of its pre-images give a partition of $d$. A branch data that arises in this
way is said to be realizable. The Hurwitz realization problem is to
characterize the branch data that are realizable. The Riemann-Hurwitz formula
gives obstructions to realizability. For a partition $A=[a_{1},\ldots,a_{l}]$,
let $v(A)=\sum_{i}(a_{i}-1)$, and for a branch data
$\mathcal{D}=\\{A_{1},\ldots,A_{k}\\}$, let $v(\mathcal{D})=\sum_{i}v(A_{i})$.
If $\mathcal{D}$ is realizable, then the Riemann-Hurwitz formula gives

$\chi(\Sigma)=2d-v(\mathcal{D}).$

In particular, $v(\mathcal{D})$ must be even and at least $2d-2$. However it
is well known that these conditions are not sufficient, see for example
[EKS84, PP06]. This problem was considered by Hurwitz in [Hur91] where he
showed the following theorem. We identify the conjugacy classes of the
symmetric group $\mathfrak{S}_{d}$ with the partitions of $\\{1,\ldots,d\\}$
as follows: the lengths of the orbits of the action of the group generated by
$\sigma\in\mathfrak{S}_{d}$ on $\\{1,\ldots,d\\}$ determine a partition of
$d$.

######  Hurwitz conditions.

The branch data $\mathcal{D}=\\{A_{1},\ldots,A_{k}\\}$ is realizable if and
only if there exists permutations $\alpha_{1}\in A_{1},\ldots\alpha_{k}\in
A_{k}$ such that

  1. (1)

$\alpha_{1}\alpha_{2}\ldots\alpha_{k}=\mathrm{id}$

  2. (2)

The group generated by the $\alpha_{i}$ acts transitively on
$\\{1,\ldots,d\\}$.

We show the following statement, that implies 6.1.

######  Proposition 6.2.

Let $d\geqslant 2$ and $1\leqslant n_{1}\leqslant\ldots\leqslant n_{k}<d$. The
branch data $\mathcal{D}=\\{A_{1},\ldots,A_{k}\\}$ where
$A_{i}=[n_{i}+1,1,\ldots,1]$ is realizable if and only
$v(\mathcal{D})=\sum_{i}n_{i}$ is even and $v(\mathcal{D})\geqslant 2d-2$.

We follow the reduction methods of [EKS84], and prove this proposition by
induction. Let us first check this proposition for $k=3$.

######  Lemma 6.3.

6.2 holds for $k=3$.

###### Proof.

We proceed by induction on $d$. If $d=2$ then $n_{1}=n_{2}=n_{3}=1$ and there
is no realizable branch data. Let us suppose that
$v(\mathcal{D})=n_{1}+n_{2}+n_{3}$ is even and that $v(\mathcal{D})\geqslant
2d-2$. If $n_{1}=n_{2}=n_{3}=d-1$, then $d$ is odd and we can take
$\alpha_{1}=\alpha_{2}$, with $\alpha_{1}=(1,\ldots,d)$. Then
$\alpha_{3}=\alpha_{1}^{-2}$ is a $d$-cycle, and these $\alpha_{i}$ satisfy
the Hurwitz conditions. Let us now suppose that
$\min\\{n_{1},n_{2},n_{3}\\}<d-1$. We may assume that $n_{3}<d-1$. Let
$n^{\prime}_{1}=n_{1}-1$, $n^{\prime}_{2}=n_{2}-1$ and $n^{\prime}_{3}=n_{3}$
and
$\mathcal{D}^{\prime}=\\{[n^{\prime}_{1}+1,1\ldots,1],[n^{\prime}_{2}+1,1\ldots,1],[n^{\prime}_{3}+1,1\ldots
1]\\}$ . We have $v(\mathcal{D}^{\prime})=v(\mathcal{D})-2$. Hence
$v(\mathcal{D}^{\prime})$ is even and at least $2d^{\prime}-1$, where
$d^{\prime}=d-1$. Therefore by induction the datum $\mathcal{D}^{\prime}$ is
realizable. Hence there exist $\alpha^{\prime}_{1}$, $\alpha^{\prime}_{2}$ and
$\alpha^{\prime}_{3}$ that are cycles in $\mathfrak{S}_{d-1}$ of length
$n^{\prime}_{1}+1$, $n^{\prime}_{2}+1$ and $n^{\prime}_{3}+1$ and that
generate a group that acts transitively on $\\{1,\ldots,d-1\\}$ and such that
$\alpha^{\prime}_{1}\alpha^{\prime}_{2}\alpha^{\prime}_{3}=\mathrm{id}$. There
exists $1\leqslant i\leqslant d-1$ that is in the support of both
$\alpha^{\prime}_{1}$ and $\alpha^{\prime}_{2}$, because
$\alpha^{\prime}_{1}\alpha^{\prime}_{2}$ is a cycle. We may suppose,
conjugating the $\alpha^{\prime}_{i}$ if necessary, that $i=d-1$ is in their
support. We let $\alpha_{1}=\alpha^{\prime}_{1}(d-1,d)$,
$\alpha_{2}=(d-1,d)\alpha^{\prime}_{2}$ and $\alpha_{3}=\alpha^{\prime}_{3}$.
The permutations $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ act transitively
on $\\{1,\ldots,d\\}$ and $\alpha_{i}\in A_{i}$ for $1\leqslant i\leqslant 3$.
Moreover $\alpha_{1}\alpha_{2}\alpha_{3}=\mathrm{id}$. ∎

We now prove 6.2.

###### Proof.

We proceed by induction on $k$. It is obvious for $k=2$ since in this case
$n_{1}=n_{2}=d-1$. It thus suffices to consider $d$-cycles $\alpha_{1}$ and
$\alpha_{2}=\alpha_{1}^{-1}$ to satisfy the Hurwitz conditions. We now assume
that $k>3$. Let us first suppose that there exist $1\leqslant i<k$ such that
$n_{i}+n_{i+1}\leqslant d-1$. There exist a $(n_{i}+1)$-cycle $\alpha$ and a
$(n_{i+1}+1)$-cycle $\beta$ such that $\alpha\beta$ is a
$(n_{i}+n_{i+1}+1)$-cycle. Indeed we can give explicit examples, or use Lemma
6.3 in $\mathfrak{S}_{n_{i}+n_{i+1}+1}$. Let $A$ be the partition associated
to $\alpha\beta$. We may assume that $i=1$. The branch data
$\mathcal{D}^{\prime}=\\{A,A_{3},\ldots,A_{k}\\}$ is realizable by induction,
since $v(\mathcal{D}^{\prime})=v(\mathcal{D})$. Therefore there exist
$\sigma\in A$, $\alpha_{3}\in A_{3},\ldots,\alpha_{k}\in A_{4}$ that act
transitively on $\\{1,\ldots,d\\}$ such that
$\sigma\alpha_{3}\ldots\alpha_{k}=\mathrm{id}$. There exists
$\gamma\in\mathfrak{S}_{d}$ such that $\sigma=\gamma\alpha\beta\gamma^{-1}$.
Let $\alpha_{1}=\gamma\alpha\gamma^{-1}$ and
$\alpha_{2}=\gamma\beta\gamma^{-1}$. The permutations
$\alpha_{1},\ldots,\alpha_{k}$ satisfy the Hurwitz conditions.

We now suppose that for every $1\leqslant i<k$, we have
$n_{i}+n_{i+1}\geqslant d$. By Lemma 6.3, there exist $\alpha\in A_{1}$ and
$\beta\in A_{2}$ such that $\alpha\beta$ is a $d$-cycle or a $(d-1)$-cycle.
This can also be found in [EKS84, Corollary 4.4, Lemma 4.5]. As before let $A$
be the partition associated to $\alpha\beta$. The branch data
$\mathcal{D}^{\prime}=\\{A,A_{3},\ldots,A_{k}\\}$ satisfies
$v(\mathcal{D}^{\prime})\geqslant d-2+n_{3}+n_{4}\geqslant 2d-2$. Therefore by
induction there exist $\sigma\in A$, $\alpha_{3}\in A_{3},\ldots,\alpha_{k}\in
A_{4}$ that act transitively on $\\{1,\ldots,d\\}$ and such that
$\sigma\alpha_{3}\ldots\alpha_{k}=\mathrm{id}$. There exists
$\gamma\in\mathfrak{S}_{d}$ such that $\sigma=\gamma\alpha\beta\gamma^{-1}$.
Let $\alpha_{1}=\gamma\alpha\gamma^{-1}$ and
$\alpha_{2}=\gamma\beta\gamma^{-1}$. The permutations
$\alpha_{1},\ldots,\alpha_{k}$ satisfy the Hurwitz conditions. ∎

###  6.2. Spherical structures

In this section we consider representations that fix a point in
$\mathbb{H}^{3}$. We can conjugate $\rho$ so that
$\rho\in\mathrm{Hom}(\Gamma,\mathrm{SO}_{3}(\mathbb{R}))$.

####  6.2.1. General obstructions

Suppose we have a branched projective structure $X$ on $\Sigma$ with holonomy
$\rho\in\mathrm{Hom}(\Gamma,\mathrm{SO}_{3}(\mathbb{R}))$. Let us denote by
$f:\tilde{\Sigma}\to\mathbb{CP}^{1}$ its developing map. The pullback
$f^{*}\omega$ of the volume form $\omega$ associated with the round metric is
$\Gamma$-invariant. Indeed for every $\gamma\in\Gamma$,

$\gamma^{*}f^{*}\omega=(f\circ\gamma)^{*}\omega=(\rho(\gamma)\circ
f)^{*}\omega=f^{*}\rho(\gamma)^{*}\omega=f^{*}\omega.$

Therefore it induces a form $\omega_{\Sigma}$ on $\Sigma$. The Gauss-Bonnet
formula gives, as observed in [MP19, Section 1.2]:

$\sum_{i=1}^{k}n_{i}-\frac{1}{2\pi}\int_{\Sigma}\omega=2g-2.$

Therefore, we must have $\sum_{i}n_{i}>2g-2$. This explains 2 in the spherical
case.

Let us suppose that $\rho$ takes its values in a finite subgroup of
$\mathrm{SO}_{3}(\mathbb{R})$ of order $n\geqslant 1$. We can consider the
cover $S$ of $\Sigma$ associated with the subgroup $\ker\rho$ of $\Gamma$. The
developing map $f$ induces a branched cover $f:S\to\mathbb{CP}^{1}$ of degree
$d$. The Riemann-Hurwitz formula gives

$n\chi(\Sigma)=2d-n\sum_{i=1}^{k}n_{i}.$

The degree $d$ must satisfy $d\geqslant\max_{i}n_{i}+1$. This explains 3.

####  6.2.2. Cyclic holonomy

We focus here on representations
$\rho\in\mathrm{Hom}(\Gamma,\mathrm{SL}_{2}(\mathbb{C}))$ with finite image
included in $\mathrm{SO}_{2}(\mathbb{R})$. Let us observe that any
representation $\rho\in\mathrm{Hom}(\Gamma,\mathrm{SO}_{2}(\mathbb{R}))$ lifts
to $\mathrm{SL}_{2}(\mathbb{C})$. Indeed the $2$-sheeted cover of
$\mathrm{SO}_{2}(\mathbb{R})=\\{\pm\begin{pmatrix}\alpha&0\\\
0&\alpha^{-1}\end{pmatrix}\mid\alpha\overline{\alpha}=1\\}$ is the subgroup
$\widetilde{\mathrm{SO}_{2}(\mathbb{R})}=\\{\begin{pmatrix}\alpha&0\\\
0&\alpha^{-1}\end{pmatrix}\mid\alpha\overline{\alpha}=1\\}$ of
$\mathrm{PSL}_{2}(\mathbb{C})$ that is abelian, hence every commutator in
$\widetilde{\mathrm{SO}_{2}(\mathbb{R})}$ is trivial. The finite subgroups of
$\mathrm{SO}_{2}(\mathbb{R})$ are cyclic and we fix $\mathbb{Z}_{n}$ such a
cyclic group of order $n\geqslant 2$. Let $\rho$ be a surjective
representation in $\mathrm{Hom}(\Gamma,\mathbb{Z}_{n})$.

######  Proposition 6.4.

Let $1\leqslant n_{1}\leqslant\ldots\leqslant n_{k}$ be such that
$\sum_{i}n_{i}=2g+2l$ for some $l\geqslant 0$. We have
$\rho\in\mathcal{\mathrm{Hol}}(\mathcal{P}(n_{1},\ldots,n_{k}))$ if and only
if $n_{k}<n(l+1)$.

Let us denote by $P_{n}$ a pyramid with basis a regular $n$-gon. We can
identify a branched projective structure with holonomy in
$\mathrm{Hom}(\Gamma,\mathbb{Z}_{n})$ with a branched structure modeled on
$P_{n}$ with holonomy in the cyclic of order $n$ isometry group of $P_{n}$,
see Figure 7.

Figure 7. The pyramid model $P_{4}$.

One benefit of this point of view is that a surface equipped with such a
structure inherits a conical Euclidean structure as studied for example in
[Tro86]: it is obtained by gluing Euclidean triangles along their boundaries.
Let us recall the Gauss-Bonnet formula in this setting.

######  Lemma 6.5 (Troyanov).

Let us consider a conical Euclidean structure on the surface $\Sigma_{g_{S}}$
of genus $g_{S}$. Let us denote by
$2\pi(\theta_{1}+1),\ldots,2\pi(\theta_{s}+1)$ the total angles of its conical
points. We have

$\sum_{i=1}^{s}\theta_{i}=2g_{S}-2.$

We refer to [Tro86, Section 3] for a short proof of Lemma 6.5. We now prove
6.4. It suffices to exhibit a branched projective structure with given branch
data and surjective holonomy in $\mathrm{Hom}(\Gamma,\mathbb{Z}_{n})$ by 3.2.

###### Proof.

We are going to explain how to construct a genus $g$ surface with holonomy a
surjective homomorphism of $\mathrm{Hom}(\Gamma,\mathbb{Z}_{n})$ and branch
data given by $n_{1},\ldots,n_{k}$. Let us start by considering $l+1$ copies
of $\mathbb{CP}^{1}$. Let us choose an annulus neighborhood of the equator in
$\mathbb{CP}^{1}$. We identify this neighborhood with a rectangle
$[0,n]\times[0,1]$ with its vertical sides identified. A horizontal
translation of length $1$ in this model corresponds to an order $n$ rotation
in $\mathbb{CP}^{1}$. We are going to cut open the corresponding annuli in
these spheres along segments and glue the boundaries back in a different
pattern. Let us begin with the case $k=1$ and $n$ even. We cut open an annulus
along a length one horizontal segment and then along another, that is obtained
by applying a length $1$ horizontal translation to the first one. We then make
$g+l-1$ other slits in the annuli, along segments that are translates of the
first one by an integer, see Figure 8. We make sure that there is at least one
slit on each annulus. We also make sure that these segments do not intersect.
Observe that we can put $\frac{n}{2}$ of these line segments in each annulus.
We have room to cut along these $g+l+1$ segments since we have $n_{1}=2g+2l$,
thus $\frac{n}{2}(l+1)>\frac{n_{1}}{2}\geqslant g+l$. We glue back the
boundary components with translations as indicated in Figure 8.

$1$
---
$1$
---
$2$
---
$2$
---
$3$
---
$3$
---
$4$
---
$4$
---
$5$
---
$5$
---
$6$
---
$6$
---
$7$
---
$7$
---
$8$
---
$8$
---
Figure 8. Construction for $(n,k,l,g)=(6,1,2,5)$.

The surface we obtain has genus $g$. Indeed let us denote by $g^{\prime}$ this
genus. Using the pyramidal model, we get a flat structure on this surface. The
sum of the angles $\theta_{i}$ on a pyramid is $-2$ by Lemma 6.5, thus again
by Lemma 6.5 we have $2g^{\prime}-2=n_{1}-2(l+1)=2g-2$. Moreover the holonomy
of the resulting surface is a surjective homomorphism of
$\mathrm{Hom}(\Gamma,\mathbb{Z}_{n})$.

Let us now turn to the case $k=1$ and $n$ odd. Observe that we can now put
$\frac{n+1}{2}$ line segments in the first annulus, but only $\frac{n-1}{2}$
ones in the others using the same techniques. If there is enough room to put
the $g+l+1$ line segments as before, then we proceed as in Figure 8. If not,
we may increase the total angle contained in two annuli by replacing the cuts
of Figure 8 with those of Figure 9.

$1$
---
$1$
---
$2$
---
$2$
---
$3$
---
$3$
---
$4$
---
$6$
---
$5$
---
$5$
---
$6$
---
$4$
---
$7$
---
$8$
---
$10$
---
$10$
---
$7$
---
$8$
---
Figure 9. Construction for $(n,k,l,g)=(5,1,2,5)$.

Each of the annuli in which this pattern is used contributes to the total
angle by $2\pi n$. Since $n_{1}<n(l+1)$, it is possible the get an angle of
$2\pi(n_{1}+1)$. It follows from Lemma 6.5 that we get a genus $g$ surface. It
has a projective structure with holonomy a surjective homomorphism of
$\mathrm{Hom}(\Gamma,\mathbb{Z}_{n})$.

Let us now turn to the case $k\geqslant 2$. We are again going to make slits
in the annuli and glue the boundaries back using a pattern that we now
describe. We consider $n_{1}+1$ vertical line segments that are obtained one
from the other by horizontal translations. We will identify their boundaries
in a cyclic way, see Figure 10. Let us suppose that $n_{1}<n_{2}$. We put
$n_{2}-n_{1}$ parallel line segments below, such that the top of exactly one
of these segments touches one of the $n_{1}+1$. We will identify these
segments in a cylic way, as indicated in Figure 10. Making slits along these
segments and identifying the boundaries following this pattern will create
three conical points: one of angle $2\pi(n_{1}+1)$, one of angle
$2\pi(n_{2}+1)$ and one of angle $2\pi(n_{2}-n_{1}+1)$ lines above. We
continue similarly: we put $n_{3}-(n_{2}-n_{1})$ line segments below, so that
each touches the one above in a single point. We continue this construction
until the gluing process gives the desired angles $n_{1},\ldots,n_{k-1}$. We
then change the last angle with horizontal slits as in Figure 10. If
$n_{1}=n_{2}$, we consider $n_{1}+1$ line segments that we identify in a
cyclic way, as in the top of Figure 10. We thus make the construction of the
pattern for $n_{3},\ldots,n_{k}$ below these segments.

$1$
---
$2$
---
$2$
---
$3$
---
$3$
---
$1$
---
$4$
---
$5$
---
$5$
---
$6$
---
$6$
---
$4$
---
$7$
---
$8$
---
$8$
---
$7$
---
$9$
---
$10$
---
$10$
---
$11$
---
$12$
---
$14$
---
$11$
---
$9$
---
$13$
---
$12$
---
$14$
---
$13$
---

Figure 10. Gluing pattern for $k=6$ and $(n_{i})_{1\leqslant i\leqslant
6}=(2,2,2,3,3,6)$.

This pattern has $g+l$ connected components, unless $k$ is even and
$n_{2i}=n_{2i-1}$ for every $1\leqslant i\leqslant\frac{k}{2}$, where it has
$g+l+1$ connected components. Indeed one can count directly or argue as
follows. If we place each connected component of this pattern on a different
sphere, cut open along the segments and glue as indicated in the pattern, we
get a genus $1$ surface with conical points $n_{1},\ldots,n_{k}$. It follows
from Lemma 6.5 that the number $C$ of connected components of this pattern
satisfies $\sum_{i}n_{i}-2C=0$. In the special case $n_{2i}=n_{2i-1}$ for each
$i$, then we get a genus $0$ surface, thus $\sum_{i}n_{i}-2C=-2$.

We put the connected components of this pattern on the annuli as in Figure 11.

$1$
---
$2$
---
$2$
---
$3$
---
$3$
---
$1$
---
$8$
---
$9$
---
$5$
---
$6$
---
$6$
---
$7$
---
$7$
---
$4$
---
$4$
---
$5$
---
$9$
---
$8$
---
Figure 11. Construction for $(n,k,l,g,n_{1},n_{2},n_{3})=(3,3,1,5,2,5,5)$.

We make sure that there is at least one line segment on each of the $l+1$
spheres. We have room to place the vertical segments in the annuli since
$n_{i}<n(l+1)$. As before, there may not be enough room to put the horizontal
segments when $n$ is odd. We can then use the pattern described in the case
$k=1$ in Figure 9 for the horizontal line segments. The resulting surface is
connected, has genus $g$ by Lemma 6.5, and has holonomy a surjective
homomorphism of $\mathrm{Hom}(\Gamma,\mathbb{Z}_{n})$. ∎

####  6.2.3. Dihedral holonomy

We now consider representations $\rho\in\mathrm{Hom}(\Gamma,\mathrm{D}_{n})$.
We show that 1, 2 and 3 are the only obstructions of being in
$\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$. Let us consider a bypiramid
$B_{n}$ obtained by gluing two pyramids $P_{n}$ along their base, see Figure
12. As in the cyclic holonomy case, a branched projective structure with
holonomy in $\mathrm{Hom}(\Gamma,\mathrm{D}_{n})$ may be identified with a
structure modeled on $B_{n}$ with holonomy in its isometry group, which is
isomorphic to $\mathrm{D}_{n}$. It thus inherits an Euclidean structure as
before.

Figure 12. The bipyramid model $B_{4}$.

######  Proposition 6.6.

Let $\rho\in\mathrm{Hom}(\Gamma,\mathrm{D}_{n})$ be a surjective homomorphism.
The representation $\rho$ is in $\mathrm{Hol}(\mathcal{P}(n_{1},\ldots
n_{k}))$ if and only if

  1. (1)

$\sum_{i}n_{i}=2g+2l$, where $l\geqslant 0$ is such that
$\max_{i}n_{i}<2n(l+1)$ if $\rho$ lifts to $\mathrm{SL}_{2}(\mathbb{C})$.

  2. (2)

$\sum_{i}n_{i}=2g+2l-1$ where $l\geqslant 0$ is such that
$\max_{i}n_{i}<n(2l+1)$ if $\rho$ does not lift to
$\mathrm{SL}_{2}(\mathbb{C})$.

##### The genus $1$ case.

Let us suppose that $g=1$. Recall that there is no surjective homomorphism in
$\mathrm{Hom}(\Gamma_{1},\mathrm{D}_{n})$ if $n\geqslant 3$ since
$\Gamma_{1}\simeq\mathbb{Z}^{2}$ is abelian. Let us also recall that the
action of $\mathrm{Aut}^{+}(\Gamma_{1})$ on the set of surjective
homomorphisms of $\mathrm{Hom}(\Gamma_{1},\mathrm{D}_{2})$ is transitive by
3.4. The surjective homomorphisms of $\mathrm{Hom}(\Gamma_{1},\mathrm{D}_{2})$
do not lift to $\mathrm{SL}_{2}(\mathbb{C})$. 3 is trivially satisfied since
$\sum_{i}n_{i}=2l+1$ thus $\max_{i}n_{i}<2(2l+1)$. Therefore we may rephrase
6.6 as follows.

######  Lemma 6.7.

Let $\rho\in\mathrm{Hom}(\Gamma_{1},\mathrm{D}_{2})$ be a surjective
homomorphism. We have $\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$
if and only if $\sum_{i}n_{i}$ is odd.

###### Proof.

Let $s=\rho(a_{1})$ and $t=\rho(b_{1})$. Let us consider the round metric on
$\mathbb{CP}^{1}\simeq\mathbb{S}^{2}$. Let $C$ be the great circle on
$\mathbb{S}^{2}$ passing through the fixed points of both $s$ and $t$. Let us
denote by $S$ and $N$ the fixed points of $t$. Let $d$ be the distance on
$\mathbb{CP}^{1}$ associated with the round metric. We decompose $C$ into $4$
arcs: $P_{N}=\\{x\in C\mid d(x,N)\leqslant 1\\}$, $P_{S}=s(P_{N})=\\{x\in
C\mid d(x,S)\leqslant 1\\}$, and $P^{\pm}$ the (closure of the) two connected
components of $C\setminus(P_{S}\cup P_{N})$ in $C$. Let us cut
$\mathbb{S}^{2}$ along $C$ to get a half-sphere with boundary $C$. We glue
back its boundary in the following way. We glue $P_{N}$ to $P_{S}$ with $s$
and $P^{-}$ to $P^{+}$ with $t$, see Figure 13. We thus get a torus with
holonomy $\rho$ and a single conical point of total angle $2\pi(1+1)$. There
exists a curve based at the conical point, namely the image of $P_{S}$ in the
torus, that develops injectively. Hence by 5.7 we have
$\rho\in\mathrm{Hol}(\mathcal{P}(2d+1))$ for every $d\geqslant 0$ and
$\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ for every $1\leqslant
n_{1}\leqslant\ldots\leqslant n_{k}$ such that $\sum_{i}n_{i}$ is odd by 5.9.
∎

$P_{N}$
---
$P_{S}$
---
$P^{-}$
---
$P^{+}$
---
Figure 13. Half-sphere.

##### The genus $g\geqslant 2$ case.

We now prove 6.6 for $g\geqslant 2$.

###### Proof.

Let $n_{1},\ldots,n_{k}$ satisfying the hypotheses of 6.6. By 3.5, it suffices
to exhibit a projective structure with branch data $n_{1},\ldots,n_{k}$ and
holonomy a surjective homomorphism of $\mathrm{Hom}(\Gamma,\mathrm{D}_{n})$.
Let us begin with the case where $\sum_{i}n_{i}$ is even. Let us consider
$l+1$ copies of $\mathbb{S}^{2}=\mathbb{CP}^{1}$. We consider an annular
neighborhood of some longitudinal line in the northern hemisphere of
$\mathbb{S}^{2}$, and its image by the flip $s$ that interchanges the north
pole with the south pole. We therefore have $2(l+1)$ annuli, see Figure 14.

Figure 14. Two annuli embedded in $\mathbb{CP}^{1}$.

We make the same slits as in the proof of 6.4. We make sure that there is a
slit in each sphere, that there are two slits in an annulus glued with an
order $n$ rotation. We also make sure that there are at least two slits, one
in the northern annulus and one in the southern annulus of one of the spheres,
that are glued together by $s$. This is possible since the pattern of the
slits described in 6.4 has at least $g+l$ connected components and $g\geqslant
2$. Moreover recall that in the case where it has exactly $g+l$ connected
components, there are two slits glued by a rotation of order $n$ in the
pattern. We get a projective structure on a surface of genus $g$ by the same
argument as in 6.4: it is a consequence of Lemma 6.5. Its holonomy is a
surjective homomorphism of $\mathrm{Hom}(\Gamma,\mathrm{D}_{n})$ that lifts to
$\mathrm{SL}_{2}(\mathbb{C})$. Indeed this follows from the fact that
$\sum_{i}n_{i}$ is even.

Let us now turn to the case where $n$ is even and $\sum_{i}n_{i}$ is odd: we
have $\sum_{i}n_{i}=2g+2l-1$. We consider $l$ copies of
$\mathbb{S}^{2}=\mathbb{CP}^{1}$ and a half-sphere: we cut $\mathbb{CP}^{1}$
along the great circle $C$ that goes through both the fixed points of $s$ and
the fixed points $N$ and $S$ of the order $2$ element of $\ker\epsilon$. Since
$\sum_{i}n_{i}$ is odd, at least one of the $n_{i}$ is odd, say $n_{i_{0}}$.
Let us decompose the boundary of the half-sphere into $4$ circular arcs as
before. One of these arcs is the set $P_{N}=\\{x\in C\mid d(N,x)<1\\}$, where
$d$ denotes the distance associated with the round metric on $\mathbb{S}^{2}$,
and $P_{S}=s(P_{N})=\\{x\in C\mid d(S,x)<1\\}$. The other two parts $P^{\pm}$
are the (closure of the) connected components of $C\setminus(P_{N}\cup
P_{S})$, see the circle in green and red of the left bypyramid of Figure 15.
We glue $P_{N}$ with $P_{S}$, these are the green arcs in Figure 15, with the
map $s$. We now make $\frac{n_{i_{0}}-1}{2}$ slits along circular arcs
obtained from $P^{-}$ by a rotation of order $n$ in the spheres and identify
the boundaries components, see the red arcs in Figure 15. We have room to make
these slits since we may put $\frac{n}{2}-1$ of them in the half-sphere, and
$n$ in each of the $l$ spheres and $n_{i_{0}}<2nl+n$. We now consider $2l$
annular neighborhood of longitudinal circles in the spheres as before, away
from these slits, see the blue annuli in the right bypiramid of Figure 15. We
also consider the same annular neighborhood of length $n$ in the half-sphere:
see the blue annulus on the left side of Figure 15.

Figure 15. Construction for $(n,k,l,g,n_{1})=(4,1,1,2,5)$.

In these annuli, we make surgeries associated with the branch data
$\\{n_{1},\ldots,n_{k}\\}\setminus\\{n_{i_{0}}\\}$ as explained above in the
case where $\rho$ lifts in $\mathrm{SL}_{2}(\mathbb{C})$. The gluing pattern
we have described has at least $\frac{1}{2}(\sum_{i}n_{i}-1)=g+l-1$ connected
components, and since $g\geqslant 2$ it is possible to put a slit on each
sphere. Since $\max_{i}n_{i}<2nl+n$, we have room to put these slits in the
annuli. We thus get a connected surface, with charts in $\mathbb{CP}^{1}$,
branch datum corresponding to $n_{1},\ldots,n_{k}$. This surface has genus $g$
since by Lemma 6.5, its genus $g^{\prime}$ satisfies
$\sum_{i}n_{i}=2g^{\prime}+2l-1$. Its holonomy is a surjective homomorphism of
$\mathrm{Hom}(\Gamma,\mathrm{D}_{n})$ that does not lift so
$\mathrm{SL}_{2}(\mathbb{C})$ since $\sum_{i}n_{i}$ is odd. ∎

####  6.2.4. Tetrahedron

A branched projective structure with holonomy in
$\mathrm{Hom}(\Gamma,\mathfrak{A}_{4})$ may be identified as before with a
branched structure modeled on the regular tetrahedron with holonomy in its
isometry group $\mathfrak{A}_{4}$. We show that the first three obstructions
are the only ones to being in $\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$,
where $1\leqslant n_{1}\leqslant\ldots\leqslant n_{k}$.

######  Proposition 6.8.

Let $\rho\in\mathrm{Hom}(\Gamma,\mathfrak{A}_{4})$ be a surjective
homomorphism. We have $\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$
if and only if

  1. (1)

$\sum_{i}n_{i}=2g+2l$, for $l\geqslant 0$ such that $\max_{i}n_{i}<12(l+1)$ if
$\rho$ lifts to $\mathrm{SL}_{2}(\mathbb{C})$

  2. (2)

$\sum_{i}n_{i}=2g+2l-1$ for $l\geqslant 0$ such that $\max_{i}n_{i}<12l+6$ if
$\rho$ does not lift to $\mathrm{SL}_{2}(\mathbb{C})$.

###### Proof.

Let us consider $n_{1},\ldots,n_{k}$ satisfying the hypotheses of 6.8. It
suffices to exhibit a branched projective structure in
$\mathcal{P}(n_{1},\ldots,n_{k})$ with holonomy a surjective homomorphism of
$\mathrm{Hom}(\Gamma,\mathfrak{A}_{4})$ thanks to 3.8. Let us start with the
case where $\sum_{i}n_{i}=2g+2l$ for some $l\geqslant 0$. Let us consider
$l+1$ copies of the regular tetrahedron $T$. We will make slits along line
segments in the tetrahedra and glue the boundaries back with isometries of
$T$. We use the pattern of Figure 10. Let us describe how to put this pattern
in the tetrahedra. Start by considering a line segment in one of the triangles
of one of the tetrahedra as in Figure 16. We consider $n_{1}$ line segments
obtained from this one by applying an isometry of $T$. We identify the
boundaries of the slits obtained by cutting open the tetrahedra along these
segments with the corresponding isometries, following the pattern of Figure
10. We continue to put the pattern of the line segments in this way.

We have room to make these slits since $\max_{i}n_{i}<12(l+1)$. We make sure
to have one slit on each tetrahedron and that one of the tetrahedra has slits
that are glued together with isometries generating $\mathfrak{A}_{4}$. This
construction gives a projective structure on a closed surface of genus $g$
with holonomy a surjective homomorphism of
$\mathrm{Hom}(\Gamma,\mathfrak{A}_{4})$. Indeed the genus $g^{\prime}$ of the
resulting surface can be computed using Lemma 6.5: it satisfies
$2g^{\prime}-2=\sum_{i}n_{i}-2(l+1)=2g-2$.

Let us now turn to the case $\sum_{i}n_{i}=2g+2l-1$ for some $l\geqslant 0$.
We consider $l$ copies of the regular tetrahedron and a half-tetrahedron: we
cut a tetrahedron in half along a plane going through the midpoints of two
pairs of opposite edges. We divide the resulting boundary into $2$ pairs as in
the dihedral case: two red line segments and two green ones. We identify the
green lines with an isometry of the tetrahedron. One of the $n_{i}$, say
$n_{i_{0}}$ must be odd. Let us make $\frac{n_{i_{0}}-1}{2}$ slits on the
tetrahedra and the half-tetrahedron, along line segments obtained by applying
isometries to one of the red boundaries. We identify the resulting boundaries
in a cyclic way. We then make the slits and gluing associated with the
branching data $\\{n_{1},\ldots,n_{k}\\}\setminus\\{n_{i_{0}}\\}$. We have
room to make all these slits because of the assumption $\max_{i}n_{i}<12l+6$.
We make sure to obtain a connected surface (each tetrahedron must have at
least one slit) and that the resulting holonomy is a surjective homomorphism
of $\mathrm{Hom}(\Gamma,\mathfrak{A}_{4})$. By Lemma 6.5, the genus of the
resulting surface is $g$ and its holonomy does not lift to
$\mathrm{SL}_{2}(\mathbb{C})$ since $\sum_{i}n_{i}$ is odd. ∎

$1$
---

$2$
---

$2$
---

$3$
---

$1$
---

$3$
---

Figure 16. Construction for $(l,g,k,n_{1},n_{2})=(0,2,2,2,2).$

####  6.2.5. Cube

We can identify the branched projective strutures with holonomy in the group
of the isometries of the cube with a branched structure modeled on the cube
with holonomy in its isometry group. Again we may geometrize a surjective
homomorphism $\rho\in\mathrm{Hom}(\Gamma,\mathfrak{S}_{4})$ that satisfies the
conditions of the first three obstructions.

######  Proposition 6.9.

Let $\rho\in\mathrm{Hom}(\Gamma,\mathfrak{S}_{4})$ be a surjective
homomorphism. We have $\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$
if and only if

  1. (1)

$\sum_{i}n_{i}=2g+2l$, for $l\geqslant 0$ such that $\max_{i}n_{i}<24(l+1)$ if
$\rho$ lifts to $\mathrm{SL}_{2}(\mathbb{C})$

  2. (2)

$\sum_{i}n_{i}=2g+2l-1$ for $l\geqslant 0$ such that $\max_{i}n_{i}<24l+12$ if
$\rho$ does not lift to $\mathrm{SL}_{2}(\mathbb{C})$.

###### Proof.

The construction is analogous to the one of 6.8, see Figure 17. ∎

Figure 17. Construction for $(l,g,k,n_{1},n_{2})=(0,11,2,10,11).$

####  6.2.6. Icosahedron

As before, a branched projective structure with holonomy in
$\mathrm{Hom}(\Gamma,\mathfrak{A}_{5})$ may be identified with a branched
structure modeled on the icosahedron. Therefore we will geometrize the
surjective representations of $\mathrm{Hom}(\Gamma,\mathfrak{A}_{5})$ by
gluing together regular icosahedra.

######  Proposition 6.10.

Let $\rho\in\mathrm{Hom}(\Gamma,\mathfrak{A}_{5})$ be a surjective
homomorphism. We have $\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$
if and only if

  1. (1)

$\sum_{i}n_{i}=2g+2l$, for $l\geqslant 0$ such that $\max_{i}n_{i}<60(l+1)$ if
$\rho$ lifts to $\mathrm{SL}_{2}(\mathbb{C})$.

  2. (2)

$\sum_{i}n_{i}=2g+2l-1$ for $l\geqslant 0$ such that $\max_{i}n_{i}<60l+30$ if
$\rho$ does not lift to $\mathrm{SL}_{2}(\mathbb{C})$.

Again, the constructions are analogous to those of 6.8 and 6.9, see Figure 18.

Figure 18. Construction for $(l,g,k,n_{1})=(0,3,1,5)$

####  6.2.7. Holonomy with infinite image in $\mathrm{SO}_{3}(\mathbb{R})$

We characterize the representations
$\mathrm{Hom}(\Gamma,\mathrm{SO}_{3}(\mathbb{R}))$ with infinite image that
are in $\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$. The infinite closed
subgroups of $\mathrm{SO}_{3}(\mathbb{R})$ are isomorphic to
$\mathrm{SO}_{2}(\mathbb{R})$, $\mathrm{O}_{2}(\mathbb{R})$ or are the whole
group $\mathrm{SO}_{3}(\mathbb{R})$. Let us fix $1\leqslant
n_{1}\leqslant\ldots\leqslant n_{k}$.

##### Holonomy with dense image in $\mathrm{SO}_{2}(\mathbb{R})$

Recall that any representation with image in $\mathrm{SO}_{2}(\mathbb{R})$
lifts to $\mathrm{SL}_{2}(\mathbb{C})$. We show that 1 and 2 are the only
obstructions in this case.

######  Proposition 6.11.

A representation $\rho\in\mathrm{Hom}(\Gamma,\mathrm{SO}_{2}(\mathbb{R})$ is
in $\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ if and only if
$\sum_{i}n_{i}$ is even and at least $2g$.

###### Proof.

Let us suppose that $\sum_{i}n_{i}$ is even and at least $2g$. By 6.4, there
exists a representation in $\mathrm{Hom}(\Gamma,\mathrm{SO}_{2}(\mathbb{R}))$
with finite image in $\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$.
Therefore since the closure $\mathrm{Aut}^{+}(\Gamma)\cdot\rho$ contains such
a representation by4.1, it follows from the Ehresmann-Thurston principle (5.6)
that $\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$. ∎

##### Holonomy with dense image in $\mathrm{O}_{2}(\mathbb{R})$

We now suppose that $\rho\in\mathrm{Hom}(\Gamma,\mathrm{O}_{2}(\mathbb{R}))$
has dense image.

######  Proposition 6.12.

The representation $\rho$ is in
$\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ if and only if
$\sum_{i}n_{i}>2g-2$ and $\sum_{i}n_{i}$ is even (resp. odd) if $\rho$ lifts
to $\mathrm{SL}_{2}(\mathbb{C})$ (resp. does not lift to
$\mathrm{SL}_{2}(\mathbb{C})$).

###### Proof.

Let $\rho\in\mathrm{Hom}(\Gamma,\mathrm{O}_{2})$ with dense image. There
exists $\rho^{\prime}\in\mathrm{Hom}(\Gamma,\mathrm{D}_{n})$ in
$\mathrm{Hol}(\mathcal{P}(n_{1}\ldots,n_{k}))$ for $n$ large enough by 6.6. By
4.2, $\rho^{\prime}$ is in the closure of $\mathrm{Aut}^{+}(\Gamma)\cdot\rho$
thus $\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ by the Ehresmann-
Thurston principle 5.6. ∎

##### Holonomy with dense image in $\mathrm{SO}_{3}(\mathbb{R})$

######  Proposition 6.13.

A representation $\rho\in\mathrm{Hom}(\Gamma,\mathrm{SO}_{3}(\mathbb{R}))$
with dense image is in $\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ if and
only if both $\sum_{i}n_{i}>2g-2$ and $\sum_{i}n_{i}$ is even if $\rho$ lifts
to $\mathrm{SL}_{2}(\mathbb{C})$ and odd otherwise.

###### Proof.

As in 6.12, there exists
$\rho^{\prime}\in\mathrm{Hom}(\Gamma,\mathrm{SO}_{3}(\mathbb{R}))$ in
$\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$. For example we can take
$\rho^{\prime}$ with its image in a dihedral group $\mathrm{D}_{n}$ by 6.6.
This also follows from [MP19, Theorem A] where the existence of spherical
metrics with conical points of prescribed angles satisfying the Gauss-Bonnet
constraint is shown. By Lemma 4.4, $\rho^{\prime}$ is in the closure of
$\mathrm{Aut}^{+}(\Gamma)\times\mathrm{SO}_{3}(\mathbb{R})\cdot\rho$.
Therefore $\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ by the
Ehresmann-Thurston principle, 5.6. ∎

###  6.3. Affine holonomy

In this subsection we geometrize affine representations. An example of a
branched projective structure with affine holonomy is an affine structure, as
studied for example in [Gha18].

######  Definition 6.14.

An affine structure is a branched projective structure with affine holonomy
whose developing map takes its values in $\mathbb{C}\subset\mathbb{CP}^{1}$.

This definition generalizes for example the notion of translation surface.

######  Definition 6.15.

A translation surface structure $X$ on $\Sigma$ is an affine structure with
holonomy in $\mathrm{Hom}(\Gamma,\mathbb{C})$, where
$\mathbb{C}\subset\mathrm{Aff(\mathbb{C})}$ is the subgroup of translations.
An even half-translation structure is an affine structure with holonomy in
$\mathrm{Hom}(\Gamma,\\{\pm 1\\}\rtimes\mathbb{C})$.

######  Remark 6.16.

This amounts to requiring that $\alpha=\mathrm{Li\circ\rho}$ is the trivial
homomorphism for the translation surface case, and that $\alpha$ has its image
in $\\{\pm 1\\}$ for the half-translation case.

######  Remark 6.17.

Our definition of even half-translation surface is equivalent to the usual
definitions of half-translation surfaces, see for example [Zor06, Lan08], with
the extra requirement that the total angle of every conical point is even in
the following sense: it is of the form $k\pi$, with $k\in 2\mathbb{N}$. In
other words, following the notations of [Lan08], we are looking at the half-
translation surfaces that are either translation surfaces or correspond to
quadratic differentials in

$\bigcup_{n_{i}\geqslant
1\mid\sum_{i}n_{i}=2g-2}\mathcal{Q}(2n_{1},2n_{2},\ldots,2n_{k}).$

This extra requirement is necessary to obtain a holonomy defined on $\Gamma$,
and not on the fundamental group of a punctured surface.

####  6.3.1. General obstructions

Let us show that the divisor of a branched projective structure with affine
holonomy has degree at least $2g-2$ and that we may read in this degree
whether such a structure is an affine one or not.

######  Proposition 6.18.

If $\rho\in\mathrm{Hom}(\Gamma,\mathrm{Aff}(\mathbb{C}))$ is the holonomy of a
projective structure in $\mathcal{P}(n_{1},\ldots,n_{k})$ with developing map
$f$, then $\sum_{i}n_{i}\geqslant 2g-2$. Moreover $\sum n_{i}=2g-2$ if and
only if $f(\tilde{\Sigma})\subset\mathbb{C}$.

The following proof of 6.18 is an adaptation of [Kap20, Section 6], where
Kapovich shows the same result in the realm of holonomies with values in the
subgroup of translations $\mathbb{C}\subset\mathrm{Aff(\mathbb{C})}$.

###### Proof.

Let us assume that $\rho$ is the holonomy of a branched projective structure
with holomorphic developing map $f:\tilde{\Sigma}\to\mathbb{CP}^{1}$ for a
complex structure on $\Sigma$. Let us choose a lift
$\tilde{\rho}\in\mathrm{Hom}(\Gamma,\mathrm{SL}_{2}(\mathbb{C}))$ of $\rho$
and define $\varphi\in\mathrm{Hom}(\Gamma,\mathbb{C}^{\star})$ by
$\varphi(\gamma)=\lambda_{\gamma}$ where
$\tilde{\rho}(\gamma)=\begin{pmatrix}\lambda_{\gamma}^{-1}&t_{\gamma}\\\
0&\lambda_{\gamma}\end{pmatrix}$. Let us consider the vector bundles
$V=\tilde{\Sigma}\times\mathbb{C}^{2}/\Gamma$ and
$\Lambda=\tilde{\Sigma}\times\mathbb{C}/\Gamma$, where $\Gamma$ acts by
$\gamma\cdot(\tilde{x},v)=(\gamma\cdot\tilde{x},\tilde{\rho}(\gamma)\cdot v)$
and $\gamma\cdot(\tilde{x},z)=(\gamma\cdot\tilde{x},\varphi(\gamma)\cdot z)$.
The projection on the second coordinate gives a morphism of holomorphic
bundles $\phi:V\to\Lambda$. The developing map gives for each point
$\tilde{x}\in\tilde{\Sigma}$ a line $f(\tilde{x})\subset\mathbb{C}^{2}$, and
by $\rho$-equivariance induces a line subbundle $L\subset V$. We restrict
$\phi$ to get a morphism of line bundles $\phi:L\to\Lambda$, which is
injective if and only if $f(\tilde{\Sigma})$ does not contain
$\infty\in\mathbb{CP}^{1}$.

######  Lemma 6.19.

We have $\deg L\leqslant\deg\Lambda$ with equality if and only if $\phi$ is
injective.

###### Proof.

Let us pick a smooth section $s:\Sigma\to L$ transverse to the zero section
whose zeroes are different from those of $\phi:L\to\Lambda$. We also assume
that $\phi\circ s$ is transverse to the zero section. The degree of $L$ is the
intersection number between $s(\Sigma)$ and the zero section: $\deg
L=\sum_{p\in s^{-1}(0)}\mathrm{sign}_{p}(s)$, where $\mathrm{sign}_{p}(s)$ is
the sign of $\det(ds_{p})$ where $s$ is viewed as
$s:U\subset\mathbb{C}\to\mathbb{C}$ in a local chart around $p$, and a local
trivialization of $L$. The zeroes of the section $\phi\circ s$ of $\Lambda$
are of two types: the zeroes of $s$ and the zeroes of $\phi$. Each zero of
$\phi$ has sign $1$, because in a local chart centered at $0$ and a local
trivialization, we have $\phi\circ s:z\mapsto\alpha(z)s(z)$ where $\alpha$ is
holomorphic, $\alpha(0)=0$ and $s(0)\neq 0$. Therefore $d(\phi\circ
s)=s(0)\cdot d\alpha_{0}$ has positive determinant. If $p$ is a zero of $s$,
we have $\alpha(0)\neq 0$ and $s(0)=0$, thus $d(\phi\circ
s)_{0}=\alpha(0)\cdot ds_{0}$ and $\mathrm{sign}_{p}(\phi\circ
s)=\mathrm{sign}_{p}(s)$. We have

$\deg\Lambda=\sum_{p\in(\phi\circ s)^{-1}(0)}\mathrm{sign}_{p}(\phi\circ s)\\\
=\sum_{p\in s^{-1}(0)}\mathrm{sign}_{p}(\phi\circ
s)+\\#\\{p\in\Sigma\mid\phi=0\\}.$

Therefore, $\deg\Lambda=\deg L+\\#\\{p\in\Sigma\mid\phi=0\\}\geqslant\deg L$
with equality if and only if $\phi$ is injective. ∎

######  Lemma 6.20.

We have $\deg\Lambda=0$.

###### Proof.

It suffices to show that this degree does not depend on the homomorphism
$\varphi\in\mathrm{Hom}(\Gamma,\mathbb{C}^{\star})$, since for
$\varphi:\gamma\mapsto 1$, the bundle $\Lambda$ is trivial. Let
$(\varphi_{t})_{t\in[0,1]}$ be a path in
$\mathrm{Hom}(\Gamma,\mathbb{C}^{\star})$, and $\Lambda_{t}$ be the bundle
over $\Sigma$ defined as $\Lambda$ with the action of $\varphi_{t}$. Let $W$
be the vector bundle over $\Sigma\times[0,1]$ defined by
$W=\tilde{\Sigma}\times\mathbb{C}\times[0,1]/\Gamma$ where $\Gamma$ acts by
$\gamma\cdot(\tilde{x},z,t)=(\tilde{\gamma}\cdot x,\varphi_{t}(\gamma)\cdot
z,t)$. Let us choose a linear connection on $W$. The parallel transport along
curves $x\times[0,1]$, $x\in\Sigma$, defines an isomorphism between the
underlying oriented real bundles $\Lambda_{0}$ and $\Lambda_{1}$. Therefore
$\Lambda_{0}$ and $\Lambda_{1}$ have the same Euler class and thus the same
degree. Since
$\mathrm{Hom}(\Gamma,\mathbb{C}^{\star})\simeq(\mathbb{C}^{\star})^{2g}$ is
path-connected, the degree of $\Lambda$ does not depend on $\varphi$. ∎

It is shown in [GKM00, Section C] that

$\deg L=g-1-\frac{1}{2}\sum_{i}n_{i}.$

Therefore we have $\sum_{i}n_{i}\geqslant 2g-2$, with equality if and only if
$\infty\notin f(\tilde{\Sigma})$. ∎

We have seen in Section 4.2.1 that the group $\mathrm{GL}_{2}^{+}(\mathbb{R})$
acts on $\mathrm{Hom}(\Gamma,\mathbb{Z}_{2}\rtimes\mathbb{C})$, the set of
homomorphisms with image in the group $\mathrm{Aff}(\mathbb{C})$ and linear
part with finite image of order at most $2$. As a corollary of 6.18, we see
that this action preserves the property of being in the image of
$\mathrm{Hol}$ if $\sum_{i}n_{i}=2g-2$.

######  Corollary 6.21.

Let $\rho\in\mathrm{Hom}(\Gamma,\mathbb{Z}_{2}\rtimes\mathbb{C})$ and
$A\in\mathrm{GL}_{2}^{+}(\mathbb{R})$. For every $1\leqslant
n_{1}\leqslant\ldots\leqslant n_{k}$ such that $\sum_{i}n_{i}=2g-2$, we have:

$\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))\iff
A\cdot\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k})).$

###### Proof.

If $\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$, then there exists a
developing map $f:\tilde{\Sigma}\to\mathbb{CP}^{1}$ that is
$\rho$-equivariant. By 6.18, $f$ takes its values in $\mathbb{C}$. Let us see
$A$ as a map $A:\mathbb{C}\to\mathbb{C}$. The map $A\circ f$ is a developing
map for a projective structure in $\mathcal{P}(n_{1},\ldots,n_{k})$ with
holonomy $A\cdot\rho$. ∎

####  6.3.2. Euclidean holonomy

We now focus on the case where $\rho$ is Euclidean. Recall that it means that
the linear part
$\alpha=\mathrm{Li}\circ\rho\in\mathrm{Hom}(\Gamma,\mathbb{C}^{\star})$ of
$\rho$ has its image in $\mathbb{S}^{1}=\\{z\in\mathbb{C}\mid
z\overline{z}=1\\}$. Let us explain 4.

######  Proposition 6.22.

Suppose that $\rho\in\mathrm{Hom}(\Gamma,\mathrm{Isom}^{+}(\mathbb{E}^{2}))$
is the holonomy of an affine structure. We have $\mathrm{Vol}(\rho)>0$.

###### Proof.

We can identify $\tilde{\Sigma}$ with $Y$, where $Y=\mathbb{H}^{2}$ if
$g\geqslant 2$ and $Y=\mathbb{C}$ if $g=1$, by the uniformization theorem and
consider a holomorphic developing map $f:Y\to\mathbb{C}$. Let us choose
$D\subset Y$ a compact fundamental domain for the action of $\Gamma$ on $Y$.
We have:

$\mathrm{Vol}(\rho)=\int_{D}f^{*}(dx\wedge dy)=\int_{D}\det(df)dx\wedge dy.$

Since $f$ is holomorphic and nonconstant, $\det(df)\geqslant 0$ everywhere and
$\det(df)$ is continuous and nonzero. ∎

##### Trivial linear part

Let us first assume that $\alpha$ is the trivial homomorphism. Let $1\leqslant
n_{1}\leqslant\ldots\leqslant n_{k}$ be such that $\sum_{i}n_{i}=2g-2$. By
6.18 we are asking if $\rho$ is the holonomy of a translation surface with
prescribed singularities. This question was addressed by the author in [LF20]
and independently by Bainbridge, Johnson, Judge and Park in [BJJP20], where
the following is proven.

######  Refined Haupt’s theorem.

We have $\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ if and only if

  1. (1)

$\mathrm{Vol}(\rho)>0$

  2. (2)

$\mathrm{Vol}(\rho)\geqslant(\max_{i}n_{i}+1)\mathrm{Area}(\mathbb{C}/\Lambda)$
if $\Lambda=\\{z_{0}\in\mathbb{C}\mid z+z_{0}\in\rho(\Gamma)\\}$ is a lattice
in $\mathbb{C}$.

Let us remark that Refined Haupt’s theorem shows that 4 and 5 are the only
obstructions of being in $\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ for a
representation $\rho\in\mathrm{Hom}(\Gamma,\mathbb{C})$ if
$\sum_{i}n_{i}=2g-2$.

We now suppose $\sum_{i}n_{i}\geqslant 2g$. Since $\sum_{i}n_{i}$ is not
minimal, the developing maps we are considering have
$\infty\in\mathbb{CP}^{1}$ in their image. Therefore our projective structures
may be obtained by performing surgeries on $\mathbb{CP}^{1}$.

######  Proposition 6.23.

Let $1\leqslant n_{1}\leqslant\ldots\leqslant n_{k}$ be such that
$\sum_{i}n_{i}$ is even and at least $2g$. There exists a branched projective
structure in $\mathcal{P}(n_{1},\ldots,n_{k})$ with holonomy $\rho$.

###### Proof.

We construct a projective structure on $\Sigma_{g}$ with branching data
$n_{1},\ldots,n_{k}$ and holonomy in
$\overline{(\mathrm{Aut}^{+}(\Gamma)\times\mathrm{Aff}(\mathbb{C}))\cdot\rho}$.
It will then follow from 5.6 that
$\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$.

Let us consider the Riemann sphere $\mathbb{CP}^{1}$. We choose
$x\in\mathbb{C}\subset\mathbb{CP}^{1}$ and cut open $\mathbb{CP}^{1}$ along
the segment lines $l_{0}=[x,x+1]$ and $l_{k}=[x+2k-1,x+2k]$ for $1\leqslant
k\leqslant g$. We glue the bottom side of $l_{i}$ to the top side of
$l_{i+1}$, in cyclic notation, as indicated in Figure 19.

$1$
---
$5$
---
$2$
---
$1$
---
$3$
---
$2$
---
$4$
---
$3$
---
$5$
---
$4$
---
$\mathbb{C}$
---
Figure 19. Construction for $g=4.$

We thus get a genus $g$ surface with charts in $\mathbb{CP}^{1}$, and a unique
branch point of order $2g$. Moreover its holonomy is given by
$(z+1,z+1,z+2,z,\ldots z+2,z)$, that lies in
$\mathrm{Aut}^{+}(\Gamma)\cdot\rho_{\infty}$, where $\rho_{\infty}$ comes from
Lemma 4.11. Hence this holonomy is in
$\overline{\mathrm{Aut}^{+}(\Gamma)\times\mathrm{Aff}(\mathbb{C})\cdot\rho}$
by Lemma 4.11. Observe that the closed curve following $l_{0}$ develops
injectively. Therefore by 5.7, $\rho_{\infty}\in\mathrm{Hol}(\mathcal{P}(2k))$
for every $k\geqslant g$. It follows from 5.9 and the Ehresmann-Thurston
principle 5.6 that $\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$. ∎

##### Nontrivial linear part

We now suppose that $\alpha$ has finite image of order $n\geqslant 2$. Let us
first suppose that $1\leqslant n_{1}\leqslant\ldots\leqslant n_{k}$ are such
that $\sum_{i}n_{i}$ is even and at least $2g$. We may perform the same
surgeries as in 6.23.

######  Proposition 6.24.

There exists a branched projective structure in
$\mathrm{Hol}(n_{1},\ldots,n_{k})$ with holonomy $\rho$.

###### Proof.

Our construction is very similar to the case $n=1$, where we cut and glue
$\mathbb{CP}^{1}$ as in Figure 19. We will just flip one of the segment lines,
see Figure 20. More precisely, we start as before with the Riemann sphere
$\mathbb{CP}^{1}$. We cut open the sphere $\mathbb{CP}^{1}$ along the line
segments $[x,x+1]$, $[x+1,x+2],[x+3,x+4],\ldots,[x+2g-3,x+2g-2]$ and $[\zeta
x,\zeta x+\zeta]$, where $x\in\mathbb{C}$ is chosen so that these line
segments do not intersect. We glue back the resulting boundary components as
indicated in Figure 20.

$1$
---
$2$
---
$1$
---
$3$
---
$2$
---
$4$
---
$3$
---

$5$
---

$4$
---

$\mathbb{C}$
---
$5$
---
Figure 20. Construction for $g=4$ and $n=4$.

We thus get a genus $g$ surface with charts in $\mathbb{CP}^{1}$, a single
branch point of order $2g$ and holonomy in
$\mathrm{Aut}^{+}(\Gamma)\cdot\rho_{\infty}$, thus in
$\overline{\mathrm{Aut}^{+}(\Gamma)\times\mathrm{Aff}(\mathbb{C})\cdot\rho}$.
The curve following the line segment $[x,x+1]$ develops injectively, thus
$\rho_{\infty}\in\mathrm{Hol}(\mathcal{P}(2k))$ for every $k\geqslant g$ by
5.7. Therefore $\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ by 5.9
and 5.6. ∎

###### Minimal degree

We suppose that $\alpha$ has finite image of order $n\geqslant 2$ and that
$1\leqslant n_{1}\leqslant\ldots\leqslant n_{k}$ are such that
$\sum_{i}n_{i}=2g-2$. Let us first make an observation on the volume of the
pullback.

######  Remark 6.25.

Suppose that $f:\Sigma\to S$ is a branched cover and that $S$ is endowed with
a branched projective structure $X$ with holonomy $\rho$. The volume of the
holonomy of $f^{*}X$ is $\deg(f)\cdot\mathrm{Vol}(\rho)$.

This observation leads to another obstruction to being in
$\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ that we now describe. Let
$\Lambda$ be the subgroup of $\mathbb{C}$ generated by the complex numbers $a$
such that the Möbius transformation $z+a$ belongs to $\ker\alpha$. Suppose
that $\Lambda$ is a lattice in $\mathbb{C}$. If $\rho$ is the holonomy of the
developing map $f$, then since $f$ has image in $\mathbb{C}$ by 6.18, it
induces a branched cover $S\to\mathbb{C}/\Lambda$, where $S$ is the cover of
$\Sigma$ associated with $\ker\alpha$. The degree $d$ of this branched cover
must satisfy $d\geqslant\max_{i}n_{i}+1$, since the branched cover has the
local form $z\mapsto z^{n_{i}+1}$ at the conical points. Moreover $d$ is also
given by $d=n\mathfrak{\mathrm{vol}(\rho)}/{\mathrm{Area(\Lambda)}}$. This
explains 5. Note that when $n=1$ this obstruction is included in the statement
of the refined Haupt’s theorem.

######  Example 6.26.

Every half-translation surface of genus $2$ with a single conical point is a
translation surface. Indeed let $\rho$ be the holonomy of an even half-
translation surface with nontrivial $\alpha=\mathrm{Li}\circ\rho$. Since
$\alpha(\Gamma)=\\{\pm 1\\}$, we can suppose that
$\rho=(-z,z,z+z_{2},z+z_{3})$ by Lemma 4.10. We have $\Lambda=\langle
z_{2},z_{3}\rangle$ and
$\mathrm{Area}(\Lambda)=\det(z_{2},z_{3})=\mathrm{Vol}(\rho)$. Thus the group
$\Lambda$ generated by $z_{2}$ and $z_{3}$ is a lattice in $\mathbb{C}$. 5
gives $2\mathrm{Vol}(\rho)\geqslant(n_{i}+1)\mathrm{Vol}(\rho)$, thus
$n_{i}=1$ for all $i$. Since $\sum_{i}n_{i}=2g-2=2$, there are $2$ conical
points. In other words, we have $\mathcal{Q}(4)=\emptyset$, following the
notation of [Lan08] ; see also [MS93, Lan08] for alternative proofs of this
fact.

Let us show that 4 and 5 are in this case the only obstructions to being in
$\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$.

######  Proposition 6.27.

If $\mathrm{Vol}(\rho)>0$ and
$n\mathrm{Vol}(\rho)\geqslant(\max_{i}n_{i}+1)\mathrm{Area}(\Lambda)$ if
$\Lambda$ is a lattice, then
$\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$.

We suppose that $\rho$ satisfies the conditions of 4 and 5. By Lemma 4.10, we
may assume that

$\rho=(e^{2\pi i/n}z,e^{2\pi i/n}z,z+x_{2},z+y_{2},\ldots,z+x_{g},z+y_{g}).$

##### Case $g=2$.

Let us first suppose that $g=2$ and $n\geqslant 3$. We consider a
parallelogram $P\subset\mathbb{C}$ whose vertices are $z_{0}$, $z_{0}+x_{2}$,
$z_{0}+y_{2}$ and $z_{0}+x_{2}+y_{2}$, where $z_{0}\in\mathbb{C}$ is such that
$0$ is in the interior of $P$. Note that it is indeed a parallelogram since
$\mathrm{Vol}(\rho)=\det(x_{2},y_{2})>0$. We are going to add a handle to the
torus obtained by gluing its opposite edges. Let $\epsilon>0$ be such that the
disk $D=\\{z\in\mathbb{C}\mid|z|<\epsilon\\}$ is included in the interior of
$P$. Let us cut open $P$ along the circular segment joining $z_{0}\in D$ and
$e^{2\pi/n}z_{0}$, and along the circular segment joining $e^{2\pi/n}z_{0}$
and $e^{4\pi/n}z_{0}$. We then glue the boundary components as indicated in
Figure 21.

$\epsilon$
---
$i\epsilon$
---
$-\epsilon$
---
$+$
---
$-$
---
$+$
---
$-$
---
Figure 21. Construction for $g=2$ and $n=4$.

The topological result of this surgery is to add a handle. Moreover the
resulting genus $2$ surface comes naturally with charts in $\mathbb{C}$ and
has holonomy $\rho$. Thus $\rho\in\mathrm{Hol}(\mathcal{P}(2))$ and also
$\rho\in\mathrm{Hol}(\mathcal{P}(1,1))$ by 5.8. If $n=2$, then we have seen in
6.26 that we cannot have $\rho\in\mathrm{Hol}(\mathcal{P}(2))$. However we
have $\rho\in\mathrm{Hol}(\mathcal{P}(1,1))$: it suffices to cut open $P$
along a line segment $l$ and along $-l$. We make sure that these line segments
do not intersect and are both in $P$. We then identify the opposites sides of
the boundary components as in Figure 22 with the map $z\mapsto-z$. Note that
this shows that $\mathcal{Q}(2,2)\neq\emptyset$.

$x$
---
$y$
---
$-x$
---
$-y$
---
$+$
---
$-$
---
$-$
---
$+$
---
Figure 22. Construction for $g=2$ and $n=2$.

######  Remark 6.28.

Observe that if $g=1$, then $\rho$ must be spherical by Lemma 4.10. Indeed, we
can suppose that $\rho=(e^{2\pi i/n}z,z)$, therefore we cannot have
$\sum_{i}n_{i}=0$. In particular this shows that
$\mathcal{Q}(\emptyset)=\emptyset$.

We now suppose that $g\geqslant 3$.

##### The generic case.

Let us first deal with the case where the group $\Lambda$ that is generated by
the $e^{2\pi ki/n}x_{j}$ and $e^{2\pi ki/n}y_{j}$, $k\in\mathbb{Z}$ is not a
lattice. We also suppose that $n\geqslant 3$. It follows from Lemma 4.16 that
we can assume that the group generated by the $x_{j}$ and $y_{j}$ is not a
lattice. We may consider a translation surface structure on $\Sigma_{g-1}$
with holonomy $(z+x_{2},\ldots,z+y_{g})$ and a single conical point
$p\in\Sigma_{g-1}$ by the refined Haupt’s theorem. We can translate the charts
so that there exist $\epsilon>0$ and a chart $\varphi:U\to V$, where $U\subset
S$ and $V\subset\mathbb{C}$ are open, such that $p\in U$, and $\varphi(p)\in
D(0,\epsilon)\setminus\\{0\\}$ and $D(0,\epsilon)\subset V$. We can then add a
handle with holonomy $(e^{2\pi i/n}z,z)$ as in the genus $g=2$ case. We make
sure that the new slit starts from the conical point of the surface. The
resulting surface has holonomy $\rho$ and a single conical point of order
$2g-2$. It follows from 5.9 that
$\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$.

Let us turn to the case $n=2$. If $g\geqslant 4$, there exists
$\rho_{\infty}\in\mathrm{Aut}^{+}(\Gamma)\cdot\rho$ such that
$\Lambda_{\infty}=\\{a\in\mathbb{C}\mid z+a\in\rho_{\infty}(\Gamma)\\}$ is a
lattice satisfying
$\mathrm{Vol}(\rho_{\infty})\geqslant(\max_{i}n_{i}+1)\mathrm{Area}(\Lambda_{\infty})$
by Lemma 4.13. We will see in the next paragraph that this implies
$\rho_{\infty}\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$. Therefore
$\rho\in\mathcal{\mathrm{Hol}}(\mathcal{P}(n_{1},\ldots,n_{k}))$ by the
Ehresmann-Thurston principle 5.6. We now turn to the case $g=3$ and $n=2$. Let
us recall a result from [LF20, Section 4, Proposition 4.2].

######  Lemma 6.29.

Let $\chi\in\mathrm{Hom}(\Gamma_{2},\mathbb{C})$ with positive volume and
image $\chi(\Gamma_{2})$ that is not a lattice. There exists
$\chi^{\prime}\in\mathrm{Aut}^{+}(\Gamma_{2})\times\mathrm{GL}_{2}^{+}(\mathbb{R})\cdot\chi$
such that $\chi^{\prime}(a_{1})=1$, $\chi^{\prime}(b_{1})=i$ and
$|\chi^{\prime}(a_{2})|<1$.

We may assume that our representation has the form
$\rho=(-z,z,x_{1},y_{1},x_{2},y_{2})$, with the representation $\chi^{\prime}$
defined by $\chi^{\prime}(a_{i})=x_{i}$ and $\chi^{\prime}(b_{i})=y_{i}$ for
$i=1,2$ as in Lemma 6.29, by Lemma 2.2. We consider a parallelogram $P$ in
$\mathbb{C}$ whose sides are given by $x_{1}$ and $y_{1}$. Let us consider a
line segment $\ell=[z_{0},z_{0}+x_{2}]$ that is contained in $P$. We may
translate both $P$ and $\ell$, so that $\ell$ is so close to $0$ that $-\ell$
is also contained in $P$. We make slits along $\ell$ and $-\ell$, and glue
back a handle given by identifying two sides of a parallelogram whose sides
are given by $x_{2}$ and $y_{2}$, as indicated in Figure 23.

$0$
---

$++$
---

$++$
---

$--$
---

$--$
---

$==$
---

$==$
---

Figure 23. Construction for $g=3$ and $n=2$.

We thus get a genus $3$ surface with holonomy given by

$(z+1,z+i,z+x,-z+y,z-x,-z).$

It follows from Lemma 4.12 that this representation is in
$\mathrm{Aut}^{+}(\Gamma)\times\mathrm{GL}_{2}^{+}(\mathbb{R})\cdot\rho$.
Therefore $\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$.

##### The lattice case.

Let us suppose that $n=4$ and $\Lambda=L=\mathbb{Z}[i]$. We consider a
rectangle $R$ whose sides are given by $m=\mathrm{Vol}(\rho)$ and $1$. Let us
identify the opposite sides of $R$ with translations. Let us draw $m$ annuli
in $R$ obtained by translating one of them by integers, see Figure 24. We make
the slits and gluing in these annuli explained in Section 6.2.2 for the
branched datum $n_{1},\ldots,n_{k}$. Since $\max_{i}n_{i}<4m$, we have room to
make the slits. Since $g\geqslant 3$, we can also make sure that the resulting
holonomy $\rho_{0}$ is a surjective homomorphism of
$\mathrm{Hom}(\Gamma,\mathbb{Z}_{4}\rtimes\mathbb{C})$, with
$\Lambda(\rho_{0})=\mathbb{Z}[i]$ and such that $\mathrm{Vol}(\rho)=m$. It
follows from Lemma 4.14 that
$\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$.

Figure 24. Annuli in $R$ for $(n,m)=(4,3)$

If $\Lambda=L=\mathbb{Z}[\omega]$, that is $n\in\\{3,6\\}$, we consider the
parallelogram whose sides are given by $m=\mathrm{Vol}(\rho)$ and
$\omega=e^{2\pi i/3}$, and draw $m$ annuli obtained by integer translations.
We then make slits and gluing in these annuli corresponding to
$n_{1},\ldots,n_{k}$.

If $n=2$, we may assume that $L=\Lambda=\mathbb{Z}[i]$: we may replace $\rho$
with $A\cdot\rho$, $A\in\mathrm{GL}_{2}^{+}(\mathbb{R})$ if necessary. The
construction is then similar to the case $n=4$.

##### Infinite linear part

Let us suppose that
$\rho\in\mathrm{Hom}(\Gamma,\mathrm{Isom}^{+}(\mathbb{E}^{2}))$ is such that
$\alpha=\mathrm{Li}\circ\rho$ is infinite. Suppose $1\leqslant
n_{1}\leqslant\ldots\leqslant n_{k}$ are such that $\sum_{i}n_{i}=2g-2$. Let
us show that here, 4 is the only obstruction to being in
$\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$.

######  Proposition 6.30.

The representation $\rho$ is in
$\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ if and only if
$\mathrm{Vol}(\rho)>0$.

###### Proof.

Let us suppose that $\mathrm{Vol}(\rho)>0$. There exists
$\rho^{\prime}\in\mathrm{Hom}(\Gamma,\mathrm{Isom}^{+}(\mathbb{E}^{2}))$ such
that $\rho^{\prime}\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ and
$\mathrm{Vol(\rho^{\prime})}=\mathrm{Vol(\rho)}$. Indeed we can for example
consider a translation surface with branch data $n_{1},\ldots,n_{k}$ and
rescale this translation surface so that it has the required volume. We then
take $\rho^{\prime}$ to be its holonomy. Since $\rho^{\prime}$ is in the
closure of
$\mathrm{Aut}^{+}(\Gamma)\times\mathrm{Isom}^{+}(\mathbb{E}^{2})\cdot\rho$ by
4.20, we have $\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ by the
Ehresmann-Thurston principle 5.6. ∎

####  6.3.3. Strictly affine holonomy

In this section we geometrize strictly affine representations. By strictly
affine, we mean that $\rho\in\mathrm{Hom}(\Gamma,\mathrm{Aff}(\mathbb{C}))$ is
such that
$\alpha=\mathrm{Li}\circ\rho\in\mathrm{Hom}(\Gamma,\mathbb{C}^{\star})$ does
not have its image contained in $\mathbb{S}^{1}\subset\mathbb{C}^{\star}$.
This is equivalent to saying that $\rho$ is not Euclidean. We show that the
only obstruction to being in $\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$
is that $\sum_{i}n_{i}$ must be even and at least $2g-2$.

######  Proposition 6.31.

Let $\rho$ be a strictly affine representation. We have
$\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ if $\sum_{i}n_{i}$ is
even and at least $2g-2$.

By Lemma 4.6 and the Ehresmann-Thurston principle 5.6, it suffices to show
that $\mathrm{Li}\circ\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$.

######  Definition 6.32.

The exponential of a representation $\chi\in\mathrm{Hom}(\Gamma,\mathbb{C})$
is the representation defined by
$\exp(\chi):\gamma\in\Gamma\mapsto\exp(\chi(\gamma))z\in\mathrm{Aff}(\mathbb{C})$.

######  Remark 6.33.

Suppose $\chi\in\mathrm{Hom}(\Gamma,\mathbb{C})$ is the holonomy of a
translation surface structure with developing map
$f:\tilde{\Sigma}\to\mathbb{C}$. The map $\exp\circ f$ is
$\exp(\chi)$-equivariant.

######  Lemma 6.34.

There exists a representation $\chi\in\mathrm{Hom}(\Gamma,\mathbb{C})$ such
that:

  1. (1)

$\exp(\chi)=\mathrm{Li}\circ\rho$

  2. (2)

$\chi$ is the holonomy of a translation surface with a single conical point.

###### Proof.

These properties are invariant by the action of $\mathrm{Aut}^{+}(\Gamma)$. We
can thus modify $\rho$ so that both $\alpha(a_{1})$ and $\alpha(a_{2})$ are
not in $\mathbb{S}^{1}$, where $\alpha=\mathrm{Li}\circ\rho$. Let us first
choose $\chi(\gamma)\in\mathbb{C}$ such that
$\alpha(\gamma)=\exp(\chi(\gamma))$ for each
$\gamma\in\\{a_{1},\ldots,b_{g}\\}$. We thus have $\exp(\chi)=\alpha$. We may
now replace $\chi(b_{1})$ with $\chi(b_{1})+2\pi ik$ where $k\in\mathbb{Z}$ so
that $\det(\chi(a_{1}),\chi(b_{1}))\neq 0$. We can then replace $\chi(b_{2})$
with $\chi(b_{2})+2\pi ik$ so that $\mathrm{Vol}(\chi)\geqslant
2g|\det(\chi(a_{1}),\chi(b_{1}))|$. Therefore if the group $\Lambda$ generated
by the $\chi(\gamma)$ for $\gamma\in\\{a_{1},\ldots,b_{g}\\}$ is a lattice,
its area is bounded above by $|\det(\chi(a_{1}),\chi(b_{1}))|$, and
$\mathrm{Vol}(\chi)\geqslant 2g\mathrm{Area}(\Lambda)$. It follows from the
refined Haupt’s theorem that $\chi$ is the holonomy of a translation surface
with a single conical point of angle $2\pi(2g-1)$. ∎

The explicit constructions made in [LF20] show that if $\chi$ is the holonomy
of a translation surface with a single conical point, then $\chi$ is also the
holonomy of another translation surface with a single conical point such that
there exists a simple closed curve $c$ that develops injectively on a line
that is not parallel to the imaginary axis. The exponential of the associated
developing map gives a translation surface with a single conical point and
holonomy $\alpha=\exp(\chi)$. The closed curve $c$ based on the conical point
develops injectively. Therefore by 5.7, we have
$\alpha\in\mathrm{Hol}(\mathcal{P}(2d))$ for each $d\geqslant g-1$, and thus
$\alpha\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ if $\sum_{i}n_{i}$ is
even and at least $2g-2$ by 5.9.

###  6.4. Dihedral holonomy

In this section, we suppose that
$\rho\in\mathrm{Hom}(\Gamma,\mathrm{PSL}_{2}(\mathbb{C}))$ is dihedral but not
affine, that is its image $\rho(\Gamma)$ fixes globally a pair of points in
$\mathbb{CP}^{1}$, but not pointwise. After conjugating $\rho$, we may assume
that $\rho(\Gamma)$ is included in the group $\\{z\mapsto cz^{\epsilon}\mid
c\in\mathbb{C},\epsilon=\pm 1\\}$. We require that $\rho$ is not conjugated
into $\mathrm{SO}_{3}(\mathbb{R})$ nor in $\mathrm{Aff}(\mathbb{C})$, that is
there exist $\gamma$ and $\delta$ in $\Gamma$ such that
$\rho(\gamma)=c_{\gamma}z$ with $|c_{\gamma}|\neq 1$ and
$\rho(\delta)=c_{\delta}z^{\epsilon(\delta)}$ with $\epsilon(\delta)=-1$.

Observe that $\epsilon:\Gamma\to\\{\pm 1\\}$ is a group homomorphism. Thus we
may assume that it satisfies $\epsilon(a_{1})=-1$ and $\epsilon(\gamma)=1$ for
$\gamma\in\\{b_{1},a_{2},\ldots b_{g}\\}$ by 3.2. We may conjugate $\rho$ with
an isometry of the form $\begin{pmatrix}\lambda&0\\\
0&\lambda^{-1}\end{pmatrix}$ so that $c_{\gamma}=1$ for $\gamma=a_{1}$. Since
$\rho([a_{i},b_{i}])=\mathrm{id}$ for $i\geqslant 2$, we must have
$\rho([a_{1},b_{1}])=\mathrm{id}$. Therefore $c_{\gamma}\in\\{1,-1\\}$ for
$\gamma=b_{1}$. It follows from Lemma 3.7 that $\rho$ lifts to
$\mathrm{SL}_{2}(\mathbb{C})$ if and only if $c_{\gamma}=1$, where
$\gamma=b_{1}$. Note that $\rho(\Gamma)$ is not abelian. Therefore we assume
that $g\geqslant 2$ in this section.

####  6.4.1. Even Stiefel Whitney class

We first assume that $\rho$ lifts to $\mathrm{SL}_{2}(\mathbb{C})$:
$c_{\gamma}=1$ for $\gamma=b_{1}$. We may extend the definition of the
exponential of a representation to the representations of the form
$\rho^{\prime}:\gamma\mapsto\epsilon(\gamma)z+c_{\gamma}$, where
$\epsilon\in\mathrm{Hom}(\Gamma,\\{\pm 1\\})$.

######  Definition 6.35.

  1. (1)

The exponential of $\rho^{\prime}$ is the representation

$\exp(\rho^{\prime}):\gamma\mapsto\exp(c_{\gamma})z^{\epsilon(\gamma)}.$

  2. (2)

The exponential of a half-translation surface $Y$ with developing map $f$ is
the projective structure $X=\exp(Y)$ with developing map $\exp\circ f$.

######  Remark 6.36.

We have $\mathrm{Hol}\circ\exp=\exp\circ\mathrm{Hol}$.

######  Proposition 6.37.

Let $X\in\mathcal{P}(n_{1},\ldots,n_{k})$ with $\sum_{i}n_{i}=2g-2$ and
holonomy $\rho$. There exists a half-translation surface structure $Y$ on
$\Sigma$ such that $X=\exp(Y)$.

###### Proof.

Let us consider a developing map $f:\tilde{\Sigma}\to\mathbb{C}$ associated
with $X$. We consider $S$ the double-cover of $\Sigma$ associated with
$\ker\epsilon\triangleleft\Gamma$. The developing map induces an affine
structure on $S$ with a branching divisor of degree $-2\chi(\Sigma)=-\chi(S)$.
Therefore the image of $f$ does not contain $\infty\in\mathbb{CP}^{1}$ by
6.18. There exists $\lambda\in\mathbb{C}^{\star}$ such that $z\mapsto\lambda
z^{-1}$ is in $\rho(\Gamma)$, thus $f(\tilde{\Sigma})$ does not contain $0$
either. Now let $\tilde{f}:\Sigma\to\mathbb{C}$ be the lift of
$f:\tilde{\Sigma}\to\mathbb{C}^{\star}$ to the universal cover $\mathbb{C}$ of
$\mathbb{C}^{\star}$. Recall that $\exp:\mathbb{C}\to\mathbb{C}^{\star}$ is
the associated covering map. For every $\gamma\in\Gamma$, and for every
$z\in\mathbb{C}$, we have $\exp(\tilde{f}(\gamma\cdot z))=f(\gamma\cdot
z)=c_{\gamma}f(z)^{\epsilon(\gamma)}$, where $\rho(\gamma):z\mapsto
c_{\gamma}z^{\epsilon(\gamma)}$. Therefore, $\tilde{f}(\gamma\cdot
z)=\epsilon(\gamma)\tilde{f}(z)+l_{\gamma}$ where
$\exp(l_{\gamma})=c_{\gamma}$. By continuity, $l_{\gamma}$ does not depend on
$z$. We thus define a map
$\rho^{\prime}:\Gamma\to\mathrm{Isom}^{+}(\mathbb{E}^{2})$ by
$\rho^{\prime}(\gamma)=\epsilon(\gamma)z+l_{\gamma}$. It follows from the
$\rho^{\prime}$-equivariance of $\tilde{f}$ that $\rho^{\prime}$ is a group
homomorphism. Indeed the relation

$\tilde{f}(\gamma_{1}\gamma_{2}\cdot
z)=\rho^{\prime}(\gamma_{1}\gamma_{2})\cdot\tilde{f}(z)=\rho^{\prime}(\gamma_{1})\rho^{\prime}(\gamma_{2})\cdot\tilde{f}(z)$

gives
$\rho^{\prime}(\gamma_{1}\gamma_{2})=\rho^{\prime}(\gamma_{1})\rho^{\prime}(\gamma_{2})$
since $\tilde{f}$ is open and any element of $\mathrm{PSL}_{2}(\mathbb{C})$ is
determined by its action on $3$ points in $\mathbb{CP}^{1}$. The half-
translation surface structure $Y$ with developing map $\tilde{f}$ satisfies
$\exp(Y)=X$. ∎

We now explain 6. Let us consider a projective structure $X$ on $\Sigma_{2}$
with branching divisor of degree $2$ and with dihedral holonomy that is not
affine. There exists a half-translation structure $Y$ on $\Sigma_{2}$ such
that $X=\exp(Y)$. We have seen in 6.26 that it is not possible for $Y$ to have
a single conical point, thus it must have two conical points of angle $4\pi$
each and so must $X$.

Let us see that this obstruction only occurs when $(g,k)=(2,1)$.

######  Proposition 6.38.

Suppose $1\leqslant n_{1}\leqslant\ldots\leqslant n_{k}$ are such that
$\sum_{i}n_{i}=2g-2$. The representation $\rho$ is in
$\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ if and only if
$(g,k)\neq(2,1)$.

###### Proof.

Let us first suppose that $g\geqslant 3$. We may suppose that
$|c_{\gamma}|\neq 1$ for $\gamma\in\\{a_{2},a_{3}\\}$, since the map
$\gamma\in\Gamma_{g-1}\mapsto\log(|c_{\gamma}|)\in\mathbb{R}$ is a non trivial
homomorphism of $\mathrm{Hom}(\Gamma_{g-1},\mathbb{R})$. We take a
representation $\rho^{\prime}:\gamma\mapsto\epsilon(\gamma)z+l_{\gamma}$ with
$\exp(\rho^{\prime})=\rho$. We may also suppose that $\rho^{\prime}$ is
geometric: changing $\rho^{\prime}(\gamma)$ with $\rho^{\prime}(\gamma)+2k\pi
i$ for $\gamma\in\\{b_{2},b_{3}\\}$, we can make sure that
$\mathrm{Vol}(\rho^{\prime})>0$ and that
$2\mathrm{Vol}(\rho^{\prime})\geqslant(n_{k}+1)\mathrm{Area}(\Lambda)$ in the
case where $\Lambda=\\{z_{0}\in\mathbb{C}\mid
z+z_{0}\in\rho^{\prime}(\Gamma)\\}$ is a lattice in $\mathbb{C}$. Therefore,
there exists by 6.27 a half-translation structure $Y$ with holonomy
$\rho^{\prime}$ and a single conical point. We consider its exponential
$X=\exp(Y)$.

Let us now turn to the case $g=2$ and $k\geqslant 2$. Observe that $k=2$ and
$n_{1}=n_{2}=1$ since $\sum_{i}n_{i}=2$. We may consider $\rho^{\prime}$ as
before such that $\rho=\exp(\rho^{\prime})$, where
$\rho^{\prime}=(z+x,z+y,-z,z)$. We may also assume that
$\mathrm{Vol}(\rho)=\det(x_{1},y_{1})$ is positive; replacing $x$ with
$x+2k\pi i$ for some $k\in\mathbb{Z}$ if necessary (or $y$ with $y+2k\pi i$).
We geometrize $\rho^{\prime}$ by 6.27 and take the exponential of the
associated developing map. ∎

We now suppose that $1\leqslant n_{1}\leqslant\ldots\leqslant n_{k}$ are such
that $\sum_{i}n_{i}$ is even and at least $2g$. We show that there are no more
obstructions to being in $\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$.

######  Proposition 6.39.

The representation $\rho$ is in
$\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$.

$1$
---
$3$
---
$3$
---
$2$
---
$\mathbb{C}$
---
$1$
---
$1+\alpha$
---
$-1$
---
$-1-\alpha$
---
$1$
---
$2$
---
Figure 25. Construction for $(g,k,n_{1})=(2,1,4)$.

###### Proof.

Let us first suppose that $g\geqslant 3$. We have seen that
$\rho\in\mathrm{Hol}(\mathcal{P}(2g-2))$. We may check on the constructions we
have made that we can suppose that there exists a curve based at the conical
point of $Y$ whose exponential develops injectively, where $Y$ comes from the
proof of 6.38. Therefore by 5.7 we have $\rho\in\mathrm{Hol}(\mathcal{P}(2d))$
for every $d\geqslant g-1$, and
$\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ by 5.8.

Let us now suppose that $g=2$. Let us denote $\rho=(z^{-1},z,c_{2}z,c_{3}z)$.
We can assume that $|c_{2}|\neq 1$, since $\rho$ is not spherical. Let us take
$\alpha$ and $\beta\in\mathbb{C}$ such that $e^{\alpha}=c_{2}$ and
$e^{\beta}=c_{3}$. We may assume that $\det(\alpha,\beta)>0$, replacing
$\beta$ with $\beta+2\pi ik$ if necessary. Let us consider a parallelogram in
$\mathbb{C}$ whose sides are given by $\alpha$ and $\beta$. We glue the sides
corresponding to $\beta$ with the map $z\mapsto z+\alpha$. This gives
projective charts on an annulus. We define new charts on this annulus by
considering the exponential of the associated developing map. Let us consider
the path $\delta_{1}=\exp\circ c$, where $c$ is the path along the line
segment in $\mathbb{C}$ joining $1$ to $1+\alpha$, and $\delta_{2}=\exp\circ
d$, where $d=-c$ is the path along the line segments from $-1$ to $-1-\alpha$.
Let us cut open $\mathbb{CP}^{1}$ along the image of $\delta_{i}$ for $i=1,2$.
We glue the resulting boundaries to those of the annulus, see Figure 25. We
obtain a genus $2$ surface with holonomy
$\rho_{1}=(c_{2}z,c_{3}z^{-1},c_{2}^{-1}z,z^{-1})=\exp(\rho^{\prime})$, where
$\rho^{\prime}=(z+\alpha,-z+\beta,z-\alpha,-z)$. It follows from Lemma 4.12
that the representation $\rho^{\prime\prime}=(-z,z,z+\alpha,z+\beta)$ is in
$\mathrm{Aut}^{+}(\Gamma)\cdot\rho^{\prime}$. Therefore,
$\rho=\exp(\rho^{\prime\prime})$ is in
$\mathrm{Aut}^{+}(\Gamma)\cdot\rho_{1}$. Since the curve $\delta$ develops
injectively, we have $\rho\in\mathrm{Hol}(\mathcal{P}(2d))$ for every
$d\geqslant 2$ by 5.7. Hence
$\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ by 5.8. ∎

####  6.4.2. Odd Stiefel Whitney class

In this subsection we suppose that $\rho$ does not lift to
$\mathrm{SL}_{2}(\mathbb{C})$, that is $c_{\gamma}=-1$ for $\gamma=b_{1}$.

######  Proposition 6.40.

We have $\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ if and only if
$\sum_{i}n_{i}$ is odd and at least $2g-1$.

###### Proof.

There exists a branched projective structure $X$ on $\Sigma_{g-1}$ with a
single branch point of order $2g-4$ and holonomy
$(c_{a_{2}}z,c_{b_{2}}z,\ldots,c_{b_{g}}z)$ by 6.31, which is obtained by
taking the exponential of a translation surface structure $Y$. Observe that we
may translate the charts of $Y$, so that the conical point is sent to
$z\in\mathbb{R}_{>0}$, such that the line segment $[-z,z]$ is included in the
image of this chart. We will now make a surgery in the corresponding chart of
$X$ to add a handle to this surface. Let us cut $\Sigma_{g-1}$ open along the
segment $[e^{-z},e^{z}]$. We will glue the resulting boundaries to the
boundary of the hemisphere
$\overline{\mathbb{H}^{2}}=\mathbb{H}^{2}\cup\mathbb{RP}^{1}\subset\mathbb{CP}^{1}$.
We identify the bottom boundary of the slit with the line segment
$[e^{-z},e^{z}]\subset\partial{\mathbb{H}^{2}}$ with the identity. We glue the
top boundary of the slit with the line segment $[-e^{z},-e^{-z}]$ of
$\partial\mathbb{H}^{2}$ with the map $z\mapsto-z$. Finally we glue the two
line segments $[-e^{-z},e^{-z}]$ and
$(\mathbb{R}\cup\\{\infty\\})\setminus[-e^{z},e^{z}]$ of
$\partial\mathbb{H}^{2}$ with the map $z\mapsto\frac{1}{z}$, see Figure 26.

$e^{-z}$
---
$e^{z}$
---
$-e^{-z}$
---
$-e^{z}$
---
$\mathbb{H}^{2}$
---
$+$
---
$-$
---
$e^{-z}$
---
$e^{z}$
---
$+$
---
$-$
---
Figure 26. $\overline{\mathbb{H}^{2}}$ on the left side. Its boundary is glued
to those on the right side.

Topologically we have added a handle to $\Sigma_{g-1}$ and thus we get a genus
$g$ surface. Moreover it has natural charts in $\mathbb{CP}^{1}$, that give a
projective structure holonomy $\rho$. We have added $3\times 2\pi$ to the
total angle of the conical point thus
$\rho\in\mathrm{Hol}(\mathcal{P}(2g-1))$. The curve on which we have glued the
handle develops injectively, hence $\rho\in\mathrm{Hol}(\mathcal{P}(2g-1+2k))$
for every $k\geqslant 0$ by 5.7. The result follows from 5.9. ∎

###  6.5. Nonelementary holonomy

The celebrated theorem of Gallo, Kapovich and Marden proven in [GKM00] is in
fact enhanced in the same paper, see [GKM00, Theorem 11.2.4].

######  Theorem 6.41 (Gallo-Kapovich-Marden).

Let $\rho\in\mathrm{Hom}(\Gamma,\mathrm{PSL}_{2}(\mathbb{C}))$ be a
nonelementary representation. We have
$\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ if and only if
$\sum_{i}n_{i}$ has the same parity as $\mathrm{sw}(\rho)$.

Their proof is based on the existence of Schottky decompositions of
nonelementary representations, shown in [GKM00, Part A]. By a Schottky
decomposition of $\rho$ we mean a pants decomposition $\Sigma=\cup_{i}P_{i}$
such that each restriction
$\rho_{|P_{i}}:\pi_{1}(P_{i})\to\mathrm{PSL}_{2}(\mathbb{C})$ is an
isomorphism onto a Schottky group. Indeed, the techniques of construction of
projective structures with given holonomy of Gallo, Kapovich and Marden allow
them to prove directly this theorem when $\rho$ has a Schottky decomposition;
see [GKM00, Part B, Theorem 11.2.4]. In genus $g=2$, there exist nonelementary
representations $\rho\in\mathrm{Hom}(\Gamma,\mathrm{PSL}_{2}(\mathbb{C}))$
that do not admit a Schottky decomposition: the pentagon representations, see
[LF19]. In the case where $\rho$ is a pentagon representation, there exists a
branched projective structure on $\Sigma_{2}$ with a single conical point of
total angle $2\pi(1+1)$ and holonomy $\rho$, see [LF19, Section 5]. Moreover
we can assume that there exists a curve based at the conical point that
develops injectively, see the red curve in [LF19, Figure 8], whose exponential
develops injectively. Thus we have $\rho\in\mathrm{Hol}(\mathcal{P}(2d+1))$
for every $d\geqslant 0$ by 5.7 and
$\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ if $\sum_{i}n_{i}$ is
odd by 5.9. Therefore 1 is the only obstruction to being in
$\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$ for nonelementary
representations.

##  7\. Geometrization in minimal degree

Let us conclude this article with a short section dedicated to the proof of
1.2. For every $\rho\in\mathrm{Hom}(\Gamma,\mathrm{PSL}_{2}(\mathbb{C}))$, we
compute the minimal degree of the branching divisor of a projective structure
having $\rho$ as holonomy. In other words, we compute the minimal value of
$\sum_{i}n_{i}$ such that
$\rho\in\mathrm{Hol}(\mathcal{P}(n_{1},\ldots,n_{k}))$. By 5.9, this number
$d(\rho)$ is the minimal $m\geqslant 0$ such that
$\rho\in\mathrm{Hol}(\mathcal{P}(\underbrace{1,\ldots,1}_{m}))$. This number
was computed by Kapovich for representations
$\rho\in\mathrm{Hom}(\Gamma,\mathbb{C})$ with values in the group of
translations $\mathbb{C}\subset\mathrm{Aff}(\mathbb{C})$ in [Kap20], and by
Gallo Kapovich and Marden for non-elementary representations and the trivial
representation in [GKM00]. This computation is a corollary of 1.1 and is given
in Figure 1.

## References

  * [BJJP20] Matt Bainbridge, Chris Johnson, Chris Judge, and Insung Park.  Haupt’s theorem for strata of abelian differentials.  arXiv, 2020.
  * [Can15] Richard D. Canary.  Dynamics on character varieties: a survey.  In Handbook of group actions. Vol. II, volume 32 of Adv. Lect. Math. (ALM), pages 175–200. Int. Press, Somerville, MA, 2015.
  * [CDF14] Gabriel Calsamiglia, Bertrand Deroin, and Stefano Francaviglia.  Branched projective structures with Fuchsian holonomy.  Geom. Topol., 18(1):379–446, 2014.
  * [CDF15] Gabriel Calsamiglia, Bertrand Deroin, and Stefano Francaviglia.  A transfer principle: from periods to isoperiodic foliations.  arXiv, 2015.
  * [CFG21] Shabarish Chenakkod, Gianluca Faraco, and Subhojoy Gupta.  Translation surfaces and periods of meromorphic differentials, 2021.  arXiv.
  * [DT06] Nathan M. Dunfield and William P. Thurston.  Finite covers of random 3-manifolds.  Invent. Math., 166(3):457–521, 2006.
  * [Edm82] Allan L. Edmonds.  Surface symmetry. I.  Michigan Math. J., 29(2):171–183, 1982.
  * [Edm83] Allan L. Edmonds.  Surface symmetry. II.  Michigan Math. J., 30(2):143–154, 1983.
  * [EKS84] Allan L. Edmonds, Ravi S. Kulkarni, and Robert E. Stong.  Realizability of branched coverings of surfaces.  Trans. Amer. Math. Soc., 282(2):773–790, 1984.
  * [EMZ03] Alex Eskin, Howard Masur, and Anton Zorich.  Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants.  Publ. Math. Inst. Hautes Études Sci., (97):61–179, 2003.
  * [Ere19] A. Eremenko.  Metrics of constant positive curvature with conic singularities. A survey, 2019.  arXiv.
  * [FM01] Benson Farb and Dan Margalit.  A Primer on Mapping Class Groups.  Pinceton University Press, 2001.
  * [Gha16] Selim Ghazouani.  Mapping class group dynamics on $\mathrm{Aff}(\mathbb{C})$-characters.  Annales de l’Institut Fourier, 66(2):729–751, 2016.
  * [Gha18] Selim Ghazouani.  Mapping class group dynamics and the holonomy of branched affine structures.  Math. Z., 289(1-2):1–23, 2018.
  * [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.
  * [Gol88] William M. Goldman.  Topological components of spaces of representations.  Invent. Math., 93(3):557–607, 1988.
  * [Gol97] William M. Goldman.  Ergodic theory on moduli spaces.  Ann. of Math. (2), 146(3):475–507, 1997.
  * [Gol03] William Goldman.  The modular group action on real $\mathrm{SL}_{2}$-characters of a one-holed torus.  Geometry and Topology, pages 443–486, 2003.
  * [Hau20] Otto Haupt.  Ein Satz über die Abelschen Integrale 1. Gattung.  Math. Z., 6(3-4):219–237, 1920.
  * [Hej75] Dennis Hejhal.  Monodromy groups and linearly polymorphic functions.  Acta Mathematica, pages 1–55, 1975.
  * [Hur91] A. Hurwitz.  Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten.  Math. Ann., 39(1):1–60, 1891.
  * [Kap17] Michael Kapovich.  Branched covers between spheres and polygonal inequalities in simplicial trees.  In preprint available at https://math.ucdavis.edu/ kapovich.EPR.covers.pdf, 2017.
  * [Kap20] Michael Kapovich.  Periods of abelian differentials and dynamics.  In Dynamics: topology and numbers, volume 744 of Contemp. Math., pages 297–315. Amer. Math. Soc., 2020.
  * [KZ03] Maxim Kontsevich and Anton Zorich.  Connected components of the moduli spaces of Abelian differentials with prescribed singularities.  Invent. Math., 153(3):631–678, 2003.
  * [Lan08] Erwan Lanneau.  Connected components of the strata of the moduli spaces of quadratic differentials.  Ann. Sci. Éc. Norm. Supér. (4), 41(1):1–56, 2008.
  * [LF19] Thomas Le Fils.  Pentagon representations and complex projective structures on closed surfaces, 2019.  arXiv.
  * [LF20] Thomas Le Fils.  Periods of abelian differentials with prescribed singularities.  Int. Math. Res. Not. IMRN, 2020.
  * [LMP09] Frank Loray and David Marín Pérez.  Projective structures and projective bundles over compact Riemann surfaces.  Astérisque, (323):223–252, 2009.
  * [MP19] Gabriele Mondello and Dmitri Panov.  Spherical surfaces with conical points: systole inequality and moduli spaces with many connected components.  Geom. Funct. Anal., 29(4):1110–1193, 2019.
  * [MS93] Howard Masur and John Smillie.  Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms.  Comment. Math. Helv., 68(2):289–307, 1993.
  * [Nie37] Jakob Nielsen.  Die Struktur periodischer Transformationen von Flächen, volume 15.  Levin & Munksgaard, 1937.
  * [Poi84] Henri Poincaré.  Sur les groupes des équations linéaires.  Acta mathematica, pages 201–311, 1884.
  * [PP06] Ekaterina Pervova and Carlo Petronio.  On the existence of branched coverings between surfaces with prescribed branch data. I.  Algebr. Geom. Topol., 6:1957–1985, 2006.
  * [PX00] Joseph P. Previte and Eugene Z. Xia.  Topological dynamics on moduli spaces. I.  Pacific J. Math., 193(2):397–417, 2000.
  * [PX02] Joseph P. Previte and Eugene Z. Xia.  Topological dynamics on moduli spaces. II.  Trans. Amer. Math. Soc., 354(6):2475–2494, 2002.
  * [Rat06] John Ratcliffe.  Foundations of hyperbolic manifolds.  Springer, 2006.
  * [Ser77] Jean-Pierre Serre.  Cours d’arithmétique.  Presses Universitaires de France, Paris, 1977.  Deuxième édition revue et corrigée, Le Mathématicien, No. 2.
  * [Thu97] William Thurston.  Three-Dimensional Geometry and Topology.  Princeton University Press, 1997.
  * [Tom15] J. Tomasini.  Realizations of branched self-coverings of the 2-sphere.  Topology Appl., 196(part A):31–53, 2015.
  * [Tro86] Marc Troyanov.  Les surfaces euclidiennes à singularités coniques.  Enseign. Math. (2), 32(1-2):79–94, 1986.
  * [Zor06] Anton Zorich.  Flat surfaces.  In Frontiers in number theory, physics, and geometry. I, pages 437–583. Springer, Berlin, 2006.