

###### Proof.

By Claim 12.12. $f_{i}|\ell$ converges to the constant map to an endpoint of
$a_{\infty}$.

Let $n_{i}\in\mathbb{Z}$ be the power of the Dehn twist along $m$ which gives
$\iota_{i}\in\pi_{1}(S)$. Thus $\rho_{i}(\gamma^{n_{i}})$ is a hyperbolic
element whose axis $a_{i}$ converges to $a_{\infty}$ and its translation
length diverge to infinity. Then the attracting fixed point of
$\rho_{i}(\gamma^{n_{i}})$ converges to the endpoint of $a_{\infty}$ which is
not $f_{\infty}(\ell)$. Thus $f_{\infty}(q)$ must be at the other endpoint. ∎

\begin{overpic}[scale={.35}]{Figure/AxisEndsAtPunctures}
\put(18.0,50.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tiny
The axis at the puncture.}}
\put(65.0,50.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tiny
The axis off the puncture; diverging.}} \end{overpic} Figure 13.
\begin{overpic}[scale={.3}]{Figure/EquidodtanceFromAxis} \end{overpic} Figure
14.

12.11

####  12.0.1. Convergence of pleated surfaces when $\rho_{\infty}(m)=I$

First we compare developing maps of $\mathbb{C}{\rm P}^{1}$-structures and the
exponential map $\exp\colon\mathbb{C}\to\mathbb{C}^{\ast}$. Let $\ell$ be the
geodesic in $\mathbb{H}^{3}$ connecting $0$ to $\infty$. Let
$\Psi\colon\mathbb{C}^{\ast}\to\ell$ be the continuous extension of the
nearest point projection $\mathbb{H}^{3}\to\ell$. Then, the composition is
$\Psi\circ\exp\colon\mathbb{C}\to\mathbb{H}^{3}$ is the Epstein map of the
$\mathbb{C}{\rm P}^{1}$-structure on $\mathbb{C}$ given by $\exp$.

Recall that, given a $\mathbb{C}{\rm P}^{1}$-structure $C=(X,q)$, for
$x\in\mathbb{C}$, $\leavevmode\nobreak\ d(x)$ is the Euclidean distance from
the zeros of the holomorphic differential $q$. Note that, if $d(x)$ is large,
then we can naturally embeds a large neighborhood of $x$ into
$\mathbb{C}(\cong\mathbb{E}^{2})$ by an isometric map onto its image, so that
vertical leaves map into horizontal lines, and horizontal leaves map into
vertical lines.

######  Proposition 12.14.

For every $\epsilon>0$, there is $R>0$, such that, if $x\in C$ satisfies
$d(x)>R$, then the Epstein map
$\Sigma\colon\tilde{C}\cong\tilde{S}\to\mathbb{H}^{3}$ is $\epsilon$-close in
the $C^{1}$-topology and the composition of the collapsing map
$\tilde{\kappa}\colon\tilde{S}\to\mathbb{H}^{2}$ and the bending map
$\beta\colon\mathbb{H}^{2}\to\mathbb{H}^{3}$ at every lift $\tilde{x}$ of $x$.

###### Proof.

######  Lemma 12.15.

For every $\epsilon>0$, there is $R>0$ such that, if $z\in\tilde{C}$ satisfies
$d(z)>R$, then the restriction of $\Sigma$ to the
$\frac{1}{\epsilon}$-neighborhood $N_{1/\epsilon}(z)$ of $z$ is
$\epsilon$-close to $w\mapsto\Psi\exp(\frac{w}{\sqrt{2}})$ on the
corresponding region in $\mathbb{C}$.

###### Proof.

By Lemma 2.4, the vertical leaves on $\Sigma$ are almost geodesic on
$\mathbb{H}^{3}$, and on the other hand, horizontal leaves on $\Sigma$ is very
short in $\mathbb{H}^{3}$. Moreover the vertical and horizontal direction are
orthogonal. The curvature estimate along the horizontal direction is given by
Lemma 4.1. Therefore $\Sigma$ is $\epsilon$-close to
$\omega\mapsto\Psi\exp(\frac{w}{\sqrt{2}})$ in both horizontal and vertical
direction, and as they stay orthogonal on $\Sigma$ (Lemma 2.4 (3)), we have
the claim. ∎

######  Lemma 12.16.

For every $\epsilon>0$, there is $R>0$, such that, if $z\in\tilde{C}$
satisfies $d(z)>R$, then the maximal ball centered at $z$ is $\epsilon$-close
to the maximal ball of the corresponding exponential map.

###### Proof.

As the Epstein maps of $C$ and $\exp$ are close, their developing maps are
also close. This implies the closeness of their maximal balls centered at $z$
and their ideal points. ∎

The proposition follows from the above lemmas.  12.14

######  Proposition 12.17.

Suppose that $\rho_{\infty}(m)=I$. Then for every diverging sequence
$t_{1}<t_{2}<\dots$, up to taking a subsequence, there are a sequence of
compatible markings $\iota_{i}\colon S\to\tau_{t_{i}}$ and a path of cylinders
$A_{t}$ in $C_{t}$ homotopy equivalent to $m$, such that in addition to
Theorem 12.1 (1), (2), (3), the following holds:

  1. (1)

$A$ maps to $A_{i}\coloneqq A_{t_{i}}$ by $\iota_{i}\coloneqq\iota_{t_{i}}$;

  2. (2)

$\beta_{t_{i}}\circ\tilde{\kappa}_{t_{i}}\colon\tilde{S}\to\mathbb{H}^{3}$
converges to a $\rho_{\infty}$-equivariant continuous map
$\tilde{S}\to\mathbb{H}^{3}\cup\mathbb{C}{\rm P}^{1}$ uniformly on compact
subsets;

  3. (3)

for each connected component $F$ of $\tilde{S}\setminus\phi^{-1}(A)$, the
restriction of $\beta_{t_{i}}\kappa_{t_{i}}$ to $F$ converges to the pleated
surface of the corresponding component of $C_{\infty}$ or the constant map to
a point of $a_{\infty}$;

  4. (4)

letting $\tilde{A}_{t}$ be a connected component of $\phi_{t}^{-1}(A)$ in
$\tilde{S}$, then $\beta_{t_{i}}\circ\tilde{\kappa}_{t_{i}}|\tilde{A}_{t}$
converges to a map onto $a_{\infty}$ uniformly on compacts in $\tilde{A}$ with
respect to a disk metric on $\mathbb{H}^{3}\cup\mathbb{C}{\rm P}^{1}$.

######  Remark 12.18.

If $\beta_{i}\circ\kappa_{i}|F$ is a constant map, then $\rho_{\infty}(F)$
must be the trivial representation by the equivariance property.

###### Proof.

For $t\gg 0$, there is a one-parameter family of loops homotopic to $m$ such
that their developments are invariant under a unique one-dimensional subgroup
$G_{t}$ of ${\rm PSL}(2,\mathbb{C})$. Then we can pick a cylinder $A_{t}$ in
$C_{t}$ homotopy equivalent to $m$, such that

  * •

$A_{t}$ is foliated by loops whose developments are invariant under $G_{t}$
for each $t\gg 0$,

  * •

$C_{t}\setminus A_{t}$ converges to $C_{\infty}$ as $t\to\infty$, and

  * •

$\operatorname{Mod}A_{t}\to\infty$ as $t\to\infty$.

By the second property, $A_{t}$ is contained in a thinner and thinner part of
$C_{t}$ as $t\to\infty$. Then the developing map of $A_{t}$ is the restriction
of $\exp\colon\mathbb{C}\to\mathbb{C}^{\ast}$ to a bi-infinite strip $T_{t}$,
i.e. a region in $\mathbb{C}$ bounded by a pair of parallel lines. Then its
deck transformation group ($\cong\mathbb{Z}$) is generated by a translation of
$T_{t}$. Then $A_{t}$ has a natural Euclidean metric by identifying
$\mathbb{C}$ with $\mathbb{E}^{2}$.

Pick a cylinder $A$ in $S$ homotopic to $m$, and fix a finite volume Euclidean
structure on $A$ with geodesic boundary (by picking a homeomorphism
$A\to\mathbb{S}^{1}\times[-1,1]$).

Pick a marking $\iota_{t}\colon S\to C_{t}$ such that

  * •

$\iota_{t}$ takes $A$ to $A_{t}$ ((1));

  * •

the restriction of $C_{t}$ to $\iota_{t}(S\setminus A)$ converges to
$C_{\infty}$;

  * •

$\iota_{t}|A$ is linear with respect to the Euclidean structures on $A$ and
$A_{t}$.

Given a component $F$ of $\tilde{S}\setminus\phi^{-1}(A)$, suppose that
$f_{t_{i}}|F$ converges to a developing map of the component of $C_{\infty}$.
Then, accordingly, $\beta_{i}\circ\tilde{\kappa}_{i}|F$ converges to a pleated
surfacer for the corresponding component of $C_{\infty}$. By Proposition
12.10, if $f_{t_{i}}|F$ does not converge to a developing map, then
$\beta_{i}\circ\tilde{\kappa}_{i}|F$ converges to a constant map. Thus we have
(3).

We can assume that the most of the twisting of $C_{t}$ along $m$ (given by
Theorem 12.1(2) ) occurs in $A_{t}$.

(Case One) Suppose that
$a_{\infty}:=\lim_{i\to\infty}\operatorname{Axis}\rho_{t_{i}}(m)$ is a
biinfinite geodesic. We have already fixed a Euclidean structure on the
cylinder $A$. Pick a connected fundamental domain $D$ in the universal cover
$\tilde{S}$ so that $D\cup\phi^{-1}(A)$ is single a single parallelogram $P$
which is a fundamental domain for $\tilde{A}$. Let $e$ and $h$ be the opposite
edges of $P$ which map onto different boundary circles of $A$. Then
$f_{t_{i}}(e)$ and $f_{t_{i}}(h)$ are arcs contained in a $G_{i}$-invariant
curve on $\mathbb{C}{\rm P}^{1}$, and, by Lemma 12.13, they converge to the
different endpoints of $a_{\infty}$. The four edges of
$P_{i}\subset\mathbb{C}$ are asymptotically parallel as $i\to\infty$ as the
twisting parameter diverges to infinity (i.e. the angles at the vertices
converges to either $0$ or $\pi$). We may, in addition, assume that
$\sup\\{\,\operatorname{Im}(z-w)\,|\,z,w\in P_{i}\,\\}\to 0$ as $i\to\infty$.
The complex length of $e$ and $h$ converges to an integer multiple of
$\pi/\sqrt{2}$ as $i\to\infty$ as $\rho_{t}(m)\to I$.

Recall that $\rho_{t_{i}}(m)$ is hyperbolic if $i$ is large enough. Then the
foliation of $\mathbb{C}$ by horizontal lines induces a foliation
$\mathcal{F}_{i}$ of $P_{i}$.

######  Claim 12.19.

For every $\epsilon>0$ is $i\in\mathbb{Z}_{>0}$ is sufficiently large, then,
for each leaf $s$ of $\mathcal{F}_{i}$, $\beta_{i}\circ\kappa_{i}(s)$ has
diameter less than $\epsilon$ and it is contained in the
$\epsilon$-neighborhood of the geodesic $\operatorname{Axis}\rho_{i}(m)$.

###### Proof.

Note the distance of $A_{t}$ from zeros of $q_{t}$ diverges to $\infty$ as
$t\to\infty$. By Lemma 2.4 (1), the diameter of each arc is less than
$\epsilon>0$.

Recall that, as the convergence $C_{t}\to C_{\infty}$, by Theorem 12.1(3c),
$f_{i}(e)$ and $f_{i}(h)$ converge to the end points of the limit axis
$a_{\infty}$. Then by Proposition 12.14, $\beta_{i}\circ\kappa_{i}(e)$ and
$\beta_{i}\circ\kappa_{i}(h)$ are $\epsilon$-close to the axis $a_{i}$ of
$\rho_{t_{i}}(m)$, by taking an appropriate $A_{t}$. Then, if a leaf $s$ of
the horizontal foliation $\mathcal{F}_{t_{i}}$ in $P_{i}$ has an endpoint at
the edge $e$ or $h$, then $\beta_{i}\circ\kappa_{i}(s)$ is $\epsilon$-close to
$\beta_{i}\circ\kappa_{i}(e)$ or $\beta_{i}\circ\kappa_{i}(h)$. Thus, as
$\sup\\{|\operatorname{Im}(z-w)|\leavevmode\nobreak\ \leavevmode\nobreak\
|\,z,w\in P_{i}\\}\to 0$ as $i\to\infty$, by the equivariant property, all
leaves of $\mathcal{F}_{t_{i}}$ have this property. (Figure 16) ∎

\begin{overpic}[scale={.3}]{Figure/TwistingTrivialNeckII} \end{overpic} Figure
15.

Then by Claim 12.19 implies (4)

\begin{overpic}[scale={.3}]{Figure/TwistingPleatedSurfaceAlongParabolicNeck}
\end{overpic} Figure 16.

(Case Two) Suppose that $a_{\infty}$ is a single point on $\mathbb{C}{\rm
P}^{1}$. Then, by Proposition 12.11, $f_{\infty}(\ell)=f_{\infty}(r)$.

Pick any horoball $B$ in $\mathbb{H}^{3}$ tangent at $a_{\infty}$. For each
$i$, pick a subset $U_{i}\subset\mathbb{H}^{3}$ converging to $B$ uniformly on
compacts, such that, if $\rho_{i}(\gamma)$ is either hyperbolic or elliptic,
then $U_{i}$ is an $r_{i}$-neighborhood of $a_{i}$ for some $r_{i}>0$, and if
$\rho_{i}(\gamma)$ is parabolic, then $U_{i}$ is a horoball centered at the
parabolic fixed point of $\rho_{i}(\gamma)$.

For sufficiently large $i$, Let $N_{i}$ be the $\epsilon$-thin part of
$\tau_{i}$ homotopy equivalent to $m$. Let $\tilde{N}_{i}$ be a component of
$\psi_{i}^{-1}(N_{i})$, where $\psi_{i}\colon\mathbb{H}^{2}\to\tau_{i}$ be the
universal covering map.

######  Lemma 12.20.

If $\epsilon>0$ is sufficiently small, then $\beta_{i}(\tilde{N}_{i})$ is
eventually contained in $U_{i}$ as $i\to\infty$. Therefore,
$\beta_{i}\circ\kappa_{i}|\tilde{A}$ converges to the constant map to the
point $a_{\infty}$.

###### Proof.

Let $P$ and $R$ be the components of $\tilde{S}\setminus\phi^{-1}(A)$ adjacent
across a lift $\tilde{A}$ of $A$ which corresponds to $\tilde{N}_{i}$. If
$\beta_{i}(\tilde{N}_{i})$ is not eventually contained in $U_{i}$, then,
either $\rho_{i}|\operatorname{Stab}P$ or $\rho_{i}|\operatorname{Stab}R$
diverges to $\infty$, up to a subsequence, which is a contradiction. ∎

12.17

####  12.0.2. Convergence of holomorphic quadratic differentials when
$\rho_{\infty}(m)=I$

In the case that $\rho_{\infty}(m)=I$, the singular Euclidean structure
$E_{t_{i}}$ contains a flat cylinder $A_{t}$ homotopic to $m$, such that
$\operatorname{Mod}A_{t}\to\infty$ and the complex length of its circumference
converges to a positive multiple of $\pi/\sqrt{2}$, by Proposition 5.2.
Therefore

######  Proposition 12.21.

Let $C_{\infty}$ be the limit of $C_{t}$ in Theorem 12.1 (3b). Then, in the
Schwarzian parameters of $C_{\infty}$ are a Riemann surface with two punctures
homeomorphic to $S\setminus m$ and a holomorphic quadratic differential
$q_{\infty}$ which has a pole of order two at each puncture and their residues
are the same non-zero integer multiple of $\sqrt{2}\pi$.

###  12.1. Non-discreteness of holonomy

######  Theorem 12.22.

Suppose that $\rho_{\infty}(m)=I$. Then $\operatorname{Im}\rho_{t}\subset{\rm
PSL}(2,\mathbb{C})$ is a non-discrete subgroup for sufficiently large $t\gg
0$.

###### Proof.

Recall that $\rho_{t}(m)\to I$ but $\rho_{t}(m)\neq I$ (Theorem 12.1(1)). For
each component $F$ of $S\setminus m$, $\rho_{t}(\pi_{1}(F))$ is nontrivial for
sufficiently large $t\gg 0$ (Proposition 12.3). Thus the subgroup of
$\operatorname{Im}\rho_{t}$ generated by
$\\{\,\rho_{t}(m)\gamma\rho_{t}(m)^{-1}\,|\,\gamma\in\rho_{t}(F)\,\\}$ is non-
elementary (Lemma 12.5 and Proposition 12.6). As $\rho_{t}(m)\to I$, by the
Margulis lemma, $\operatorname{Im}\rho_{t}$ can not be discrete. ∎

##  13\. Examples of exotic degeneration

We construct examples of a path $C_{t}=(f_{t},\rho_{t})$ of $\mathbb{C}{\rm
P}^{1}$-structures pinched along a loop $m$ such that $\rho_{\infty}(m)=I$ and
$[\rho_{t}]$ converges in $\chi$ as $t\to\infty$, as in the second part of
Theorem 8.2. Indeed, we construct two examples: one with $\rho_{t}(m)$
hyperbolic and one with $\rho_{t}(m)$ elliptic for all $t\gg 0$.

###  13.1. Hyperbolic $\rho_{t}(m)$ converging to $I$.

Let $E$ be the singular Euclidean surface obtained from an $L$-shaped polygon
by identifying the opposite edges (Figure 17). Then $E$ has exactly one cone
point, and its cone angle is $6\pi$. Let $F$ be the underlying topological
surface of $E$, which is a closed surface of genus two. Let $E^{\prime}$
denote $E$ minus the cone point, and let $F^{\prime}$ denote the underlying
topological surface of $E^{\prime}$.

\begin{overpic}[scale={.1}]{Figure/LShape}
\put(25.0,60.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$a_{1}$}}
\put(72.0,20.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$a_{2}$}}
\put(40.0,52.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$b_{1}$}}
\put(50.0,14.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$b_{2}$}}
\end{overpic} Figure 17.

Let $\ell_{p}$ be the (oriented) peripheral loop around the removed cone
point. Let $\xi\colon\pi_{1}(F^{\prime})\to{\rm PSL}(2,\mathbb{C})$ be the
holonomy of $E$ minus the cone point; then, as $F^{\prime}$ has a Euclidean
structure, we can assume that its image consists of upper triangular matrices
with 1’s on the diagonal. In particular $\xi(\ell_{p})=I$ (as before, by abuse
of notation, we regard $\ell_{p}$ also as a fixed element of $\pi_{1}(S)$).
Notice that there is a point in the universal cover $\tilde{E}$ of $E$
corresponding to $\ell_{p}\in\pi_{1}(S)$.

######  Proposition 13.1.

There is a path of $\mathbb{C}{\rm P}^{1}$-structures,
$D_{t}=(h_{t},\xi_{t})$, on $F^{\prime}$ converging to $E=(h,\xi)$ as
$t\to\infty$, such that $\xi_{t}(\ell_{p})$ is a hyperbolic translation whose
axis converging to a geodesic connecting the parabolic fixed point of
$\xi_{\infty}$ and the $h$-image of the corresponding singular point of
$\tilde{E}$.

Note that elements of $\operatorname{Im}\xi$ are translations of $\mathbb{C}$.
Pick non-separating simple closed curves $a_{1},b_{1},a_{2},b_{2}$ on $E$ (as
in Figure 17) forming a standard generating set of $\pi_{1}(F)$ so that

  * •

for each $i=1,2,$ $a_{i}$ and $b_{i}$ intersect in a single point, and
$[a_{1},b_{1}][a_{2},b_{2}]=I$,

  * •

the translation directions of $a_{1}$ and $a_{2}$ are the same and the
translation direction of $b_{1}$ and $b_{2}$ are the same, and

  * •

the translation directions of $a_{i}$ and $b_{i}$ are orthogonal for each
$i=1,2$.

Let $c$ be a separating loop on $E$ which separates $\\{a_{1},b_{1}\\}$ and
$\\{a_{2},b_{2}\\}$. Then let $F_{1}$ and $F_{2}$ be the components of
$F\setminus c$ which are homeomorphic to a torus minus a disk.

######  Lemma 13.2.

Let $q_{i}$ be any geodesic in $\mathbb{H}^{3}$ starting from the parabolic
fixed point $p\in\mathbb{C}{\rm P}^{1}$ of $\operatorname{Hol}E$, and let
$H_{i}$ be the hyperbolic plane, in $\mathbb{H}^{3}$, containing an $\langle
a_{i}\rangle$-orbit of $q_{i}$. For each $i=1,2$, given any path
$h_{i,t}\,(t\geq 0)$ of hyperbolic elements in ${\rm PSL}(2,\mathbb{C})$ such
that

  1. (1)

the axis of $h_{i,t}$ is orthogonal to $H_{i}$ at a point in $q_{i}$ for all
$t\geq 0$, and

  2. (2)

$h_{t,i}\to I$ as $t\to\infty$.

Then, there is a path $\zeta_{i,t}\colon\pi_{1}(F_{i})\to{\rm
PSL}(2,\mathbb{C})$ of homomorphisms converging to
$\operatorname{Hol}(E)|\pi_{1}(F_{i})$ such that $\zeta_{i,t}(c)=h_{i,t}$.

###### Proof.

The point $p$ is contained in the ideal boundary of $H_{i}$. Let $r_{i}$ be a
geodesic in $H_{i}$, such that $R(r_{i})R(q_{i})=\xi(a_{i})$, where $R(r_{i})$
and $R(q_{i})$ are the $\pi$-rotations of $\mathbb{H}^{3}$ about $r_{i}$ and
$q_{i}$, respectively (Figure 18).

\begin{overpic}[scale={.5}]{Figure/PathsOfAxies}
\put(22.0,15.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}$q_{t,i}$}}
\put(33.0,16.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}$q_{t,i}^{\prime}$}}
\put(12.0,20.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}$r_{t,i}$}}
\put(86.0,24.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}$q_{i}$}}
\put(69.0,20.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}$r_{i}$}}
\par\put(32.0,50.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$p$}}
\put(60.0,30.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}$H_{i}$}}
\put(78.0,16.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}$H_{i}^{\perp}$}}
\end{overpic} Figure 18.

Let $H_{i}^{\perp}$ be the hyperbolic plane in $\mathbb{H}^{3}$ orthogonal to
$H_{i}$ in $q_{i}$. As $\operatorname{Axis}(h_{i,t})$ is in $H_{i}$ and
orthogonal to $q_{i}$, we let $q_{i,t},q_{i,t}^{\prime}$ be continuous paths
of geodesics in $H_{i}^{\perp}$ such that
$R(q_{i,t})R(q_{i,t}^{\prime})=h_{i,t}$, the geodesics $q_{i,t}$ and
$q_{i,t}^{\prime}$ converge to $q_{i}$ as $t\to\infty$, and the $\pi$-rotation
$R(q_{i})$ exchanges $q_{i,t}$ and $q_{i,t}^{\prime}$. There is a path of
geodesics $r_{i,t}$ such that, for all $t\gg 0$,

  * •

there is a hyperbolic plane intersecting $r_{t,i},q_{t,i},q_{t,i}^{\prime}$
orthogonally, and

  * •

$d_{\mathbb{H}^{3}}(r_{t,i},q_{t,i})=d_{\mathbb{H}^{3}}(r_{t,i},q_{t,i}^{\prime})$.

Thus by the symmetry,
$\operatorname{tr}R(q_{i,t})R(r_{i})=\operatorname{tr}R(q_{i,t}^{\prime})R(r_{i})\in\mathbb{R}\setminus[-2,2]$.

The surface $F_{i}\setminus a_{i}$ is a pair of pants, and two of its boundary
components correspond to $a_{i}$. Consider the path of homomorphisms
$\zeta_{i,t}\colon\pi_{1}(F_{i}\setminus a_{i})\to{\rm PSL}(2,\mathbb{C})$ for
$t>0$, such that the two boundary components corresponding to $a_{i}$ map to
$R(q_{i,t})R(r_{i})$ and $R(r_{i})R(q_{i,t}^{\prime})$— thus the other
boundary corresponding to $\partial F_{i}$ maps to
$R(q_{i,t})R(q_{i,t}^{\prime})=h_{i,t}$ (see [Gol09]). Then, by Theorem 5.7,
there is a path of $\mathbb{C}{\rm P}^{1}$-structures on $F_{i}\setminus
a_{i}$ with holonomy $\zeta_{i,t}$ which converges to its component of
$E\setminus(c\cup a_{i})$ as $t\to\infty$ corresponding to $F_{i}\setminus
a_{i}$. As the holonomy along the two boundary components are conjugate, for
large enough $t>0$, there is a path of $\mathbb{C}{\rm P}^{1}$-structures
$\Sigma_{i,t}$ on $F_{i}$ which converges to the component of $E\setminus c$,
so that $\operatorname{Hol}\Sigma_{i,t}|\pi_{1}F_{i}=\zeta_{i,t}$. In
particular, the holonomy of $\Sigma_{i,t}$ round the puncture is the
hyperbolic element $R(q_{i,t})R(q_{i,t}^{\prime})=h_{i,t}$. ∎

\begin{overpic}[scale={.07}]{Figure/ComposingSmallHyperbolicsIl}
\put(37.0,25.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@cmyk@stroke{0}{1}{1}{0}\pgfsys@color@cmyk@fill{0}{1}{1}{0}$v_{1,t}$}}
\put(70.0,24.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@cmyk@stroke{0}{1}{1}{0}\pgfsys@color@cmyk@fill{0}{1}{1}{0}$u_{1,t}$}}
\put(58.0,25.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}$v_{2,t}$}}
\put(23.0,21.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}$u_{2,t}$}}
\put(52.0,2.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@cmyk@stroke{0}{0}{0}{1}\pgfsys@color@cmyk@fill{0}{0}{0}{1}$q$}}
\end{overpic} Figure 19.

Notice that $H_{1}$ and $H_{2}$ are totally geodesic hyperbolic planes in
$\mathbb{H}^{3}$ tangent at $p$. We in addition assume that $H_{1}$ and
$H_{2}$ are different and $H_{1}^{\perp}=H_{2}^{\perp}\eqqcolon H$. Pick a
geodesic $q$ in $H$ initiating from $p$ contained in the region bounded by the
geodesics $q_{1}=H\cap H_{1}$ and $q_{2}=H\cap H_{2}$.

######  Proposition 13.3.

We can choose, the path of the hyperbolic isometries $h_{1,t},h_{2,t}$ (given
by Lemma 13.2) so that their composition $h_{1,t}h_{2,t}$ is eventually a
hyperbolic element whose axis is $q$ as $t\to\infty$.

###### Proof.

Pick $h_{1,t}$ and $h_{2,t}$ such that their axes converge to the parabolic
fixed point $p$. Since $h_{1,t}$ and $h_{2,t}$ converge to $I$, their product
$h_{1,t}h_{2,t}$ also converges to $I$ in ${\rm PSL}(2,\mathbb{C})$.

For each $i=1,2$, let $u_{i,t}$ be the attracting fixed and $v_{i,t}$ be the
repelling fixed point of $h_{i,t}$. We may first assume that the endpoints of
$\operatorname{Axis}h_{1,t},\operatorname{Axis}h_{2,t}$ lie on $\partial H$ in
this cyclic order $u_{2,t},v_{1,t},v_{2,t},u_{1,t}$ (Figure 19). The
composition $h_{1,t}h_{2,t}$ fixes a point on the arc in $\partial H$ between
$v_{1,t}$ and $v_{2,t}$ for each $t>0$. Note that the segment contains $p$.
Then as $\operatorname{Axis}(h_{1,t}),\operatorname{Axis}(h_{2,t})$ converge
to the parabolic fixed point $p$, there is a fixed point of $h_{1,t}h_{2,t}$
converging to $p$. Moreover, as $h_{1,t}\to I$, one can continuously adjust
the translation length of $h_{2,t}$ so that $h_{1,t}h_{2,t}$ also fixes the
other endpoint of $q$ for sufficiently large $t>0$.

Therefore $h_{1,t}h_{2,t}$ is a hyperbolic element with the axis $q$ as
$t\to\infty$. ∎

Let $h_{1,t},h_{2,t}\in{\rm PSL}(2,\mathbb{C})$ be the paths given by
Proposition 13.3. Then, by Lemma 13.2, for each $i=1,2$, we have a path of
homomorphisms $\zeta_{i,t}\colon\pi_{1}(F_{i})\to{\rm PSL}(2,\mathbb{C})$ such
that $\zeta_{i,t}(\ell)=h_{i,t}$ for $t\gg 0$. Then there is a unique path
$\zeta_{t}\colon\pi(F^{\prime})\to{\rm PSL}(2,\mathbb{C})$ so that
$\zeta_{t}|\pi_{1}(F_{i})=\zeta_{i,t}$ for $i=1,2$; thss
$\zeta_{t}(\ell_{p})=h_{1,t}h_{2,t}$. Then there is a path $D_{t}$ of
$\mathbb{C}{\rm P}^{1}$-structures on $F^{\prime}$ with holonomy $\zeta_{t}$
for $t\gg 0$ such that $D_{t}$ converges to $E^{\prime}$ as $t\to\infty$.

####  13.1.1. Constructing a closed surface from punctures surfaces

In order to make a desired example of exotic degeneration, we take two copies
$D_{t}$ of $\mathbb{C}{\rm P}^{1}$-surfaces with a single puncture from the
previous section, and glue them together with a lot of twists.

######  Theorem 13.4.

There is a path of $\mathbb{C}{\rm P}^{1}$-structures $C_{t}=(f_{t},\rho_{t})$
on a closed surface $S$ of genus four with following properties:

  * •

The conformal structure $X_{t}$ is pinched along a separating loop $m$; let
$F_{1}$ and $F_{2}$ be the connected components of $S\setminus m$.

  * •

$\rho_{t}\colon\pi_{1}(S)\to{\rm PSL}(2,\mathbb{C})$ converges in the
representation variety as $t\to\infty$.

  * •

Pick an element $\gamma\in\pi_{1}(S)$ whose free homotopy class is $m$. Then
$\rho_{\infty}(\gamma)=I$, and, for all $t>0$, the holonomy $\rho_{t}(\gamma)$
is a hyperbolic element such that its axis $a_{t}$ converges to a geodesic
$a_{\infty}$ in $\mathbb{H}^{3}$ as $t\to\infty$.

  * •

Let $\tilde{F}_{1},\tilde{F}_{2}$ be the connected components of
$\tilde{S}\setminus\phi^{-1}(m)$ which are adjacent across the lift
$\tilde{m}$ of $m$ invariant under $\gamma$. Then $C_{t}|\tilde{F}_{1}$
converges to the developing map of a $\mathbb{C}{\rm P}^{1}$-structure on a
genus two surface minus a point such that the puncture maps to an endpoint of
$a_{\infty}$ as $t\to\infty$.

  * •

$f_{t}|\tilde{F}_{2}$ converges to the constant map to the other endpoint of
$a_{\infty}$ uniformly on compacts, and $\rho_{\infty}|\pi_{1}(F_{2})$ is the
trivial representation.

###### Proof.

Recall that $\ell_{p}$ be the peripheral loop around $p$. Recall that the
holonomy of $E^{\prime}$ is parabolic, and the holonomy around the puncture of
$D_{t}$ is a hyperbolic element whose axis $a_{t}$ is converging to a geodesic
starting from the parabolic fixed point of $\operatorname{Hol}E^{\prime}$ to
the image of the puncture of $\tilde{E}$ corresponding to $\ell_{p}$.

For sufficiently large $t>0$, there is a cusp neighborhood $N_{t}$ of $D_{t}$
such that $N_{t}$ is foliated by admissible loops whose developments are
invariant under a one-dimensional subgroup $G_{t}$ of ${\rm
PSL}(2,\mathbb{C})$ containing $\xi_{t}(\ell_{p})$. We can assume that $N_{t}$
changes continuously in $t$ and asymptotically the empty set on $E$ as
$t\to\infty$. As $G_{t}$ is a one-dimensional subgroup $\rho_{t}(m)$ of ${\rm
PSL}(2,\mathbb{C})$, integer powers $\rho_{t}(m)^{n}$ for $n\in\mathbb{Z}$
continuously extends to real powers.

Let $\tilde{E}^{\prime}$ be the universal cover of $E^{\prime}$. Pick any path
of compact subsets $K_{t}$ of the universal cover $\tilde{D}_{t}$ converges to
a compact subset in $\tilde{E}^{\prime}$. Then there is a continuous function
$r_{t}\in\mathbb{R}$ diverging to $\infty$ such that

  * •

$\rho_{t}(m)^{r_{t}}\operatorname{dev}D_{t}|K_{t}$ is asymptotically the
constant map to the fixed point of $\rho_{t}(m)$ corresponding to the singular
point of $\tilde{E}$,

  * •

$\rho_{t}(m)^{r_{t}}\cdot\operatorname{Hol}(D_{t})$ is asymptotically the
trivial representation, as $t\to\infty$.

Take two copies $\Sigma_{1,t},\Sigma_{2,t}$ of $D_{t}\setminus N_{t}$ and glue
them together along their boundary components so that the amount of twisting
is $r_{t}$. Then we obtain a continuous family of the resulting
$\mathbb{C}{\rm P}^{1}$-structures $C_{t}$ on $S$. By the properties of
$r_{t}$, we can normalize the developing pair of $C_{t}$ so that

  * •

$C_{t}|\Sigma_{t,1}$ converges to a developing pair of $E^{\prime}$, and

  * •

the developing pair of $C_{t}|\Sigma_{t,2}$ converges to a pair of the
constant map (uniformly on compacts) and the trivial representation.

Thus we obtained a desired path of $\mathbb{C}{\rm P}^{1}$-structures. ∎

###  13.2. Elliptic $\rho_{t}(m)$ converging to the identity

In this section, we construct an example of $C_{t}=(f,\rho_{t})$ in Theorem C
such that $\rho_{t}(m)$ is an elliptic element for all sufficiently large
$t\geq 0$ and it converges to $I$ as $t\to\infty$.

Given an elliptic element $e\in{\rm PSL}(2,\mathbb{C})$, normalize the unite
disk model $\mathbb{D}^{3}\subset\mathbb{R}^{3}$ of $\mathbb{H}^{3}$ centered
at the origin, so that $\operatorname{Axis}(e)$ is contained in the axis of
the third coordinate. Let $\zeta\in(0,2\pi)$ be the rotation angle of $e$.
Then define $b_{e}\colon\mathbb{R}\to\partial\mathbb{H}^{3}$ by
$x\mapsto(\cos(\zeta x)\sin\zeta,\sin(\zeta x)\sin\zeta,\cos\zeta)$ which is
equivariant under $\mathbb{Z}\to\langle e\rangle$.

######  Lemma 13.5.

Let $r$ be a geodesic in $\mathbb{H}^{3}$. Pick a parallel vector field
$V\subset T\mathbb{H}^{3}$ along $r$ such that $V$ is orthogonal to $r$. Then
there are a path of (nontrivial) elliptic elements $e_{t}\in{\rm
PSL}(2,\mathbb{C})$ and a continuous function $\theta_{t}\in\mathbb{R}_{\geq
0}$ satisfying the following:

  * •

$e_{t}\to I$ as $t\to\infty$.

  * •

$\operatorname{Axis}(e_{t})$ orthogonally intersects $r$, and
$\operatorname{Axis}(e_{t})$ converges to an endpoint of $r$ on
$\mathbb{C}{\rm P}^{1}$ as $t\to\infty$.

  * •

Letting $\theta_{t}\in\mathbb{R}$ be a continuous function such that the angle
between $\operatorname{Axis}e_{t}$ (oriented) and $V$ is $\theta_{t}\mod
2\pi$.

  * •

Let $u_{t}=2\theta_{t}$. Then the rotation angle of $e_{t}^{u_{t}}$ is $\pi$
for all $t\geq 0$ (so that $e_{t}^{u_{t}}$ takes $r$ to itself with the
opposite orientation).

###### Proof.

It is easy to construct an example satisfying the first three conditions. Then
adjust the rotation angle of $e_{t}$ so that it also satisfies the last
condition.  ∎

######  Lemma 13.6.

Let $e_{t}$ be as in Lemma 13.5. Let $p$ be the endpoint of $r$ to which
$\operatorname{Axis}(e_{t})$ converges. Pick a round disk $D$ in
$\mathbb{C}{\rm P}^{1}$ containing $p$ but not the other endpoint of $r$, so
that $D\setminus\\{p\\}$ is a $\mathbb{C}{\rm P}^{1}$-structure on a punctured
disk. Then there is a path $A_{t}$ of $\mathbb{C}{\rm P}^{1}$-structures on an
annulus $A$ with smooth boundary for sufficiently large $t\gg 0$, such that

  * •

$A_{t}$ converges to $D\setminus\\{p\\}$, and

  * •

letting $\ell_{t}$ be the boundary component of $A_{t}$ converging to $p$, the
development of $\ell_{t}$ is equivalent to $b_{e_{t}}$.

###### Proof.

For sufficiently large $t>0$, one can easily construct the fundamental
membrane for $A_{t}$ for sufficiently large $t>0$ (Figure 20).

\begin{overpic}[scale={.35}]{Figure/AlmostPuncturedDisk} \end{overpic} Figure
20.

∎

######  Lemma 13.7.

Let $P$ be a pair of pants, and let $\ell$ be a boundary component of $P$. Let
$\tilde{\ell}$ be a lift of $\ell$ to the universal cover of $P$. Consider a
(flat) Euclidean cylinder with geodesic boundary, and let $P_{\infty}$ be the
surface obtained by removing an interior point $p$ of $P_{\infty}$. Then
regard $P_{\infty}$ as a $\mathbb{C}{\rm P}^{1}$-structure on $P$, and let
$(h,\xi)$ be its developing pair, so that $h$ takes $\tilde{\ell}$ to a single
point $v$ on $\mathbb{C}{\rm P}^{1}$. Let $r$ be a geodesic in
$\mathbb{H}^{3}$ connecting $v$ and the parabolic fixed point of $h$, and let
$e_{t}\in{\rm PSL}(2,\mathbb{C})$ be a path of (non-trivial) elliptic elements
given by Lemma 13.5.

Then there is a path of $\mathbb{C}{\rm P}^{1}$-structures
$P_{t}=(h_{t},\xi_{t})$ on $P$ satisfying the following:

  1. (1)

For all $t>0$, $\xi_{t}(\ell)=e_{t}$.

  2. (2)

$P_{t}$ converges to $P_{\infty}$ as $t\to\infty$. Let $\gamma_{t}\in{\rm
PSL}(2,\mathbb{C})$ be a path of hyperbolic elements with the axis $r$, such
that $\gamma_{t}\operatorname{Axis}(e_{t})$ converges to a geodesic
$g_{\infty}$ in $\mathbb{H}^{3}$ orthogonal to $r$ as $t\to\infty$. Let $H$ be
the hyperbolic plane orthogonal to $r$ which contains $g_{\infty}$. Then the
modified developing pair $\gamma_{t}(h_{t},\xi_{t})$ converges to a developing
pair for a disk minus a point, where the removed point is the attracting fixed
point $p$ of $\gamma_{t}$ and the disk is the component of $\mathbb{C}{\rm
P}^{1}\setminus\partial H$ containing $p$.

  3. (3)

Let $\ell_{t}$ be the boundary component of $P_{t}$ corresponding to $\ell$.
Then $\operatorname{dev}P_{t}$ along a lift of $\ell_{t}$ is $b_{\ell_{t}}$
(up to ${\rm PSL}(2,\mathbb{C})$).

  4. (4)

Let $\alpha$ be a boundary component of $P$ not equal to $\ell$. Then
$\xi_{t}(\alpha)$ is hyperbolic element for all $t\gg 0$ (converging to a
parabolic element as $t\to\infty$).

###### Proof.

First we construct an appropriate path of representations
$\xi_{t}\colon\pi_{1}(P)\to{\rm PSL}(2,\mathbb{C})$. Let $a_{t}$ denote
$\operatorname{Axis}(e_{t})$. Pick a pair of geodesics $q_{t},q_{t}^{\prime}$
in $\mathbb{H}^{3}$ for each $t>0$ such that

  * •

$R(q_{t})R(q_{t}^{\prime})=e_{t}$, where $R(q_{t}),R(q_{t}^{\prime})\in{\rm
PSL}(2,\mathbb{C})$ are the $\pi$-rotations of $\mathbb{H}^{3}$ about
$q_{t},q_{t}^{\prime}$, respectively.

  * •

$q_{t}$ and $q_{t}^{\prime}$ change continuously in $t>0$,

  * •

$q_{t}$ and $q_{t}^{\prime}$ intersect at the intersection $a_{t}\cap r$,

  * •

$q_{t}$ and $q_{t}^{\prime}$ are symmetric about $r$,

  * •

$q_{t}$ and $q_{t}^{\prime}$ are orthogonal to $a_{t}$, and

  * •

$q_{t}$ and $q_{t}^{\prime}$ converge to $p$ as $t\to\infty$ (see Figure 21).

There is a path of geodesics $h_{t}\,(t\geq 0)$ in $\mathbb{H}^{3}$ such that

  * •

$h_{t}$ is disjoint from $q_{t}$ and $q_{t}^{\prime}$ for all $t\geq 0$, and

  * •

$h_{t}$ converges to a geodesic $h$ in $\mathbb{H}^{3}$ sharing an endpoint
with $r$ as $t\to\infty$, such that the composition $R(r)R(h)$ is the
parabolic holonomy along a boundary geodesic of $P_{\infty}$.

Let $\xi_{t}\colon\pi_{1}(P)\to{\rm PSL}(2,\mathbb{C})$ be such that the
holonomy along boundary components are
$R(h_{t})R(q_{t}),R(q_{t})R(q_{t}^{\prime}),R(q_{t}^{\prime})R(h_{t})$. Note
that $R(h_{t})R(q_{t}),R(q_{t}^{\prime})R(h_{t})$ are hyperbolic elements, as
the rotation axes are disjoint, and they converge to the parabolic holonomy
along the boundary geodesics of $P_{\infty}$.

Pick a round disk $D$ on $P_{\infty}$ containing $p$ such that $\partial D$ on
$\mathbb{C}{\rm P}^{1}$ bounds a hyperbolic plane in $\mathbb{H}^{3}$
orthogonal to $r$. Then, apply Lemma 13.6 to $D$, let $D_{t}$ be a path of
$\mathbb{C}{\rm P}^{1}$-structures on an annulus converging to
$D\setminus\\{p\\}$, so that it gives the desired path only near the punctured
of $P_{\infty}$.

Pick a smaller closed regular neighborhood of the puncture $p$ of
$P_{\infty}$, so that its complement $K$ in $P_{\infty}$ and the interior of
$D\setminus\\{p\\}$ form an open cover of $P_{\infty}$. Then $K$ is
topologically a pair of pants. There is a path of $\mathbb{C}{\rm
P}^{1}$-structures on a pair of pants $K_{t}$ for sufficiently large $t>0$
such that $K_{t}$ converges to $K$ and $e_{t}$ is the holonomy of $K_{t}$
around the boundary component corresponding to $\ell$.

Then, by gluing $K_{t}$ and $D_{t}$, we obtained a desired path of
$\mathbb{C}{\rm P}^{1}$-structures $P_{t}$.

\begin{overpic}[scale={.2}]{Figure/GeneratingEllipticNeck}
\put(37.0,30.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}$q_{t}$}}
\put(55.0,30.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}$q_{t}^{\prime}$}}
\put(49.0,18.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@cmyk@stroke{0}{0}{0}{1}\pgfsys@color@cmyk@fill{0}{0}{0}{1}$r$}}
\put(32.0,72.0){{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}$a_{t}$}}
\put(66.0,55.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@cmyk@stroke{0}{0}{0}{1}\pgfsys@color@cmyk@fill{0}{0}{0}{1}$\mathbb{H}^{3}$}}
\end{overpic} Figure 21. Realize $e_{t}$ as the compositions of the
$\pi$-rotations about $q_{t}$ and $q_{t}^{\prime}$.

∎

######  Proposition 13.8.

Let $P_{t}=(h_{t},\xi_{t})$ be a path of $\mathbb{C}{\rm P}^{1}$-structures on
a pair of pants from Lemma 13.7. Then there is a path $\Sigma_{t}$ of
$\mathbb{C}{\rm P}^{1}$-structures on a closed surface $F$ minus a point which
satisfies the following:

  * •

There is a subsurface $A$ of $F$ whose interior contains $p$, such that $A$ is
homeomorphic to a pair of pants, and $\Sigma_{t}|A=P_{t}$ for large enough
$t>0$.

  * •

$\Sigma_{t}$ converges to a $\mathbb{C}{\rm P}^{1}$-structure $F_{\infty}$ on
$F$ as $t\to\infty$.

###### Proof.

First we construct the limit structure $F_{\infty}$. Take any complete
hyperbolic surface $\tau$ with a single cusp, such that $\tau$ is homeomorphic
to a closed surface minus a point, denoted by $F^{\prime}$. Pick a cusp
neighborhood $N$ of $\tau$ which is a horoball quotient. The pair of pants
$P_{\infty}$ has two boundary components and one puncture. As the two boundary
components of $P_{\infty}$ lift to horocycles, we can glue a copy of
$\tau\setminus N$ along each boundary component of $P_{\infty}$. We thus
obtained a $\mathbb{C}{\rm P}^{1}$-structure on a closed surface with a single
puncture so that $P_{\infty}$ is its subsurface.

There are paths $\zeta_{1,t}$ and $\zeta_{2,t}$ of representations
$\pi_{1}(\tau)\to{\rm PSL}(2,\mathbb{C})$ which converge to the holonomy of
$\tau$ as $t\to\infty$, such that their images of the peripheral loop are
$R(r_{t})R(q_{t}^{\prime})$ and $R(q_{t})R(r_{t})$, respectively, which are
hyperbolic elements (c.f. [Gol09]). Let $\tau_{1,t},\tau_{2,t}$ be paths of
$\mathbb{C}{\rm P}^{1}$-structures homeomorphic to $\tau\setminus N$ for $t\gg
0$ such that $\operatorname{Hol}(\tau_{1,t})=\zeta_{1,t}$, and
$\operatorname{Hol}(\tau_{2,t})=\zeta_{2,t}$ and $\tau_{1,t},\tau_{2,t}$
converge to $\tau\setminus N$. We may in addition assume that the boundary
components of $\tau_{1,t},\tau_{2,t}$ are invariant under one-dimensional
subgroups of ${\rm PSL}(2,\mathbb{C})$ containing $R(r)R(q_{t}^{\prime})$ and
$R(q_{t})R(r)$, respectively.

Then by gluing $\tau_{t,1},\tau_{t,2},P_{t}$ along their boundary, we obtain a
desired path $\Sigma_{t}$ of $\mathbb{C}{\rm P}^{1}$-structures. ∎

Let $\Sigma_{t}$ be the path of $\mathbb{C}{\rm P}^{1}$ -structures, obtained
from Proposition 13.8, on a compact surface with one boundary component. Let
$R_{t}$ be the $\pi$-rotation of $\mathbb{H}^{3}$ around the axis $a_{t}$ of
the elliptic $e_{t}$. By Lemma 13.7(2, 3) , we can glue two copies of
$\Sigma_{t}$ by the involution $R_{t}$, and we obtain a path of
$\mathbb{C}{\rm P}^{1}$-structures $C_{t}$ on a closed surface, so that two
copies of $\Sigma_{t}$ are embedded in $C_{t}$ disjointly up to an isotopy.
Let
$\Sigma_{t}^{1}=(h_{t}^{1},\rho_{t}^{1}),\Sigma^{2}_{t}=(h_{t}^{1},\rho_{t}^{2})$
are the subsurfaces of $C_{t}$ corresponding to $\Sigma_{t}$. Then we have, by
Lemma 13.5, we have

######  Proposition 13.9.

Let $C_{t}=(f_{t},\rho_{t})$ be the path of $\mathbb{C}{\rm P}^{1}$-structures
as above, and let $m$ be the loop on $C_{t}$ corresponding to the boundary
components of $\Sigma_{t}^{1}$ and $\Sigma_{t}^{2}$. Let $N$ be the regular
neighborhood of $m$. Then

  * •

$\rho_{t}(m)$ converges to $I$ as $t\to\infty$, and $\rho_{t}(m)$ is an
elliptic element for all $t>0$;

  * •

the axis of $\rho_{t}(m)$ converges to the point $p$ of $\mathbb{C}{\rm
P}^{1}$;

  * •

$f_{t}\colon\tilde{S}\setminus\phi^{-1}(N)\to\mathbb{C}{\rm P}^{1}$ converges
to a $\rho_{\infty}$-equivariant continuous map
$f_{\infty}\colon\tilde{S}\setminus\phi^{-1}(N)\to\mathbb{C}{\rm P}^{1}$, such
that $f_{\infty}$ is a local homeomorphism in the interior;

  * •

for each connected component $\tilde{N}$ of $\phi^{-1}(N)$, the boundary
components of $\tilde{N}$ map to the corresponding limit $p$ of the axis
$a_{t}$.

\begin{overpic}[scale={.2}]{Figure/RotatingAxis}
\put(90.0,20.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@cmyk@stroke{0}{0}{0}{1}\pgfsys@color@cmyk@fill{0}{0}{0}{1}$a_{t}$}}
\put(60.0,30.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@cmyk@stroke{0}{0}{0}{1}\pgfsys@color@cmyk@fill{0}{0}{0}{1}$e_{t}^{u_{t}}$}}
\put(14.0,30.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@cmyk@stroke{0}{0}{0}{1}\pgfsys@color@cmyk@fill{0}{0}{0}{1}$a_{t}$}}
\put(75.0,1.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@cmyk@stroke{0}{0}{0}{1}\pgfsys@color@cmyk@fill{0}{0}{0}{1}$\Sigma_{t}^{1}$}}
\put(69.0,46.0){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@cmyk@stroke{0}{0}{0}{1}\pgfsys@color@cmyk@fill{0}{0}{0}{1}$e_{t}^{-u_{t}}\Sigma_{t}^{2}$}}
\end{overpic} Figure 22. The rotation of the axis $a_{t}$ yields twist.

13.9

## References

  * [AB] Dylan Allegretti and Tom Bridgeland.  The monodromy of meromorphic projective structures.  Preprint, arXiv:1802.02505v1.
  * [Bab] Shinpei Baba.  On thurston’s parametrization of $\mathbb{C}{\rm P}^{1}$-structures.  Preprint, arXiv:1904.00588.
  * [Bab10] Shinpei Baba.  A Schottky decomposition theorem for complex projective structures.  Geom. Topol., 14(1):117–151, 2010.
  * [Bab17] Shinpei Baba.  $2\pi$-grafting and complex projective structures with generic holonomy.  Geom. Funct. Anal., 27(5):1017–1069, 2017.
  * [Ber74] Lipman Bers.  Spaces of degenerating Riemann surfaces.  pages 43–55. Ann. of Math. Studies, No. 79, 1974.
  * [Dum09] David Dumas.  Complex projective structures.  In Handbook of Teichmüller theory. Vol. II, volume 13 of IRMA Lect. Math. Theor. Phys., pages 455–508. Eur. Math. Soc., Zürich, 2009.
  * [Dum17] David Dumas.  Holonomy limits of complex projective structures.  Adv. Math., 315:427–473, 2017.
  * [Ear81] Clifford J. Earle.  On variation of projective structures.  In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume 97 of Ann. of Math. Stud., pages 87–99. Princeton Univ. Press, Princeton, N.J., 1981.
  * [Eps] Charles Epstein.  Envelopes of horospheres and weingarten surfaces in hyperbolic 3-space.  Preprint.
  * [FM12] Benson Farb and Dan Margalit.  A primer on mapping class groups, volume 49 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 2012.
  * [GKM00] Daniel Gallo, Michael Kapovich, and Albert Marden.  The monodromy groups of Schwarzian equations on closed Riemann surfaces.  Ann. of Math. (2), 151(2):625–704, 2000.
  * [GM] Subhojoy Gupta and Mahan Mj.  Meromorphic projective structures, grafting and the monodromy map.  Available at https://arxiv.org/abs/1904.03804.
  * [Gol87] William M. Goldman.  Projective structures with Fuchsian holonomy.  J. Differential Geom., 25(3):297–326, 1987.
  * [Gol09] William M. Goldman.  Trace coordinates on Fricke spaces of some simple hyperbolic surfaces.  In Handbook of Teichmüller theory. Vol. II, volume 13 of IRMA Lect. Math. Theor. Phys., pages 611–684. Eur. Math. Soc., Zürich, 2009.
  * [GW] Subhojoy Gupta and Michael Wolf.  Meromorphic quadratic differentials and measured foliations on a Riemann surface.  Math. Ann., 373.
  * [Hej75] Dennis A. Hejhal.  Monodromy groups and linearly polymorphic functions.  Acta Math., 135(1):1–55, 1975.
  * [HP04] Michael Heusener and Joan Porti.  The variety of characters in ${\rm PSL}_{2}(\mathbb{C})$.  Bol. Soc. Mat. Mexicana (3), 10(Special Issue):221–237, 2004.
  * [Hub81] John H. Hubbard.  The monodromy of projective structures.  In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume 97 of Ann. of Math. Stud., pages 257–275. Princeton Univ. Press, Princeton, N.J., 1981.
  * [Hub06] John Hamal Hubbard.  Teichmüller theory and applications to geometry, topology.  Matrix Editions, Ithaca, NY, 2006.
  * [Kap95] Michael Kapovich.  On monodromy of complex projective structures.  Invent. Math., 119(2):243–265, 1995.
  * [Kap01] Michael Kapovich.  Hyperbolic manifolds and discrete groups, volume 183 of Progress in Mathematics.  Birkhäuser Boston Inc., Boston, MA, 2001.
  * [KP94] Ravi S. Kulkarni and Ulrich Pinkall.  A canonical metric for Möbius structures and its applications.  Math. Z., 216(1):89–129, 1994.
  * [KT92] Yoshinobu Kamishima and Ser P. Tan.  Deformation spaces on geometric structures.  In Aspects of low-dimensional manifolds, volume 20 of Adv. Stud. Pure Math., pages 263–299. Kinokuniya, Tokyo, 1992.
  * [Luo93] Feng Luo.  Monodromy groups of projective structures on punctured surfaces.  Invent. Math., 111(3):541–555, 1993.
  * [LZ] John Loftin and Tengren Zhang.  Coordinates on the augmented moduli space of convex $\mathbb{R}{\rm P}^{2}$-structures.  Preprint, arXiv:1812.11389.
  * [New] Peter Newstead.  Geometric invariant theory 3-space.  CIMAT Lectures, 2006.
  * [Str84] Kurt Strebel.  Quadratic differentials, volume 5 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)].  Springer-Verlag, Berlin, 1984.
  * [Tan99] Harumi Tanigawa.  Divergence of projective structures and lengths of measured laminations.  Duke Math. J., 98(2):209–215, 1999.