# On the existence of holomorphic curves in compact quotients of
$\mathrm{SL}(2,\mathbb{C})$

Indranil Biswas  School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Mumbai 400005, India<EMAIL_ADDRESS>,
Sorin Dumitrescu  Université Côte d’Azur, CNRS, LJAD, France
<EMAIL_ADDRESS>,  Lynn Heller  Institute of Differential Geometry, Leibniz
Universität Hannover, Welfengarten 1, 30167 Hannover<EMAIL_ADDRESS>hannover.de and  Sebastian Heller  Institute of Differential Geometry, Leibniz
Universität Hannover, Welfengarten 1, 30167 Hannover<EMAIL_ADDRESS>
###### Abstract.

We prove the existence of a pair $(\Sigma,\,\Gamma)$, where $\Sigma$ is a
compact Riemann surface with $\text{genus}(\Sigma)\,\geq\,2$, and
$\Gamma\,\subset\,{\rm SL}(2,\mathbb{C})$ is a cocompact lattice, such that
there is a generically injective holomorphic map
$\Sigma\,\longrightarrow\,{\rm SL}(2,\mathbb{C})/\Gamma$. This gives an
affirmative answer to a question raised by Huckleberry and Winkelmann [HW] and
by Ghys [Gh].

###### Key words and phrases:

Local system, character variety, cocompact lattice, monodromy, parabolic
bundle.

###### 2020 Mathematics Subject Classification:

34M03, 34M56, 14H15, 53A55

###### Contents

  1. 1 Introduction
  2. 2 Statements of the main theorems and strategy of proof
  3. 3 Abelianization on Symmetric Riemann surfaces
  4. 4 Grafting and Spin Structures
  5. 5 Holomorphic connections with $\operatorname{SL}(2,\mathbb{R})$-monodromy
  6. 6 The dodecahedral example

##  1\. Introduction

Compact complex manifolds with holomorphically trivial tangent bundle are
known to be biholomorphic to a quotient of a complex Lie group $G$ by a
discrete cocompact subgroup $\Gamma$ [Wa]. These manifolds, also known as
parallelizable complex manifolds, are Kähler if and only if the Lie group $G$
is abelian (in which case the manifold is a compact complex torus).

Whenever $G$ is a semi-simple Lie group, and $\Gamma\,\subset\,G$ is a
cocompact lattice, a theorem of Huckleberry and Margulis [HM] says that
$G/\Gamma$ does not admit any complex analytic hypersurface. In particular,
the algebraic dimension of $G/\Gamma$ is zero, meaning $G/\Gamma$ does not
admit any nonconstant meromorphic function.

An important class of examples consists of compact quotients of
$G\,=\,\text{SL}(2,\mathbb{C})$ by cocompact Kleinian subgroups
$\Gamma\,\subset\,\text{SL}(2,\mathbb{C})$. Since $\text{PSL}(2,\mathbb{C})$
is the group of orientation preserving isometries of the hyperbolic $3$-space
$\mathbb{H}^{3}$, the compact quotient $\text{SL}(2,\mathbb{C})/\Gamma$ is an
unramified double cover of the $\text{SO}(3,\mathbb{R})$-bundle of oriented
orthonormal frames of the compact hyperbolic $3$-manifold
$\mathbb{H}^{3}/\Gamma$. While the embedding of $\Gamma$ into
$\text{SL}(2,\mathbb{C})$ is known to be rigid by Mostow’s Theorem, the
flexibility of the complex structure of $\text{SL}(2,\mathbb{C})/\Gamma$ was
discovered by Ghys [Gh] where he showed that the corresponding Kuranishi space
has positive dimension for all $\Gamma$ with positive first Betti number. The
corresponding compact hyperbolic $3$-manifolds can be constructed using
Thurston’s hyperbolisation Theorem (see, for instance, [Th] and [Gh, Lemme
6.2] for constructions of compact hyperbolic $3$-manifolds with prescribed
rational cohomology ring). Nevertheless, much of the interplay between the
geometry of the compact hyperbolic $3$-manifold $\mathbb{H}^{3}/\Gamma$ and
the complex structure of its oriented orthonormal frame bundle
$\text{PSL}(2,\mathbb{C})/\Gamma$ remains to be explored.

In course of his studies of the deformation space of the complex structures of
$\text{SL}(2,\mathbb{C})/\Gamma$ Ghys [Gh] encountered a problem previously
raised by Huckleberry and Winkelmann [HW] which would generalize the
Huckleberry-Margulis Theorem [HM] to holomorphic curves: Does there exist a
compact 3-manifold $\text{SL}(2,\mathbb{C})/\Gamma$ admitting a compact
holomorphic curve of genus $g\geq 2$? Note that the case of elliptic curves
covered by one-parameters groups in $\text{SL}(2,\mathbb{C})$ are well-known
to exist in certain quotients $\text{SL}(2,\mathbb{C})/\Gamma$.

In this paper we give an affirmative answer to this open question following a
strategy due to Ghys, see [BDHH, CDHeL]. We construct on the trivial
holomorphic bundle of rank two over a Riemann surfaces $\Sigma$ an irreducible
holomorphic $\text{SL}(2,\mathbb{C})$-connection such that the image of the
corresponding monodromy homomorphism lies in a cocompact lattice $\Gamma$ in
$\text{SL}(2,\mathbb{C})$. The parallel frame of this connection then gives
rise to a holomorphic map from $\Sigma$ into the quotient
$\text{SL}(2,\mathbb{C})/\Gamma$ which, due to the irreducibility of the
connection, does not factor through any elliptic curve. A first step towards
realizing Ghys’ strategy was previously made in [BDHH] where holomorphic
connections with (real) Fuchsian monodromy were constructed. In this paper we
first show that every irreducible $\text{SL}(2,\mathbb{R})$-representation
with sufficient many symmetries can be realized as the monodromy of a
holomorphic $\text{SL}(2,\mathbb{C})$-connection. Then an example of such a
symmetric representation contained in a cocompact lattice $\Gamma$ in
$\text{SL}(2,\mathbb{C})$ is given. We end with an outlook on the relationship
between holomorphic curves in SL$(2,\mathbb{C})/\Gamma$ and surfaces of
constant mean curvature $H\,=\,1$ in the hyperbolic 3-space.

## Acknowledgements

The authors would like to thank Bertrand Deroin for many useful joint
discussions on the topic, which initiated our project. SH would like to thank
Nick Schmitt for stimulating discussions about surfaces in hyperbolic
3-manifolds.

IB is partially supported by a J. C. Bose Fellowship. SD was partially
supported by the French government through the UCAJEDI Investments in the
Future project managed by the National Research Agency (ANR) with the
reference number ANR2152IDEX201. LH is supported by the DFG grant HE 7914/2-1
of the DFG priority program SPP 2026 Geometry at Infinity. SH is supported by
the DFG grant HE 6829/3-1 of the DFG priority program SPP 2026 Geometry at
Infinity.

##  2\. Statements of the main theorems and strategy of proof

Let $\Gamma\,\subset\,\operatorname{SL}(2,\mathbb{C})$ be a cocompact lattice
and consider the compact complex $3$-manifold
$N\,=\,\operatorname{SL}(2,\mathbb{C})/\Gamma$. This three manifold can be
viewed as the double covering of the $\mathrm{SO}(3)$-frame bundle of the
corresponding hyperbolic 3-orbifold $\mathbb{H}^{3}/\Gamma$ and it is called
the unitary frame bundle. Let $\Sigma$ be a Riemann surface of genus
$g\,\geq\,2$, fix a base point $x_{0}\,\in\,\Sigma$ and consider an
irreducible representation

$\rho_{\Sigma}\,\colon\,\pi_{1}(\Sigma,\,x_{0})\,\longrightarrow\,\Gamma\,\subset\,\operatorname{SL}(2,\mathbb{C}).$

Following an idea of Ghys, we aim at realizing such a representation
$\rho_{\Sigma}$ as the monodromy representation of a holomorphic flat
connection $\nabla$ on the trivial holomorphic $\mathbb{C}^{2}$-bundle over
$\Sigma$. Then the corresponding parallel frame
$\Psi\colon\Sigma\rightarrow\operatorname{SL}(2,\mathbb{C}),$ with
$\Psi(x_{0})=$ Id induces a well-defined holomorphic map $f_{\rho_{\Sigma}}$
from $\Sigma$ into $N$ (with monodromy representation $\rho_{\Sigma}$). The
map $f_{\rho_{\Sigma}}$ does not factor through an elliptic curve since
$\rho_{\Sigma}$ is irreducible.

In the following we impose various symmetries on $\Sigma$ and $\rho_{\Sigma}$.
Let $\Sigma$ be the covering of $\mathbb{C}P^{1}$ of degree $g+1$ defined by
the equation

$y^{g+1}\,=\,\frac{(z-p_{1})(z-p_{2})}{(z-p_{3})(z-p_{4})},$ (2.1)

where

$p_{1}\,=\,e^{i\varphi},\quad p_{2}\,=\,-e^{-i\varphi},\quad
p_{3}\,=\,-e^{i\varphi},\quad p_{4}\,=\,e^{-i\varphi}$ (2.2)

with $\varphi\,\in\,(0,\,\pi/2)$, i.e., $\Sigma$ is totally branched over
$p_{1},\,p_{2},\,p_{3},\,p_{4}$. The representations
$\rho_{\Sigma}\,=\,\widehat{\rho}$ we consider are compatible with the
covering, i.e., it is induced by the monodromy representation $\rho$ of a
particular rank $2$ logarithmic connection $\nabla$ (described in Section 3.1)
over the 4-punctured sphere

$S_{4}\,:=\,\mathbb{C}P^{1}\setminus\\{p_{1},\,\dots,\,p_{4}\\}.$ (2.3)

Fix a base point $s_{0}\,\in\,S_{4}$, and let
$\gamma_{p_{j}}\,\in\,\pi_{1}(S_{4},\,s_{0})$ be the curve that goes around
$p_{j}$ anti-clockwise for $j=1,...,4$, such that
$\gamma_{p_{4}}\gamma_{p_{3}}\gamma_{p_{2}}\gamma_{p_{1}}\,=\,1$ (see Section
3.2). Let $M_{j}$ denote the monodromy of $\nabla$ along $\gamma_{p_{j}}$.
Then $\rho$ is assumed to satisfy the following RSR condition.

######  Definition 2.1.

An irreducible representation
$\rho\,\colon\,\pi_{1}(S_{4},\,s_{0})\,\longrightarrow\,\operatorname{SL}(2,\mathbb{C})$
is called RSR-representation if it has the following three properties:

  * •

Real: $\rho$ takes values in $\operatorname{SL}(2,\mathbb{R})$ and

$x\,=\,\operatorname{tr}(M_{1}M_{2})\,<\,-2,\
\,y\,=\,\operatorname{tr}(M_{2}M_{3})\,<\,-2,\
\,z\,=\,\operatorname{tr}(M_{1}M_{3})\,<\,-2,$

where $M_{j}\,=\,\rho(\gamma_{p_{j}})$.

  * •

Symmetric: the four monodromies $M_{1},\,\cdots,\,M_{4}$ lie in the same
conjugacy class determined by $\text{diag}(e^{2\pi i\widetilde{r}},\,e^{-2\pi
i\widetilde{r}})$ for some
$\widetilde{r}\,\in\,(\tfrac{1}{4},\,\tfrac{1}{2})$.

  * •

Rectangular: $\text{tr}(M_{1}M_{3})\,=\,\text{tr}(M_{2}M_{4})$.

Every $\rho\,\in\,{\rm
Hom}(\pi_{1}(S_{4},\,s_{0}),\,\operatorname{SL}(2,\mathbb{C}))$ such that
$\rho(\gamma_{p_{j}})^{k}\,=\,1$ for all $j=1,\,...,\,4,$ for some integer
$k\,>\,0$, lifts to a representation of $\pi_{1}(\Sigma,\,x_{0})$ for the
Riemann surface in (2.1) of suitable genus $g$, see Lemma 3.16 for the
details. This motivates the definition of the genus of a RSR-representation,
see also [BDHH, Section 3].

######  Definition 2.2.

Let $\rho$ be a RSR-representation such that one (and hence all, as they have
same conjugacy class) of $M_{1},\,\cdots,\,M_{4}$ have order
$k\,\in\,\mathbb{N}.$ Then, the genus of $\rho$ is $k-1,$ if $k$ is odd, and
it is $\tfrac{k}{2}-1$ if $k$ is even.

Since the symmetries are analogous to those of the Lawson minimal surfaces of
genus $g$, we define:

######  Definition 2.3.

Any homomorphism
$\widehat{\rho}\,:\,\pi_{1}(\Sigma,\,x_{0})\,\longrightarrow\,\operatorname{SL}(2,\mathbb{R})$
induced by some RSR-representation $\rho$ (defined in (2.1)) is called real
Lawson-symmetric (RL).

Our main theorem is the following.

######  Theorem 2.4 (Main Theorem).

Let $\rho$ be a RSR-representation of genus $g$, and let $\Gamma$ be a
compatible cocompact lattice in ${\rm SL}(2,\mathbb{C})$, meaning $\Gamma$
contains the image of the corresponding RL-representation $\widehat{\rho}$.
Then there exists a genus g Riemann surface $\Sigma$ of the form (2.1) and a
holomorphic map from $\Sigma$ to the compact complex 3-manifold
SL$(2,\mathbb{C})/\Gamma$. Moreover, the holomorphic map does not factor
through a torus.

######  Remark 2.5.

The techniques presented in this paper produce in fact infinitely many
holomorphic maps from Riemann surfaces $\Sigma^{n},$ $n\in\mathbb{N},$ of the
same genus into $\operatorname{SL}(2,\mathbb{C})/\Gamma$ inducing the same RL-
representation $\widehat{\rho}.$ To enhance clarity we discuss here only the
simplest case arising from grafting once.

######  Remark 2.6.

The assumption of the theorem is for example fulfilled if the 3-manifold
$\mathbb{H}^{3}/\Gamma$ contains a totally geodesic surface of genus $g\geq 2$
with enough symmetries such that the induced monodromy representation
$\widehat{\rho}$ is RL. Note that in the example we give below, the
representation $\widehat{\rho}$ is not Fuchsian.

We prove the following (see Theorem 6.1):

######  Theorem 2.7.

Let $\Gamma$ be the cocompact lattice in $\mathbb{H}^{3}$ given by the
dodecahedron tiling $\\{5,3,4\\}$ of the hyperbolic 3-space. Then there exist
a holomorphic curve of genus $4$ in the compact 3-manifold
SL$(2,\mathbb{C})/\Gamma.$

######  Remark 2.8.

The RSR-representation $\rho$ is obtained from the pentagon tiling of
$\mathbb{H}^{2}\subset\mathbb{H}^{3}$ which can be extended to the
dodecahedron tiling of $\mathbb{H}^{3}.$ The genus of the RSR-representation
is 4. The holomorphic map obtained from this example has 4 simple branch
points and by Riemann-Hurwitz it cannot factor through a lower genus surface.

The symmetry assumptions in Theorem 2.4 ensure that the moduli space
$\mathcal{M}_{\mathbb{R},sym}(\Sigma)$ of compatible equivariant
$\operatorname{SL}(2,\mathbb{R})$-representations, which contains $\rho$, is
only (real) $1$-dimensional. Moreover, the compatible Riemann surface
structures on $\Sigma$ are determined by rectangular tori
$T^{2}_{\tau}\,=\,\mathbb{C}/\mathbb{Z}\oplus i\tau\mathbb{Z}$ for
$\tau\,\in\,\mathbb{R}_{>0}$ with one puncture $[0]\,\in\,T^{2}_{\tau}$ given
by an appropriate quotient of a Hitchin cover of $\Sigma$. The corresponding
flat connections and their representations can therefore be investigated on
$T^{2}_{\tau}$ instead of $\Sigma$. The details of the setup are explained in
Section 3. To obtain a holomorphic map into
$\operatorname{SL}(2,\mathbb{C})/\Gamma$ we show that every element of
$\mathcal{M}_{\mathbb{R}}^{r}(T^{2}_{\tau})\cong\mathcal{M}_{\mathbb{R},sym}(\Sigma)$
can be realized as the monodromy representation of a logarithmic connection
$\nabla^{H}$ on a specific rank 2 parabolic bundle $H$. Then, this connection
over $T^{2}_{\tau}$ is shown to lift to a holomorphic connection on
$\mathcal{O}^{\oplus 2}$ over the Riemann surface $\Sigma$ with induced RL-
monodromy $\widehat{\rho}.$

In a first step we therefore construct a logarithmic connection
$\nabla^{H}(\tau)$ with $\operatorname{SL}(2,\mathbb{R})$-monodromy for every
punctured torus $T^{2}_{\tau}\setminus\\{o\\}$, compare with Theorem 5.3, with
the prescribed parabolic structure $H$. This theorem is of independent
interest and it is proven in Section 5. The main observation is that grafting,
a procedure generating new real projective structures 111complex projective
structures with real monodromy from old ones, changes the induced spin
structure on the surface. In this context it is very important to note that we
perform grafting not on flat projective bundles, but on their lifts to flat
vector bundles which yield the different spin structures. We recall grafting
for compact Riemann surfaces in Section 4 together with its straight forward
generalization to the case of a 1-punctured torus. Using abelianization on a
fixed $T^{2}_{\tau}$ and the fact that (every connected component of) the
moduli space $\mathcal{M}_{\mathbb{R}}^{r}$ is diffeomorphic to
$\mathbb{R}_{>0}$, we obtain (via the intermediate value theorem) that there
exists a holomorphic connection $\nabla^{H}(\tau)$ (on the prescribed
parabolic bundle) with real monodromy between the uniformization connection
$\nabla^{U}$ of $T^{2}_{\tau}$ and another oper connection $\nabla^{G}$ of
$T^{2}_{\tau}$ obtained from a simple-grafting of the uniformization
connection of a different Riemann surface $T^{2}_{\tau^{uG}}$.

In a second step, fixing a given representation
$\rho\in\mathcal{M}^{r}_{\mathbb{R}}$, we start at an appropriate initial
configuration of $\nabla^{U},\nabla^{H}$ and $\nabla^{G}$ and vary the Riemann
surface structure $\tau\in\mathbb{R}_{>0}$ of $T^{2}_{\tau}$. We show in Lemma
4.7 and Lemma 4.8 that both connections $\nabla^{U}(\tau)$ and
$\nabla^{G}(\tau)$ sweep out the 1-dimensional moduli space of real
representations $\mathcal{M}_{\mathbb{R}}^{r}$ (which contains $\rho$).
Moreover, $\mathcal{M}_{\mathbb{R}}^{r}$ is an ordered space and the ordering
between the three connections $\nabla^{U}$, $\nabla^{H}$ and $\nabla^{G}$ is
preserved through a continuous deformation. Since furthermore the Riemann
Hilbert mapping is a local diffeomorphism on the 1-punctured torus, see Lemma
5.7, the dependence of $\nabla^{H}(\tau)$ in $\tau$ can be chosen to be
continuous. Up to technicalities (which are taken care of), due to the fact
that $\tau$ is not necessarily a global coordinate on the submanifold
$\mathcal{M}^{r}_{\mathbb{R}}\subset\mathcal{M}_{1,1}^{r},$ the connection
$\nabla^{H}(\tau)$ must also sweep out the moduli space
$\mathcal{M}^{r}_{\mathbb{R}}$. In particular, there exists a value $\tau_{0}$
such that the monodromy representation of $\nabla^{H}(\tau_{0})$ is the
prescribed representation $\rho\in\mathcal{M}^{r}_{\mathbb{R}}$. By replacing
$\nabla^{U}(\tau)$ and $\nabla^{G}(\tau)$ with multiple graftings (which
differ by a simple-grafting), our arguments show that for every
$\rho\in\mathcal{M}_{\mathbb{R}}^{r}$ there exist infinitely many different
$\tau_{n}\in\mathbb{R}_{>0},$ $n\in\mathbb{N}$, with holomorphic connections
$\nabla^{H}(\tau_{n})$ having the same monodromy $\rho.$ To explain Remark
2.5, note that the Riemann surface structures of the tori $T^{2}_{\tau_{n}}$
and the Riemann surfaces (2.1) both degenerate as $n\to\infty$ and we obtain
infinitely many different holomorphic curves in the quotient of
$\mathrm{SL}(2,\mathbb{C})$ by the compatible cocompact lattice $\Gamma$.

Theorem 2.4 would worth little if we could not prove the existence of at least
one RSR-representation compatible with a cocompact lattice $\Gamma.$ In the
last section (Theorem 6.1) we explicitly construct an RSR-representation and
show that it is compatible with the cocompact lattice of the dodecahedron
$\\{5,3,4\\}$ tessellation of the hyperbolic 3-space. We expect many more
examples by investigating the existence of totally geodesic surfaces inside
compact hyperbolic $3$-manifold with RSR-monodromy.

##  3\. Abelianization on Symmetric Riemann surfaces

###  3.1.  Logarithmic connections and parabolic bundles

Consider a compact connected Riemann surface $\Sigma$. Its canonical line
bundle will be denoted by $K_{\Sigma}$, while ${\mathcal{O}}_{\Sigma}$ will
denote the sheaf of holomorphic functions on $\Sigma$. A holomorphic
$\rm{SL}(2,\mathbb{C})$–bundle over $\Sigma$ is a rank two holomorphic vector
bundle $V\,\longrightarrow\,\Sigma$ such that the determinant line bundle
$\det V\,:=\,\bigwedge^{2}V$ is holomorphically trivial.

Let ${\mathbf{D}}\,=\,p_{1}+\ldots+p_{n}$ be an effective reduced divisor,
i.e., the points $p_{j}\,\in\,\Sigma$ are pairwise distinct. Denote by
${\overline{\partial}}_{V}$ the Dolbeault operator for a holomorphic
$\rm{SL}(2,\mathbb{C})$–bundle $V$ on $\Sigma$; so the kernel of
${\overline{\partial}}_{V}$ defines the sheaf $\mathcal{V}$ of holomorphic
sections of $V$. A logarithmic $\rm{SL}(2,\mathbb{C})$–connection
$\nabla\,=\,\overline{\partial}_{V}+\partial^{\nabla}$ on $V$ with polar part
contained in the divisor $\mathbf{D}$ is a holomorphic differential operator

$\partial^{\nabla}\,:\,\mathcal{V}\,\longrightarrow\,\mathcal{V}\otimes
K_{\Sigma}\otimes{\mathcal{O}}_{\Sigma}({\mathbf{D}})$

such that

  * •

the Leibniz rule
$\partial^{\nabla}(fs)\,=\,f\partial^{\nabla}(s)+s\otimes\partial{f}$ holds
for all $s\in\mathcal{V}$ and $f\in{\mathcal{O}}_{\Sigma}$, and

  * •

the induced holomorphic connection on $\det V\,=\,{\mathcal{O}}_{\Sigma}$
coincides with the de Rham differential $d$ on ${\mathcal{O}}_{\Sigma}$.

Note that all logarithmic connections on $\Sigma$ are necessarily flat. At
every point $p_{j}$ in the singular divisor ${\mathbf{D}}$ of a logarithmic
$\rm{SL}(2,\mathbb{C})$–connection $\nabla$ on $V$, the associated residue

$\text{Res}_{p_{j}}(\nabla)\,\in\,\text{End}(V_{p_{j}})$

is tracefree. Let $\rho_{j}$ and $-\rho_{j}$ be the eigenvalues of
$\text{Res}_{p_{j}}(\nabla)$; the logarithmic connection $\nabla$ is called
non-resonant if $2\rho_{j}\,\notin\,\mathbb{Z}$ for all
$j\,=\,1,\,\cdots,\,n$. In the non-resonant case, the local monodromy of
$\nabla$ around $p_{j}$ is conjugate to the diagonal matrix with entries
$\exp(\pm 2\pi i\rho_{j})$ (see [De, p. 53, Théorème 1.17]).

A parabolic structure  $\mathcal{P}$ on a $\rm{SL}(2,\mathbb{C})$–bundle $V$
over the divisor ${\mathbf{D}}$ is defined by a collection of complex lines
$L_{j}\,\subset\,V_{p_{j}}$ together with parabolic weights
$r_{j}\,\in\,]0,\,\tfrac{1}{2}[$ for all $j\,=\,1,\,\cdots,\,n$. For a
parabolic structure $\mathcal{P}$, the divisor ${\mathbf{D}}$ is called the
parabolic divisor and $\\{L_{j}\\}_{j=1}^{n}$ are called the quasiparabolic
lines. The parabolic degree of a holomorphic line subbundle $W\,\subset\,V$ is
defined to be

$\text{par-deg}(W)\,:=\,{\rm deg}(W)+\sum_{j=1}^{n}r^{W}_{j}\,,$

where $r^{W}_{j}\,=\,r_{j}$ if $W_{p_{j}}\,=\,L_{j}$, and
$r^{W}_{j}\,=\,-r_{j}$ if $W_{p_{j}}\,\neq\,L_{j}$.

######  Definition 3.1 ([MS, MY]).

A parabolic structure $\mathcal{P}$ on the $\rm{SL}(2,\mathbb{C})$–bundle $V$
is called stable (respectively, semistable) if $\text{par-deg}(W)\,<\,0$
(respectively, $\text{par-deg}(W)\,\leq\,0$) for every holomorphic line
subbundle $W\,\subset\,V$. A semistable parabolic bundle that is not stable is
called strictly semistable. A parabolic bundle which is not semistable is
called unstable.

Any non-resonant logarithmic $\rm{SL}(2,\mathbb{C})$–connection $\nabla$ on
$V$ for which all the residues have their eigenvalues in the interval
$(-\tfrac{1}{2},\,\tfrac{1}{2})$ induces a parabolic structure $\mathcal{P}$
on $V$. The parabolic divisor of $\mathcal{P}$ is the singular locus
$\mathbf{D}\,=\,p_{1}+\ldots+p_{n}$ of $\nabla$. The parabolic weight at
$p_{j}$ is the positive eigenvalue $\rho_{j}$ of $\text{Res}_{p_{j}}(\nabla)$
and the quasiparabolic line at $p_{j}$ is the eigenline for $\rho_{j}$.

A strongly parabolic Higgs field on a parabolic $\rm{SL}(2,\mathbb{C})$–bundle
$(V,\,{\mathcal{P}})$ is a holomorphic section

$\Phi\,\in\,H^{0}(\Sigma,\,\text{End}(V)\otimes
K_{\Sigma}\otimes{\mathcal{O}}_{\Sigma}({\mathbf{D}}))$

such that $\text{tr}(\Phi)\,=\,0$ and

$\Phi(p_{j})(V_{p_{j}})\,\subset\,L_{j}\otimes(K_{\Sigma}\otimes{\mathcal{O}}_{\Sigma}({\mathbf{D}}))_{p_{j}}$

for all $j\,=\,1,\,\cdots,\,n$. These conditions imply that $\Phi(p_{j})$ is
nilpotent with the quasiparabolic lines $L_{j}\,\subset\,{\rm
kernel}(\Phi(p_{j}))$ for all $j=1,...,n$.

Two non-resonant logarithmic $\rm{SL}(2,\mathbb{C})$–connections $\nabla_{1}$
and $\nabla_{2}$ on $V$ with polar part contained in
${\mathbf{D}}\,=\,p_{1}+\ldots+p_{n}$ induce the same parabolic structure on
$V$ if and only if $\nabla_{1}-\nabla_{2}$ is a strongly parabolic Higgs field
for the parabolic structure given by $\nabla_{1}$ (or equivalently, for the
parabolic structure given by $\nabla_{2}$).

A general result of Mehta and Seshadri [MS, p. 226, Theorem 4.1(2)], and
Biquard [Biq, p. 246, Théorème 2.5] (see also [Pi, Theorem 3.2.2]) implies
that the above construction of associating a parabolic bundle to a logarithmic
connection actually produces a bijection between the space of isomorphism
classes of irreducible flat ${\rm SU}(2)$–connections on
$\Sigma\setminus{\mathbf{D}}$ and the stable parabolic
$\rm{SL}(2,\mathbb{C})$–bundles on $(\Sigma,\,{\mathbf{D}})$. As a
consequence, every logarithmic connection $\nabla$ on $V$ giving rise to a
stable parabolic $\rm{SL}(2,\mathbb{C})$–structure $\mathcal{P}$ admits a
unique strongly parabolic Higgs field $\Phi$ on $(V,\,{\mathcal{P}})$ such
that the monodromy representation of $\nabla+\Phi$ is unitary.

###  3.2. Flat $\rm{SL}(2,\mathbb{C})$–connections on the $4$-punctured sphere

Let ${\mathcal{S}}_{4}$ denote the Riemann sphere $\mathbb{C}P^{1}$ with four
unordered marked points

${\mathcal{S}}_{4}\,:=\,(\mathbb{C}P^{1},\,\\{p_{1},\,\cdots,\,p_{4}\\})\,,$

with $p_{j}$ as in (2.2) and recall that

$S_{4}\,=\,\mathbb{C}P^{1}\setminus\\{p_{1},\,\cdots,\,p_{4}\\}\,$ (3.1)

is the underlying topological four-punctured sphere. Fix a base point
$s_{0}\,\in\,S_{4}$. For every $j=1,\,...,\,4$, consider a simply closed,
oriented, and $s_{0}$-based loop $\gamma_{p_{j}}$ going around a single
puncture $p_{j}$. The fundamental group $\pi_{1}(S_{4},\,s_{0})$ is generated
by these curves $\gamma_{p_{j}}$ with $j=1,\,...,\,4$ and they satisfy the
relation $\gamma_{p_{4}}\gamma_{p_{3}}\gamma_{p_{2}}\gamma_{p_{1}}\,=\,1$.

Convention. For convenience, the composition of loops generating the
fundamental group operation is considered to be from right to left, i.e.,
$\gamma_{2}\gamma_{1}$ denotes the loop obtained by first performing the loop
$\gamma_{1}$ and then $\gamma_{2}$.

Every $\operatorname{SL}(2,\mathbb{C})$-representation of
$\pi_{1}(S_{4},\,s_{0})$ is determined by the images
$M_{j}\,\in\,\operatorname{SL}(2,\mathbb{C})$ of the generators
$\gamma_{p_{j}}\,\in\,\pi_{1}(S_{4},\,s_{0})$, for $j=1,\,...,\,4$ and we have

$M_{4}M_{3}M_{2}M_{1}\,=\,I.$

We restrict to the symmetric case where

$\operatorname{tr}(M_{j})\,=\,2\cos(2\pi\widetilde{r}),\;\quad\forall\,\,\,\;j=1,\,...,\,4,$

with $\widetilde{r}\in(\frac{1}{4},\,\frac{1}{2}).$ We denote by
$\mathcal{M}_{0,4}^{\widetilde{r}}$ the space of equivalence classes of
$\operatorname{SL}(2,\mathbb{C})$-representation of $\pi_{1}(S_{4},\,s_{0})$
with local monodromy satisfying the above condition at all punctures. This
$\mathcal{M}_{0,4}^{\widetilde{r}}$ is identified with the space of flat
$\operatorname{SL}(2,\mathbb{C})$-connections on the four-punctured sphere
such that all four local monodromies are in the same conjugacy class given by

${\left(\begin{matrix}e^{-2\pi i\widetilde{r}}&0\\\ 0&e^{2\pi
i\widetilde{r}}\end{matrix}\right)}\,\in\,{\rm SL}(2,{\mathbb{C}})\,.$ (3.2)

For $\rho\in\mathcal{M}_{0,4}^{\widetilde{r}}$, we denote by

$\widetilde{x}\,=\,\operatorname{tr}(M_{2}M_{1}),\,\
\widetilde{y}\,=\,\operatorname{tr}(M_{3}M_{2}),\,\
\widetilde{z}=\operatorname{tr}(M_{3}M_{1})$

its trace coordinates. They satisfy

$\widetilde{x}^{2}+\widetilde{y}^{2}+\widetilde{z}^{2}+\widetilde{x}\widetilde{y}\widetilde{z}-2\mu^{2}(\widetilde{x}+\widetilde{y}+\widetilde{z})+4(\mu^{2}-1)+\mu^{4}\,=\,0.$
(3.3)

The corresponding affine variety is called a (relative) character variety. The
following result of characterizing a representation by its image in the
corresponding relative character variety is well-known and dates back to
Fricke and Klein, see [Go88, BeG].

######  Lemma 3.2.

Let $(\widetilde{x},\widetilde{y},\widetilde{z})\in\mathbb{C}^{3}$ satisfying
equation (3.3) and
$(\widetilde{x}-2)(\widetilde{y}-2)(\widetilde{z}-2)\,\neq\,0$. Then, there
exist a unique $\rho\in\mathcal{M}_{0,4}^{\widetilde{r}}$ such that
$(\widetilde{x},\widetilde{y},\widetilde{z})$ are the trace coordinates of
$\rho$.

Moreover, a totally reducible representation is conjugate to a
$\mathrm{SU}(2)$-representation if and only if
$\widetilde{x},\,\widetilde{y},\,\widetilde{z}\,\in\,[-2,\,2]$, while it is
conjugate to an $\operatorname{SL}(2,\mathbb{R})$-representation if
$\widetilde{x},\,\widetilde{y},\,\widetilde{z}\,\in\,\mathbb{R}$ are real and
at least one of them lying in $\mathbb{R}\setminus[-2,\,2]$.

######  Remark 3.3.

For the parabolic weight $\widetilde{r}\,\in\,(0,\,\frac{1}{4})$, there is a
natural biholomorphic map between the character varieties for $\widetilde{r}$
and $\tfrac{1}{2}-\widetilde{r}$. In fact, this biholomorphism is induced by
$M_{k}\,\longmapsto\,-M_{k}$, which gives the identity map in terms of the
respective $(\widetilde{x},\,\widetilde{y},\,\widetilde{z})$-trace
coordinates. Note that

$2\text{cos}(2\pi\widetilde{r})\,=\,-2\text{cos}(2\pi(\tfrac{1}{2}-\widetilde{r})),$

and therefore also equation (3.3) does not change.

###  3.3. Flat $\rm{SL}(2,\mathbb{C})$–connections on the 1-punctured torus

For $\tau\,\in\,\mathbb{R}_{>0}$ let

$T_{\tau}^{2}\,:=\,\mathbb{C}/\Gamma,\quad\text{with}\quad\Gamma\,=\,{\mathbb{Z}}+\tau
i{\mathbb{Z}}\,\subset\,\mathbb{C}$ (3.4)

be a rectangular torus. Moreover, let $o\,=\,[0]\,\in\,T_{\tau}^{2}$ and
$p_{0}\,:=\,\frac{1+\tau i}{4}\,\in\,T_{\tau}^{2}$ and consider
$\pi_{1}(T_{\tau}^{2}\setminus\\{o\\},\,p_{0})$ the fundamental group of the
one-punctured torus $T_{\tau}^{2}\setminus\\{o\\}$ with basepoint $p_{0}$.
This is a free group with two generators
$\gamma_{x},\,\gamma_{y}\,\in\,\pi_{1}(T_{\tau}^{2}\setminus\\{o\\},\,p_{0}),$
where

$\gamma_{x}\,\colon\,[0,\,1]\,\longrightarrow\,T_{\tau}^{2}\setminus\\{o\\};\
\,s\,\longmapsto\,s+\frac{1+\tau i}{4}$ (3.5)

and

$\gamma_{y}\,\colon\,[0,\,1]\,\longrightarrow\,T_{\tau}^{2}\setminus\\{o\\};\
\,s\,\longmapsto\,\tau is+\frac{1+\tau i}{4}.$ (3.6)

The commutator
$\gamma_{y}^{-1}\gamma_{x}^{-1}\gamma_{y}\gamma_{x}\,\in\,\pi_{1}(T_{\tau}^{2}\setminus\\{o\\},\,p_{0})$
corresponds to a simple loop going around the marked point $o$ anti-clockwise.

For $r\in(0,\,\tfrac{1}{2})$ let $\mathcal{M}_{1,1}^{r}$ be the moduli space
of flat $\mathrm{SL}(2,\mathbb{C})$-connections on the 1-punctured torus
$T^{2}_{\tau}\setminus\\{o\\}$ with local monodromy around the marked point
$o$ lying in the conjugacy class of the matrix

${\left(\begin{matrix}e^{-2\pi ir}&0\\\ 0&e^{2\pi
ir}\end{matrix}\right)}\,\in\,{\rm SL}(2,{\mathbb{C}})\,.$ (3.7)

As for the 4-punctured sphere, the conjugacy class is determined by the value
of its trace $2\cos(2\pi r)$, see [Go03]. For an element in
$\mathcal{M}_{1,1}^{r}$ let $X,\,Y$ be the monodromies along the curves

$\gamma_{x},\,\gamma_{y}\,\in\,\pi_{1}(T^{2}_{\tau}\setminus\\{0\\},\,p_{0})$

(defined in (3.5) and (3.6)), and let

$x\,=\,\operatorname{tr}(X),\quad y\,=\,\operatorname{tr}(Y),\quad
z\,=\,\operatorname{tr}(YX)$

be the corresponding trace coordinates satisfying the equation

$x^{2}+y^{2}+z^{2}-xyz-2-2\cos(2\pi r)\,=\,0\,.$ (3.8)

The corresponding affine variety is also called the (relative) character
variety of the 1-punctured torus. By a result of Fricke, the moduli space
$\mathcal{M}_{1,1}^{r}$ is diffeomorphic to the character variety defined in
(3.8) by associating to a monodromy representation the traces
$x\,=\,\operatorname{tr}(X),\,y\,=\,\operatorname{tr}(Y)$ and
$z\,=\,\operatorname{tr}(YX)$ (see, [Go03, Section 2.1]).

######  Remark 3.4.

For given $x$ and $y$, the equation (3.8) is quadratic in $z$, and hence there
are two (possibly equal) solutions of (3.8) in $z$ which we refer to as
$z_{1}$ and $z_{2}$. If $r$, $x$ and $y$ are all real, then $z_{1}$ and
$z_{2}$ are complex conjugate to each other.

######  Theorem 3.5 ([Go03]).

For $r\,\in\,(0,\,\tfrac{1}{2})$ fixed, the space of all real points of the
character variety defined by (3.8) has 5 connected components: one compact
component characterized by the condition $x,\,y,\,z\,\in\,[-2,\,2]$ and four
non-compact components (which are all diffeomorphic to each other). The
compact component consists of $\mathrm{SU}(2)$-representations, while the non-
compact components consist of $\mathrm{SL}(2,\mathbb{R})$-representations.

######  Remark 3.6.

The non-compact components are interchanged by tensoring with a flat
$\mathbb{Z}_{2}$-bundle. These are called sign-change automorphisms by Goldman
[Go03].

### A map between the character varieties

By [BDHH, Theorem 4.9] (see also [HH]), there exists for every
$r\,\in\,(0,\,\tfrac{1}{2})$ a degree 4 birational map between the moduli
space $\mathcal{M}^{r}_{1,1}$ of flat $\mathrm{SL}(2,\mathbb{C})$-connections
on the one-punctured torus $T^{2}_{\tau}$ (as defined in (3.4)) and the moduli
space $\mathcal{M}_{0,4}^{\widetilde{r}}$ of flat
$\mathrm{SL}(2,\mathbb{C})$-connections on the four-punctured sphere $S_{4}$
(as defined in (2.3)) with

$\widetilde{r}\,=\,\tfrac{1+2r}{4}\,\in\,(\tfrac{1}{4},\,\tfrac{1}{2}).$

On the level of character varieties this map
$\mathcal{M}^{r}_{1,1}\longrightarrow\mathcal{M}_{0,4}^{\widetilde{r}}$ is
given by

$(x,\,y,\,z)\,\longmapsto\,(\widetilde{x},\,\widetilde{y},\,\widetilde{z})\,=\,(2-x^{2},\,2-y^{2},\,2-z^{2}).$
(3.9)

The construction of the above map in [BDHH, Theorem 4.9] uses the rectangular
torus $T^{2}_{\tau}$, with $\tau\in\mathbb{R}_{>0},$ being a double cover of
$\mathbb{C}P^{1}$ branched over the four points $p_{1},\,\cdots,\,p_{4}$
(defined in (2.2)), i.e., $T^{2}_{\tau}$ is given by

$y^{2}\,=\,\frac{(z-p_{1})(z-p_{2})}{(z-p_{3})(z-p_{4})}.$

Define

${\mathcal{S}}_{4}(\tau)\,:=\,(\mathbb{C}P^{1},\,\\{p_{1},\,\cdots,\,p_{4}\\})\,,$
(3.10)

with $p_{j}$ as in (2.2), chosen to define a rectangular torus $T^{2}_{\tau}$,
with $\tau\,\in\,\mathbb{R}_{>0}$. Since $T_{\tau}^{2}$ is rectangular, the
reflection along one edge

$\eta\,\colon\,T_{\tau}^{2}\,\longrightarrow\,T_{\tau}^{2},\quad[w]\,\longmapsto\,[\overline{w}],$
(3.11)

where $w$ is the global coordinate on $\mathbb{C},$ defines a real involution
on $T_{\tau}^{2}$.

######  Remark 3.7.

Since the real involution $\eta$ considered here is different than in [BDHH],
we also use slightly different coverings of the 4-punctured sphere, see Lemma
3.16. Nevertheless, the main results in [BDHH] to obtain Fuchsian
representations on the holomorphically trivial bundle remain true by analogous
arguments.

###  3.4. Abelianization

Every element in $\mathcal{M}_{1,1}^{r}$ can be represented (meaning it lies
in the same smooth gauge class) by a logarithmic flat connection with a simple
pole at $o$. Abelianization yields particularly well-behaved coordinates
$a,\,\chi\,\in\,\mathbb{C}$ on ${\mathcal{M}}_{1,1}^{r}$ as follows, see also
[BDH, Section 4], or [HH]. For $L$ being the $C^{\infty}$-trivial bundle
$T_{\tau}^{2}\times{\mathbb{C}}\,\longrightarrow\,T_{\tau}^{2}$ the generic
logarithmic connection in $\mathcal{M}_{1,1}^{r}$ is given by

$\nabla\,=\,\nabla^{a,\chi,r}\,=\,{\left(\begin{matrix}\nabla^{L}&\gamma^{-}_{\chi}\\\
\gamma^{+}_{\chi}&\nabla^{L^{*}}\end{matrix}\right)}\,,$ (3.12)

where

$\nabla^{L}\,=\,d+adw+\chi d\overline{w}$ (3.13)

is the flat connection on $L$ for constant $a,\,\chi\in\mathbb{C}$ and $w$
being the global holomorphic coordinate on $T_{\tau}^{2}$ (see [BDH, Section
4]). Moreover, $\nabla^{L^{*}}$ is the dual connection of $\nabla^{L}$, and
the induced holomorphic structure (by (3.12)) on $L$ is given by the Dolbeault
operator $\overline{\partial}^{0}+\chi d\overline{w}$, where
$\overline{\partial}^{0}\,=\,d^{\prime\prime}$ is the $(0,1)$-part of the de
Rham differential operator $d$. In this generic case, characterized by $\chi$
not being a half-lattice point of Jac$(T^{2}_{\tau})$, i.e.,
$L^{2}\neq\mathcal{O}_{T_{\tau}^{2}},$ $\gamma^{+}_{\chi}$ and
$\gamma^{-}_{\chi}$ are meromorphic sections with respect to the holomorphic
structures given by the Dolbeault operators

$\overline{\partial}^{0}-2\chi d\overline{w}\quad\text{ and
}\quad\overline{\partial}^{0}+2\chi d\overline{w},$

respectively, with simple poles at $o\,\in\,T^{2}_{\tau}$ and residues
determined by the eigenvalue of the local monodromy $r\in(0,\frac{1}{2})$.

In the non-generic case, the underlying rank two holomorphic bundle is a non-
trivial extension of a spin bundle $S$ by itself. With respect to the
$C^{\infty}$-splitting $S\oplus S^{*}$ the Dolbeaut operator is given by

$\overline{\partial}^{\nabla}\,=\,{\left(\begin{matrix}\overline{\partial}^{0}+\chi
d\overline{w}&d\overline{w}\\\ 0&\overline{\partial}^{0}-\chi
d\overline{w}\end{matrix}\right)}$ (3.14)

for a half lattice point $\chi\in$ Jac$(T^{2}_{\tau})$, while the
$\partial$-part

$\partial^{\nabla}\,=\,{\left(\begin{matrix}\partial^{S}&bdw\\\
cdw&(\partial^{S})^{*}\end{matrix}\right)}\,$ (3.15)

is singular at $o$, i.e., $\partial^{S}$ and its dual $(\partial^{S})^{*}$ are
line bundle connections singular at $o$ and $b$ is a function singular at $o,$
but $c\in\mathbb{C}^{*}$ is a non-zero constant. Thus, in the non-generic
case, the connection takes the form of an (orbifold) oper, compare with (4.1)
below.

######  Remark 3.8.

For given $\nabla$ the holormorphic structure on $L$, denoted by
$\chi(\nabla)\in$ Jac$(T^{2}_{\tau})$ by abuse of notation, is only well
defined up to taking the dual.

######  Remark 3.9.

The parabolic weight at the puncture $o$ induced by the logarithmic connection
$\nabla$ is given by $r.$ For $\nabla=\nabla^{a,\chi,r}$, where $\chi$ is not
a half-lattice point, the parabolic line $l_{o}$ is uniquely determined, up to
a holomorphic automorphism of $L\oplus L^{*}$, by the condition that it is
neither the line $L_{o}$, nor the line $L^{*}_{o}$ (i.e., the off-diagonal is
non-zero). In the non-generic case, the parabolic line is either given by
$l_{o}=S_{0}$ and the parabolic bundle is unstable, or the parabolic line is
not contained in the unique holomorphic line subbundle of degree 0, and the
parabolic bundle is stable.

Note that $o$ is contained in the fix point set of the reflexion $\eta$ in
(3.11). Since $r\in(0,\,\frac{1}{2})$ is real, $\eta$ induces a real
involution of the corresponding de Rham moduli space

$\widehat{\eta}\,\colon\,\mathcal{M}_{1,1}^{r}\,\longrightarrow\,\mathcal{M}_{1,1}^{r},\
\quad[\nabla]\,\longmapsto\,[\eta^{*}\overline{\nabla}]$

and we have

######  Lemma 3.10.

On the rectangular torus $T_{\tau}^{2}$ the gauge class of a connection
$\nabla\,=\,\nabla^{a,\chi,r}$, with $\chi\notin\tfrac{1}{2}\Gamma$, is fixed
by the involution $\eta$ if and only if one of the following four conditions
holds:

$\begin{split}&\chi\in\mathbb{R},\;\text{ and }\,\;a\,\in\,\mathbb{R},\\\
\text{or }\quad&\chi+k\frac{\pi i}{2}\,\in\,\mathbb{R}\,\;\text{ and
}\;a-k\frac{\pi i}{2}\,\in\,\mathbb{R}\;\;\text{for some
}\;k\,\in\,\mathbb{Z},\\\ \text{or }\quad&\chi\,\in\,i\mathbb{R}\,\;\text{ and
}\;a\,\in\,i\mathbb{R},\\\ \text{ or
}\quad&\chi+k\frac{\pi}{2\tau}\,\in\,i\mathbb{R}\,\;\text{ and
}\;a-k\frac{\pi}{2\tau}\,\in\,i\mathbb{R}\;\;\text{for some
}\;k\,\in\,\mathbb{Z}.\end{split}$ (3.16)

###### Proof.

We have

$\eta^{*}dw\,=\,d\overline{w}\quad\text{and}\quad\eta^{*}d\overline{w}\,=\,dw.$

Hence, for $\chi\,\in\,\mathbb{R}$ and $a\,\in\,\mathbb{R}$, the connection
$\nabla^{L}$ in (3.13) satisfies the condition

$\eta^{*}\overline{\nabla^{L}}\,=\,\nabla^{L}.$

By [HH, BDH] the sections $\gamma^{\pm}_{\chi}$ in (3.12) are unique up to
scaling. Moreover, the quadratic residue of

$\gamma^{+}_{\chi}\gamma^{-}_{\chi}(dw)^{2}$

is $r^{2}$, and hence this residue determines the conjugacy class of the
monodromy of $\nabla$ around the singular point $o\,\in\,T_{\tau}^{2}$. Thus,
we obtain constants $c^{+},\,c^{-}$ with

$\eta^{*}\overline{\gamma^{\pm}_{\chi}dw}\,=\,c^{\pm}\gamma^{\pm}_{\chi}$

and consequently the two connections $\nabla$ and $\eta^{*}\overline{\nabla}$
are gauge equivalent. The argument for the other $3$ cases in (3.16) works
analogously.

Conversely, if the pull-back $\eta^{*}\overline{\nabla}$ is gauge equivalent
to $\nabla$ then (by [HH, BDHH]) $\eta^{*}\overline{\nabla}^{L}$ is gauge
equivalent to either $\nabla^{L}$ or its dual. This yields that $a$ and $\chi$
must satisfy one of the conditions in (3.16) ∎

For latter purposes, we denote the space of $\eta$-invariant representations
by

$\mathcal{M}_{1,1}^{r,\eta}\,:=\,\\{[\nabla]\,\in\,\mathcal{M}_{1,1}^{r}\,\,\mid\,\,\,[\eta^{*}\overline{\nabla}]\,=\,[\nabla]\\}.$

###  3.5. The hidden symmetries of RSR-representations

######  Proposition 3.11.

For $\widetilde{r}\,\in\,(\tfrac{1}{4},\,\tfrac{1}{2})$ consider
$\widetilde{\rho}\in\mathcal{M}_{0,4}^{\widetilde{r}}$ and let
$(\widetilde{x},\,\widetilde{y},\,\widetilde{z})$ be the corresponding trace
coordinates satisfying (3.3) and
$(\widetilde{x}-2)(\widetilde{y}-2)(\widetilde{z}-2)\,\neq\,0$ (as in Lemma
3.2). Then there exist an element $(x,\,y,\,z)$ in $\mathcal{M}_{1,1}^{r}$,
unique up to signs, satisfying (3.8) with $r\,=\,2\widetilde{r}-\frac{1}{2}$
such that

$\widetilde{x}\,=\,2-x^{2},\quad\widetilde{y}\,=\,2-y^{2},\quad\widetilde{z}\,=\,2-z^{2}.$
(3.17)

###### Proof.

As in the proof of Theorem 4.9 in [BDHH], the Klein-Fricke equation (3.3)
factors into the product of the Klein-Fricke equation (3.8) for $(x,\,y,\,z)$
and the Klein-Fricke equation (3.8) for $(x,\,y,\,-z)$ when applying (3.17).
Hence, for given $(\widetilde{x},\,\widetilde{y},\,\widetilde{z})$ either the
corresponding $(x,\,y,\,z)$ or $(x,\,y,\,-z)$ solves equation (3.8). ∎

######  Remark 3.12.

Proposition 3.11 shows the existence of additional symmetries of
representations $\widetilde{\rho}\in\mathcal{M}_{0,4}^{\widetilde{r}}$ on the
4-punctured sphere, see equation (4.19) in [BDHH] or [Go97, $\S~{}6$].

######  Corollary 3.13.

For $\widetilde{r}\,\in\,(\tfrac{1}{4},\,\tfrac{1}{2})$ let
$\rho\in\mathcal{M}_{0,4}^{\widetilde{r}}$ be a RSR-representation. Then
$\rho$ corresponds via abelianization to a real representation in
$\mathcal{M}_{1,1}^{r},$ with $r\,=\,2\widetilde{r}-\frac{1}{2}$. By abuse of
notation we will denote the induced representation in $\mathcal{M}_{1,1}^{r},$
by $\rho$ as well.

###### Proof.

By Definition 2.1, the image of $\rho$ in the character variety (3.3) is a
point $(\widetilde{x},\,\widetilde{y},\,\widetilde{z})$ such that
$\widetilde{x}\,<\,-2$, $\widetilde{y}\,<\,-2$ and $\widetilde{z}\,<\,-2$.
Hence the corresponding solution $(x,\,y,\,z)$ of (3.17) is a real point in
the character variety of the the 1-punctured torus defined by (3.8). By [Go03]
, see Theorem 3.5, this solution $(x,\,y,\,z)$ corresponds to a real element
in $\mathcal{M}_{1,1}^{r}$, where $r\,=\,2\widetilde{r}-\frac{1}{2}$. ∎

###  3.6. Strictly semi-stable parabolic bundles on the 4-punctured sphere

On ${\mathcal{S}}_{4}$, the Riemann sphere $\mathbb{C}P^{1}$ with four
unordered marked points, fix the parabolic weight
$\widetilde{r}\,\in\,(\tfrac{1}{4},\,\tfrac{1}{2})$ to be the same at each
puncture. Then, up to isomorphism, there are exactly 3 strictly semi-stable
parabolic rank 2 bundles with trivial determinant and given parabolic weights.
These 3 parabolic bundles are defined on the holomorphic rank 2 bundle
$\mathcal{O}\oplus\mathcal{O}\,\longrightarrow\,\mathbb{C}P^{1}$, and
determined by the choice of signs
$\sigma_{2},\,\sigma_{3},\,\sigma_{4}\,\in\,\\{\pm 1\\}$ with

$1+\sigma_{2}+\sigma_{3}+\sigma_{4}\,=\,0,$

induced by the reducible Fuchsian systems

$D\,=\,d+\begin{pmatrix}\rho&0\\\
0&-\rho\end{pmatrix}\left(1\frac{dz}{z-p_{1}}+\sigma_{2}\frac{dz}{z-p_{2}}+\sigma_{3}\frac{dz}{z-p_{3}}+\sigma_{4}\frac{dz}{z-p_{4}}\right).$
(3.18)

Moreover, they admit strongly parabolic and offdiagonal Higgs fields $\Phi$
with non-zero determinant, e.g., for $\sigma_{2}=1,\sigma_{3}=\sigma_{4}=-1$
we have

$\Phi\,=\,\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\left(\frac{dz}{z-p_{1}}-\frac{dz}{z-p_{2}}\right)+\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\left(\frac{dz}{z-p_{3}}-\frac{dz}{z-p_{4}}\right).$ (3.19)

######  Lemma 3.14.

In the abelianization coordinates, the holomorphic structures, i.e., the
$\chi$ coordinates, corresponding to the 3 strictly semistable parabolic
bundles with parabolic weight $\widetilde{r}$ at every puncture are given by
the $4\times$3=12 non-trivial 4-spin bundles on the torus $T^{2}_{\tau}$,
i.e., by the line bundles $L\in{\rm Jac}(T^{2}_{\tau})$ with $L^{\otimes
4}\,=\,{\mathcal{O}}_{T^{2}_{\tau}}$ and $L^{\otimes
2}\,\neq\,{\mathcal{O}}_{T^{2}_{\tau}}$.

###### Proof.

The case $\sigma_{3}=1,\sigma_{2}=\sigma_{4}=-1$ is already considered in
[HHSch], and [BDHH]. We only give the proof for $\sigma_{2}\,=\,1$ and
$\sigma_{3}\,=\,\sigma_{4}\,=\,-1$, as the case $\sigma_{4}\,=\,1$ and
$\sigma_{2}\,=\,\sigma_{3}\,=\,-1$ work analogously. Let

$\pi\,\colon\,\Sigma_{2}\,\longrightarrow\,\mathcal{S}_{4}$

be a double covering of the sphere $\mathbb{C}P^{1}$ branched over the four
marked points $p_{1},...,p_{4}$, defined by the equation

$\widetilde{y}^{2}\,=\,\prod_{i=1}^{4}(z-p_{j}).$

Let $w_{j}:=\pi^{-1}(p_{j})$ for $i=1,...,4$ and consider the reducible
Fuchsian system $D$ (3.18) on $\mathcal{S}_{4}.$ We will show that the line
bundle $L$ which determines via abelianization (3.12) the gauge class of the
connection $[\pi^{*}D]$ is given by
$L\,=\,L(w_{1}-w_{2})\,\longrightarrow\,\Sigma_{2}$. That $L$ corresponds to a
half-lattice point of the Jacobian translates to the condition $L=L^{*}.$

######  Remark 3.15.

In fact $\Sigma_{2}\,\cong\,\mathbb{C}/(2\mathbb{Z}+2\tau i\mathbb{Z})$ is a
4-fold covering of $T^{2}_{\tau}$. The holomorphic structure on the spin
bundle $L(w_{1}-w_{2})\,\longrightarrow\,\Sigma_{2}$ is given by

$\overline{\partial}^{0}-\frac{\pi}{4\tau}d\overline{w}\,\equiv\,\overline{\partial}^{0}+\frac{\pi}{4\tau}d\overline{w},$

see e.g. [HH, Section 3]. In particular, for the choice of signs
$\sigma_{2}\,=\,1$ and $\sigma_{3}\,=\,\sigma_{4}\,=\,-1$ the
$\chi$-coordinate of $[\pi^{*}D]\,\in\,\mathcal{M}_{1,1}^{r}$ is a real half
lattice point of Jac$(\Sigma_{2})$. Since $\Sigma_{2}$ is a 4-fold covering of
$T^{2}_{\tau}$, the $1/4$-lattice points of Jac$(T^{2}_{\tau})$ pull back to
half lattice points of Jac$(\Sigma_{2})$, i.e., to non-trivial spin bundles of
$\Sigma_{2}$.

#### Proof of Lemma 3.14 continued

The Higgs field $\Phi$ as defined in (3.19) has eigenvalues $\pm
c\frac{dz}{\widetilde{y}}$ for some $c\in\mathbb{C}^{*}$ and a direct
computation shows that its eigenline bundles $E^{\pm}\to\Sigma_{2}$ are given
by holomorphic inclusion maps

$\begin{pmatrix}s_{\pm}\\\ t_{\pm}\end{pmatrix}\colon
E^{\pm}\to\mathcal{O}\oplus\mathcal{O}$

with divisors

$(s_{+})=(s_{-})\,=\,w_{3}+w_{4}\quad\text{and}\quad(t_{+})=(t_{-})\,=\,w_{1}+w_{2}$

(moreover, $s_{+}=-s_{-}$ and $t_{+}=t_{-}$ up to scaling), see [HH, Theorem
2]. Therefore,

$E^{\pm}\,=\,L(-w_{1}-w_{2})\,=\,L(-w_{3}-w_{4}).$

Hence, the corresponding holomorphic line bundle $L$ of degree 0 obtained
after tensoring with $L(2w_{1})$ satisfies $L=L^{*}$ and is given by

$L\,=\,L(w_{1}-w_{2})=L(2w_{1}-w_{3}-w_{4}).$

∎

The Riemann surface $\Sigma$ considered in this paper is obtained by a
covering of the 4-punctured sphere $\mathcal{S}_{4}$ with the number of sheets
depending on the parabolic weight $\widetilde{r}.$ More specifically, let
$\widetilde{r}=\frac{l}{k}\in(\frac{1}{2},\frac{1}{4})$ with coprime integers
$l,k\in\mathbb{N}$. Fix $\sigma_{2}\,=\,1$ and
$\sigma_{3}\,=\,\sigma_{4}\,=\,-1$. The Riemann surface $\Sigma=\Sigma_{g}$
given by a $(g+1)$-fold covering
$\pi_{g}:\Sigma_{g}\rightarrow\mathcal{S}_{4}$ defined by the equation

$\Sigma_{g}\colon y^{g+1}\,=\,\frac{(z-p_{1})(z-p_{2})}{(z-p_{3})(z-p_{4})},$
(3.20)

where $g=k-1$ if $k$ is odd and $g=k/2-1$ for $k$ even. Using this covering
the singularities of the connections on $\mathcal{S}_{4}$ become apparent on
$\Sigma_{g}.$ In other words, there exist a singular gauge under which the
connections become smooth connections on $\Sigma_{g}$. Likewise

$\widetilde{\Sigma}_{g}\colon
y^{g+1}=\frac{(z-p_{1})(z-p_{4})}{(z-p_{2})(z-p_{3})}$ (3.21)

is the compact surface with respect to the sign choice $\sigma_{4}=1$ and
$\sigma_{2}=\sigma_{3}=-1$. The trivial holomorphic structure on $\Sigma_{g}$
(and analogously for $\widetilde{\Sigma}_{g}$) can be easily identified
according to the following Lemma.

######  Lemma 3.16.

Let $\Sigma_{g}$ (3.20) and $D$ (3.18) be defined as above (for the same
choices of $\sigma_{2},\sigma_{3},\sigma_{4}$). Then there exist a unique
$\mathbb{Z}_{2}$-connection $\nabla^{\mathbb{Z}_{2}}$ on $\Sigma_{g}$ with
local monodromies $(-1)^{k-1}$ around each preimage of the marked points
$p_{1},\dots,p_{4}$ such that the flat $\mathrm{SL}(2,\mathbb{C})$-connection
$\nabla^{\mathbb{Z}_{2}}\otimes(\pi_{g})^{*}D$ has trivial monodromy on
$\Sigma_{g},$ i.e. it is gauge equivalent to the trivial (smooth) connection
on the compact Riemann surface $\Sigma_{g}$. If $k$ is odd,
$\nabla^{\mathbb{Z}_{2}}$ is trivial.

###### Proof.

The proof is completely analogous to the proof of Proposition 3.1 in [BDHH]
(see also [HHSch, Theorem 3.2(5)]) by adapting the covering (2.1) and the
reducible Fuchsian systems (3.18) considered to the different sign choice
$\sigma_{2}=1$ and $\sigma_{3}=\sigma_{4}=-1$ in this paper. The same holds
for the choice $\sigma_{4}=1$ and $\sigma_{2}=\sigma_{3}=-1$. ∎

######  Lemma 3.17.

Let $\nabla=\nabla^{a,\chi,r}$ be a connection on $T_{\tau}^{2}$ with
$[\eta^{*}\overline{\nabla}]=[\nabla].$ Then

$x=\operatorname{tr}(X)\in\mathbb{R},\quad\text{and}\quad
y=\operatorname{tr}(Y)\,\in\mathbb{R}\quad\text{and}\quad
z_{1}=\operatorname{tr}(YX)\,=\,\overline{z}_{2}=\overline{\operatorname{tr}(Y^{-1}X)}.$

In particular, the representation is real if and only if
$z_{1}=z_{2}\in\mathbb{R}$.

###### Proof.

The first part is completely analogous to the proof of Lemma [BDHH, Lemma
4.6]222The reader should be aware of the different real involutions $\eta$
used in [BDHH] and in this paper.. The second part is a direct consequence of
Theorem 3.5. ∎

######  Remark 3.18.

The Lemma implies that an $\eta$-invariant representation is real if and only
if the discriminant of (3.8), as quadratic equation in $z,$ is zero. To be
more explicit this gives the extra equation

$x^{2}y^{2}-4x^{2}-4y^{2}+8(1+\text{cos}(2\pi r))=0.$ (3.22)

The main advantage of considering this very symmetric case is that the space
of real and $\eta$-invariant representations becomes real 1-dimensional, see
Figure 1. The four different non-compact real components of the character
variety (see Theorem 3.5) correspond to the four different spin bundles over
the torus. The trace coordinates of the four non-compact components differ
only by signs, and these four components are mapped into the same component of
real representations of the 4-punctured sphere via abelianization (3.9).

Figure 1. The space of real representations invariant under $\eta$, for
$r=\tfrac{1}{10}$ ($\widetilde{r}=\tfrac{3}{10}$), which is the parabolic
weight of our dodecahedron example.

###  3.7. Hitchin section

Consider a spin structure

$S\longrightarrow T_{\tau}^{2}$

on the rectangular torus $T_{\tau}^{2},$ i.e., $S^{\otimes 2}\cong
K_{T^{2}_{\tau}}$, and the strictly stable and strongly parabolic Higgs bundle
$(E,\widetilde{r},l_{0},\Phi)$ given by the data

$E=S\oplus S;\quad r\in(0,\tfrac{1}{2});\quad l_{o}=\mathbb{C}S_{o}\oplus
0;\quad\Phi=\begin{pmatrix}0&0\\\ dw&0\end{pmatrix}.$

Then Hitchin-Kobayashi correspondence on non-compact curves [Si1] yields a
compatible flat connection $\nabla$ satisfying

$[\nabla]\in\mathcal{M}_{1,1}^{r}$

with $\mathrm{SL}(2,\mathbb{R})$-monodromy. The underlying holomorphic bundle
is hereby the non-trivial extension of $S$ by itself and the parabolic line
$l_{0}$ is contained in the unique holomorphic line subbundle $S$ 333The later
follows from the fact that the harmonic metric solving the self-duality
equations must be diagonal by uniqueness..

For every $q\in\mathbb{C}$ a strongly parabolic Higgs field

$\Phi^{q}=\begin{pmatrix}0&qdw\\\ dw&0\end{pmatrix}$

on the parabolic bundle $(E,r,l_{o})$ corresponds to a compatible flat
connection $\nabla^{q}$ with $[\nabla^{q}]\in\mathcal{M}_{1,1}^{r}$ having
$\mathrm{SL}(2,\mathbb{R})$-monodromy (see Simpson [Si1]) – this is a
particular instance of the so-called parabolic Hitchin-Kobayashi
correspondence.

######  Lemma 3.19.

For fixed parabolic weight $r\in(0,\tfrac{1}{2})$, the element
$[\nabla^{q}]\in\mathcal{M}_{1,1}^{r}$ is $\eta$-invariant if and only of
$q\in\mathbb{R}.$

###### Proof.

The parabolic Higgs pairs $(E,r,l_{o},\Phi^{q})$ and
$\eta^{*}\overline{(E,r,l_{o},\Phi^{q})}$ are gauge equivalent if and only if
$q\in\mathbb{R}$. Therefore the Lemma follows from the Hitchin-Kobayashi
correspondence for parabolic bundles. ∎

######  Lemma 3.20.

Let $\widetilde{r}=\frac{l}{k}\in(\frac{1}{4},\frac{1}{2})$ be rational with
coprime $l,k\in\mathbb{N}$ and $r=2\widetilde{r}-\frac{1}{2}=\frac{4l-k}{2k}$.
The connection $\nabla^{a,\chi,r}$ (on $T_{\tau}^{2}$) induces on $\Sigma_{g}$
as defined in (3.20) the trivial holomorphic structure if
$\chi=\tfrac{\pi}{4\tau}\in\mathbb{R}$ modulo sign and lattice points.
Likewise, the connection $\nabla^{a,\chi,r}$ (on $T_{\tau}^{2}$) induces on
$\widetilde{\Sigma}_{g}$ (3.21) the trivial holomorphic structure if
$\chi=i\tfrac{\pi}{4}\in\mathbb{R}$ modulo sign and lattice points.

###### Proof.

The proof is completely analogous to the proof of [BDHH, Proposition 3.1]. In
fact it is just a reformulation of Lemma 3.16 using Lemma 3.14. ∎

In view of the above lemma we define

######  Definition 3.21.

We denote by $H\in$ Jac$(T^{2}_{\tau})$ the holomorphic structure (or the
corresponding parabolic bundle on $\mathcal{S}_{4}$) that lifts to the trivial
holomorphic structure over the associated compact Riemann surface $\Sigma$,
i.e., either $H=\tfrac{\pi}{4\tau}$ for the sign choice
$\sigma_{2}=1,\sigma_{3}=\sigma_{4}=-1$ and $\Sigma=\Sigma_{g},$ or
$H=i\tfrac{\pi}{4}$ for $\sigma_{4}=1,\sigma_{2}=\sigma_{3}=-1$ and
$\Sigma=\widetilde{\Sigma}_{g}.$

######  Remark 3.22.

If $H$ lies in the image of the map

$q\in\mathbb{R}\longmapsto\chi(\nabla^{q})\in\text{Jac}(T^{2}_{\tau}),$

the two Lemmas 3.19 and 3.20 show that the pull-back of the corresponding
connection $\nabla^{q_{H}}$ to $\Sigma$ has trivial holomorphic structure and
real monodromy representation.

##  4\. Grafting and Spin Structures

###  4.1. Complex projective structures

Complex projective structures (or simply projective structures) on Riemann
surfaces are classical objects in the theory of Riemann surfaces, see [Gu1]
and the references therein. Consider an atlas
$(U_{\alpha},z_{\alpha})_{\alpha\in\mathcal{U}}$ of a Riemann surface for
which all the transition functions

$z_{\beta}\circ z_{\alpha}^{-1}(z)=\frac{az+b}{cz+d}\quad\text{for some
(constant)}\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\text{GL}(2,\mathbb{C})$

are Möbius transformations. Such an atlas is called a (complex) projective
atlas. Two projective atlases are equivalent if their union remains a
projective atlas. An equivalence classes of projective atlases is a (complex)
projective structure.

Naturally, the complex projective space $\mathbb{C}P^{1}$ itself is equipped
with its standard projective structure. For elliptic curves the natural
projective structure is obtained by identifying it with the flat torus
$\mathbb{C}/\Lambda$. All transitions functions are in this case translations.

On a compact surface $\Sigma$ of genus $g\geq 2$ a special projective
structure is provided by the uniformization theorem. In this case, there is a
global biholomorphism from the universal cover of $\Sigma$ to Poincaré’s
upper-half plane $\mathbb{H}^{2}\subset\mathbb{C}P^{1}$ which is equivariant
with respect to a group homomorphism from the fundamental group of $\Sigma$
into the group of PSL$(2,\mathbb{R})$-valued Möbius transformations (with
image a Fuchsian subgroup in PSL$(2,\mathbb{R})$). This map to the hyperbolic
plane $\mathbb{H}^{2}\subset\mathbb{C}P^{1}$ coincides with the developing map
of the unique hyperbolic metric (i.e. having constant curvature $-1$) on
$\Sigma$ compatible with the complex structure.

In general, a complex projective structure on $\Sigma$ gives rise to a
developing map $dev$ from the universal cover of $\Sigma$ to $\mathbb{C}P^{1}$
which is a local biholomorphism (but not necessarily a proper injective map).
This developing map is equivariant with respect to a group homomorphism from
the fundamental group of $\Sigma$ into PSL$(2,\mathbb{C})$ (uniquely defined
up to conjugation in the Möbius group) which is referred to as the monodromy
of the complex projective structure. By abuse of notation a complex projective
structure is called a real projective structure if the corresponding monodromy
takes values in PSL$(2,\mathbb{R})$ (up to conjugation in PSL$(2,\mathbb{C}))$
[Falt, Tak].

###  4.2. $\operatorname{SL}(2,\mathbb{C})$-Opers

A projective structure on a compact Riemann surface $\Sigma$ of genus $g\geq
2$ can also be described using particular flat
$\operatorname{SL}(2,\mathbb{C})$-connections, called opers. Let $\nabla$ be a
flat $\operatorname{SL}(2,\mathbb{C})$-connection on the rank two trivial
smooth bundle $V=\underline{\mathbb{C}}^{2}\to\Sigma$ such that its induced
holomorphic structure $\overline{\partial}^{\nabla}$ admits a holomorphic sub-
line bundle $S$ of maximal degree $(g-1).$ Take a complementary
$C^{\infty}$-bundle $S^{*}\subset V$ and write

$\nabla=\begin{pmatrix}\nabla^{S}&\psi\\\ \varphi&\nabla^{S^{*}}\end{pmatrix}$
(4.1)

with respect to $V=S\oplus S^{*}.$ As $S$ is a holomorphic subbundle $\varphi$
is a $(1,0)$-form with values in Hom$(S,S^{*})$. Moreover, the flatness of
$\nabla$ implies that

$\varphi\in H^{0}(\Sigma,K_{\Sigma}(S^{*})^{2}).$

If $\varphi\equiv 0$, then $S$ is a parallel sub-line bundle of $V$ with
respect to the connection $\nabla$ and, consequently, it must have degree zero
(and not $g-1$). Therefore, for $g\geq 2$ the holomorphic section $\varphi$ is
not identically zero. Moreover, since the degree of $K_{\Sigma}(S^{*})^{2}$ is
zero, the section $\varphi$ is nowhere vanishing. Therefore $S$ is a spin
bundle, i.e., $S^{2}=K_{\Sigma}$ as holomorphic line bundles, and the section
$\varphi$ can be identified with the constant section $\bf 1$ of the trivial
holomorphic line bundle $\underline{\mathbb{C}}\rightarrow\Sigma$.

######  Definition 4.1.

A flat $\operatorname{SL}(2,\mathbb{C})$-connection of the form (4.1) on a
compact Riemann surface is called an oper.

Given an oper $\nabla$ on the Riemann surfaces $\Sigma$ the induced projective
structure is obtained as follows. Consider, on an open simply connected subset
$U\subset\Sigma$, two linear independent $\nabla$-parallel sections of
$V=S\oplus S^{*}$

$\Psi_{1}=\begin{pmatrix}x_{1}\\\
y_{1}\end{pmatrix},\quad\Psi_{2}=\begin{pmatrix}x_{2}\\\ y_{2}\end{pmatrix}.$

Then $y_{1}$ and $y_{2}$ are holomorphic sections of $S^{*}$ as the projection
$V\to V/S$ is holomorphic. The quotient $z=y_{1}/y_{2}$ defines a holomorphic
map to $\mathbb{C}P^{1}$. Choosing two other linear independent parallel
sections

$\widetilde{\Psi}_{1}=\begin{pmatrix}\widetilde{x}_{1}\\\
\widetilde{y}_{1}\end{pmatrix}=a\Psi_{1}+b\Psi_{2}\quad\widetilde{\Psi}_{2}=\begin{pmatrix}\widetilde{x}_{2}\\\
\widetilde{y}_{2}\end{pmatrix}=c\Psi_{1}+d\Psi_{2}$

(with $ad-bc=1)$ amounts into

$z=y_{1}/y_{2}\longmapsto\widetilde{z}=\widetilde{y}_{1}/\widetilde{y}_{2}=\frac{ay_{1}+by_{2}}{cy_{1}+dy_{2}}=\frac{az+b}{cz+d},$

which is a Möbius transformation. Because $\varphi$ is nowhere vanishing, the
map $z$ is unbranched, i.e., $z$ is a local (holomorphic) diffeomorphism, and
we obtain a projective atlas.

###  4.3. Grafting

Grafting, or more precisely $2\pi$-grafting of the uniformization, introduced
by Maskit [Mas], Hejhal [Hej] and Sullivan-Thurston [ST], is a procedure to
obtain infinitely many distinct real projective structures. Our short
description here follows Goldman [Go87].

Consider the real projective structure given by the uniformization (Fuchsian)
representation of a Riemann surface $X$ and its developing map $dev$ to
$\mathbb{H}^{2}\subset\mathbb{C}P^{1}$. Every non-trivial element of the first
fundamental group $[\gamma]\in\pi_{1}(X)$ can be represented by a unique
geodesic $\gamma\subset X$ with respect to the constant curvature $-1$ metric
on $X$ (up to orientation). Under the developing map $\gamma$ is mapped to a
circular arc. The corresponding full circle $C$ intersects the boundary at
infinity of the hyperbolic disc at two points. The monodromy of the
uniformization representation along $\gamma$ is given by an element
$A\in\operatorname{SL}(2,\mathbb{R})$, unique up to sign. The sign depends on
the lift of the monodromy representation from ${\rm PSL}(2,\mathbb{R})$ to
$\operatorname{SL}(2,\mathbb{R})$.

On the other hand, a hyperbolic transformation conjugated to
$A\in\operatorname{SL}(2,\mathbb{R})$ gives rise to a Hopf torus $T_{A}$
endowed with a projective structure as follows. There exist two (unique)
circles $S_{1}$ and $S_{2}\subset\mathbb{C}P^{1}$ that are invariant under the
transformation $A.$ Let $C_{1}$ be another circle in $\mathbb{C}P^{1}$
intersecting both $S_{1}$ and $S_{2}$ perpendicularly, and consider
$C_{2}=A(C_{1}).$ Since $C_{1}$ and $C_{2}$ have no intersection points, they
bound an annulus $\mathbb{A}$. The torus $T_{A}$ is then obtained from gluing
$C_{1}$ and $C_{2}$ via $A$ and possesses by construction a projective
structure. The monodromy of the corresponding projective structure on $T_{A}$
is trivial along $C_{1}$, while it is $A$ along the curve obtained from
projecting the arc on $S_{1}$ between the intersection points with $C_{1}$ and
$C_{2}=A(C_{1})$. When cutting $T_{A}$ along $S_{1}$ (rather than $C_{1}$) we
obtain another annulus and we denote this cylinder together with its two
boundary components by $|T_{A}|.$

Grafting along the geodesic $\gamma\subset X$ glues $X$ and the appropriate
Hopf torus $T_{A}$ for the hyperbolic $A\in\operatorname{SL}(2,\mathbb{R})$
representing the monodromy along $\gamma$. More explicitly, let $S_{1}$ be the
unique circle in $\mathbb{C}P^{1}$ containing the image of the geodesic
$\gamma$ under the uniformization map, and $S_{2}$ be the boundary of
$\mathbb{H}^{2}.$ Then each of the two boundary components of $|T_{A}|$ can be
identified with the image of the geodesic $\gamma.$ Therefore,
$X\setminus\gamma$ and $|T_{A}|$ can be glued to obtain a new Riemann surface
$X^{G}$ without boundary (the notation $X^{G}$ stands for grafted $X$).
Moreover, the induced new projective structure has the same monodromy as the
uniformization of $X$. In particular, it is a real projective structure. Note
that (the developing map $dev_{G}$ of) the projective structure on $X^{G}$
induces a curvature -1 metric away from a singularity set. The singularity set
is given by the intersection of the annulus $\mathbb{A}$ with the circle
$S_{2}$ considered as the boundary of the hyperbolic plane $\mathbb{H}^{2}$
consisting of two smooth curves that are both, considered as curves in
$X^{G}$, closed and homotopic to $\gamma$.

Iteration leads to the construction of infinitely many Riemann surfaces with
distinct real projective structures, but their monodromy remains the same
uniformization monodromy of the initial Riemann surface $X$. Altogether,
starting with the isotopy class $\mathcal{C}$ of a simple closed curve on $X$,
and applying the grafting construction along the closed curve yields
(infinitely many) different Riemann surfaces of the same genus with real
projective structures. We will refer to grafting along a simply closed
geodesic $\gamma$ once as simple-grafting and neglect the possibility of
grafting multiple times along the same geodesic in the following.

######  Remark 4.2.

In what follows, $\Sigma$ will be the Riemann surface $X^{G}$ obtained from
grafting (once) the Riemann surface $\Sigma^{uG}:=X$, where the superscript
${uG}$ stands for ungrafting.

###  4.4. Spin structures

Let $\Sigma$ be a compact Riemann surface of genus $g$. A spin structure is a
choice of a holomorphic line bundle $S$ with

$S^{\otimes 2}=K_{\Sigma}.$

Two spin bundles differ by a holomorphic line bundle which squares to the
trivial holomorphic line bundle. If we equip this holomorphic line bundle with
its unique flat unitary connection, its monodromy takes values in
$\mathbb{Z}_{2}$ (as the monodromy squares to the identity). Such a
holomorphic line bundle together with its flat connection is called a
$\mathbb{Z}_{2}$-bundle. It is determined by its monodromy representation
which is a group homomorphism from the fundamental group of $\Sigma$ into the
abelian group $\mathbb{Z}_{2}.$ Hence the space of spin structures is an
affine space with underlying translation vector space
$H_{1}(\Sigma,\mathbb{Z}_{2})$.

From a topological point of view, a spin structure is given by a quadratic
form

$Q\colon H_{1}(\Sigma,\mathbb{Z}_{2})\to\mathbb{Z}_{2},$

whose underlying bilinear form is the intersection form (mod 2), see [John,
At]. The relationship between these two viewpoints can be explained as
follows: Fix a given line bundle $S$ with $S^{\otimes 2}=K_{\Sigma}.$ Then,
for every closed and immersed curve $\delta\colon S^{1}\to\Sigma$ there is a
unique $\omega\in\Gamma(S^{1},\delta^{*}K_{\Sigma})$ with

$\omega(\delta^{\prime})=1.$

Let $\widetilde{\delta}\,\colon\,\mathbb{R}\,\longrightarrow\,\Sigma$ be the
immersed curve defined as the lift of $\delta$ to the universal covering
$\mathbb{R}\to S^{1}$. Consider the pull-back $\widetilde{\omega}$ of $\omega$
to a (non-vanishing) section of $\widetilde{\delta}^{*}K_{\Sigma}$. Up to
sign, there exists a unique section

$\widetilde{s}\,\in\,\Gamma(\mathbb{R},\,\widetilde{\delta}^{*}S)$

such that $\widetilde{s}^{2}\,=\,\widetilde{\omega}$. The
$\mathbb{Z}_{2}$-monodromy of $S$ along $\delta$ is $1$ if $\widetilde{s}$
(or, equivalently, $-\widetilde{s}$) is invariant by the action of the
fundamental group of $S^{1}$ on $\mathbb{R}$ by deck-transformations. The
$\mathbb{Z}_{2}$-monodromy of $S$ along $\delta$ is $-1$ if $\widetilde{s}$ is
transformed in $-\widetilde{s}$ by the action of the generator of the
fundamental group of $S^{1}$ on $\mathbb{R}$. This $\mathbb{Z}_{2}$-monodromy
only depends on the class of $\delta$ in $H_{1}(\Sigma,\mathbb{Z}_{2})$.
Associating to the class of $\delta$ in $H_{1}(\Sigma,\mathbb{Z}_{2})$ the
above monodromy with values in $\mathbb{Z}_{2}$ uniquely determines a
quadratic form $Q\colon H_{1}(\Sigma,\mathbb{Z}_{2})\to\mathbb{Z}_{2}$ whose
underlying bilinear form is the intersection form (mod 2), and hence a
topological spin structure, see [John] and also [P] and [Bob, Section 10].

####  4.4.1. Spin structures and opers

Let $\nabla$ be an oper on $V\to\Sigma$ given by (4.1) corresponding to real
projective structure, i.e., its monodromy takes values in

$P\mathrm{SL}(2,\mathbb{R})=\text{Isom}(\mathbb{H}^{2}).$

The isomorphism between $S^{\otimes 2}$ and $K_{\Sigma}$ can be made explicit
in two equivalent ways.

First, consider for $p\in\Sigma$ an arbitrary $s_{p}\in S_{p}\subset V_{p},$
then

$\nabla s_{p}\wedge s_{p}\in(K_{\Sigma})_{p}$

is well-defined and gives rise to a bilinear map $S_{p}\times
S_{p}\to(K_{\Sigma})_{p},$ since $S$ is a holomorphic subbundle of $V.$ This
bilinear form is non-degenerate, as $\varphi$ is non-vanishing, and defines a
holomorphic isomorphism between $S^{\otimes 2}$ and $K_{\Sigma}.$ Likewise,
consider the (locally defined) parallel sections

$\Psi_{1}=\begin{pmatrix}x_{1}\\\
y_{1}\end{pmatrix},\Psi_{2}=\begin{pmatrix}x_{2}\\\
y_{2}\end{pmatrix}\in\Gamma(U,S\oplus S^{*}).$

determined by the initial condition

$(\Psi_{1})_{p}=\begin{pmatrix}s_{p}\\\
0\end{pmatrix}\quad\text{and}\quad(\Psi_{2})_{p}=\begin{pmatrix}0\\\
t_{p}\end{pmatrix}$

with

$(t_{p},s_{p})=1.$

Then, a direct computation shows

$s_{p}\otimes s_{p}=\omega_{p}\in(K_{\Sigma})_{p}$

where

$\omega\,=\,-d(\tfrac{y_{1}}{y_{2}}).$

Another way to obtain the spin bundle $S$ is the following. The standard
projective structure on $\mathbb{C}P^{1}$ is induced by the trivial connection
$d$ on the trivial holomorphic rank 2 bundle. Its spin bundle is the
tautological bundle $\mathcal{O}(-1),$ i.e, the fiber at $l\in\mathbb{C}P^{1}$
is the line $l.$ Consider for a general Riemann surface $\Sigma$ a projective
structure and developing map $dev$ induced by an oper $(V,\nabla)$. Let $M$ be
the monodromy homomorphism of $\nabla$. Then the bundle $V$ is given by the
twisted bundle

$V\,=\,(\widetilde{\Sigma}\times\mathbb{C}^{2})/\sim,$

where $(p,v)\sim(\widetilde{p},\widetilde{v})$ if and only if
$\widetilde{p}=\gamma_{*}p$ and $\widetilde{v}=M_{\gamma}v$ for a
$\gamma\in\pi_{1}(\Sigma).$ The spin bundle $S$ is then given by the twisted
pull-back of the tautological bundle
$\mathcal{O}(-1)\rightarrow\mathbb{C}P^{1}$ via the developing map

$S=dev^{*}\mathcal{O}(-1)/\sim.$ (4.2)

Due to its topological invariance, continuous deformations of the Riemann
surface and the oper do not change the (topological) spin structure. (For
closed curves in the moduli space of Riemann surfaces that are not null-
homotopic, the deformation of the spin structure along the curve might have
monodromy, see for example [At]. This corresponds to a non-trivial action of
the mapping class group.)

Note that the difference between two quadratic forms on
$H_{1}(\Sigma,\mathbb{Z}_{2})$ corresponding to spin structures is given by a
linear form on $H_{1}(\Sigma,\mathbb{Z}_{2})$, see [John].

######  Lemma 4.3.

Let $\Sigma$ be a compact Riemann surface. The uniformization connection
$\nabla^{U}$ and the simple-grafting connection $\nabla^{G}$, along an isotopy
class $\mathcal{C}$ of a simple non-null-homotopic curve $\mathcal{C}$ on
$\Sigma$, (whose monodromy lies in the same connected component of real
representations) induce different spin structures on the Riemann surface
$\Sigma.$ More precisely, the difference of the corresponding spin structures
is determined by the linear form on $H_{1}(\Sigma,\mathbb{Z}_{2})$ obtained by
inserting a representative of the class $\mathcal{C}\in
H_{1}(\Sigma,\mathbb{Z}_{2})$ into the intersection form mod 2 on $\Sigma$.

######  Remark 4.4.

Clearly, the lemma generalizes to multiple graftings as well.

###### Proof.

Consider the Riemann surface $\Sigma^{uG}=X$ (obtained by ungrafting), i.e.,
its uniformization connection is gauge equivalent to $\nabla^{G}$. Since the
two complex structures on $\Sigma^{uG}$ and $\Sigma,$ viewed as two points in
the Teichmüller space of genus $g$ surfaces, can be connected by a smooth
curve, both uniformization connections (in the same connected component of
real points in the de Rham moduli space) on $\Sigma^{uG}$ and $\Sigma$ induce
the same topological spin structure. It remains to show that grafting once
(from the uniformization oper of $\Sigma^{uG}$ to the oper $\nabla^{G}$ on
$\Sigma$) changes this spin structure.

Recall that we can compute the value of the quadratic form $Q$ associated to
the spin structure on the class in $H^{1}(\Sigma,\mathbb{Z}_{2})$ of an
immersed curve $\delta\colon S^{1}\to\Sigma$ by considering the lifting
$\widetilde{\omega}\,\in\,\Gamma(\mathbb{R},\,\widetilde{\delta}^{*}K_{\Sigma})$
to the universal cover of $S^{1}$ of the unique section
$\omega\in\Gamma(S^{1},\delta^{*}K_{\Sigma})$ defined by
$\omega(\delta^{\prime})=1$ and then considering the
$\mathbb{Z}_{2}$-monodromy defined by a section
$\widetilde{s}\,\in\,\Gamma(\mathbb{R},\,\widetilde{\delta}^{*}S)$ such that
$\widetilde{s}^{2}\,=\,\widetilde{\omega}$.

Along closed curves representing an element of $\pi_{1}(\Sigma)$ that do not
intersect the (simple) grafting curve $\gamma$, the developing map does not
change. Therefore, by (4.2) the quadratic form of the spin structure
specialized on those curves remains the same.

Consider the closed curve $\delta_{1}$ in $\Sigma^{uG}$ which intersects the
grafting curve $\gamma$ once. The developing map along the corresponding
closed curve $\delta$ on $\Sigma$ (which intersects the grafting curve
$\gamma$ once) is obtained from the developing map along $\delta_{1}$ by
precomposing it with the circle $C_{1}$ along the Hopf cylinder $T_{A}$ 555We
implicitly assume that $C_{1}$ does not pass through $\infty$. If it does, we
can replace $C_{1}$ by $C_{2}$ without altering the remaining arguments.. With
respect to the holomorphic 1-form $dz$ on
$\mathbb{C}P^{1}\setminus\\{\infty\\}$ and for
$C_{1}=\\{ae^{i\theta}+b\;|\;\theta\in[0,2\pi]\\}$ for appropriate
$a>0,b\in\mathbb{C}$, the section $\omega$ of the canonical bundle, along
$C_{1}$, is given by

$\omega_{\mid\theta}=-ia^{-1}e^{-i\theta}dz.$

Using (4.2) we can use the holomorphic section $\sqrt{dz}=(z,1)$ on
$\mathbb{C}P^{1}\setminus\\{\infty\\}$ along $C_{1}$ and observe that a
section $s$ of $S$, along $C_{1}$, such that $s^{2}=\omega$ is given by

$s(\theta)=\sqrt{-ia^{-1}}e^{-\tfrac{i\theta}{2}}\sqrt{dz}.$

Hence, specialized on the closed curve $\delta$, the quadratic form associated
to the spin structure of the oper $\nabla^{G}$ on $\Sigma$ differs from the
quadratic form of the spin structure induced by the uniformization oper of
$\Sigma^{G}$ by a $-1$ factor. This completes the proof. ∎

###  4.5. Grafting on the 1-punctured torus $T^{2}_{\tau}$

Fix the parabolic weight $r\in(0,\frac{1}{2})$ and consider the real subspace
$\mathcal{M}_{1,1}^{r}(\mathbb{R})$ of $\mathcal{M}_{1,1}^{r}$ corresponding
to the real character variety. By [Go03, Theorem 3.4.1] each non-compact
connected component of the real character variety
$\mathcal{M}_{1,1}^{r}(\mathbb{R})$ is in one-to-one correspondence to
hyperbolic structures on the one-punctured torus with conical angle $4\pi r$
at the marked point (as the rotation angle satisfies $\theta=2r$). Therefore,
we refer to elements of $\mathcal{M}_{1,1}^{r}(\mathbb{R})$ as conical
hyperbolic structures for short. For conical hyperbolic structures [Bu,
Theorem 1.5.2] shows that every free homotopy class of curves on the torus can
be represented by a simply closed geodesic.

Fix a real representation $\rho\in\mathcal{M}_{1,1}^{r}(\mathbb{R})$ and
denote by $X,\,Y\in\operatorname{SL}(2,\mathbb{R})$ its values along
$\gamma_{x},\,\gamma_{y}\,\in\,\pi_{1}(T^{2}_{\tau}\setminus\\{0\\},\,p_{0}),$
respectively (see (3.5) and (3.6)). Then the corresponding conical hyperbolic
structure constructed in [Go03, Theorem 3.4.1] is obtained by gluing the
opposite edges of a particular hyperbolic quadrilateral
$P_{1}P_{2}P_{3}P_{4}.$ The point $P_{4}$ hereby is a fixed point in the
hyperbolic plane $\mathbb{H}^{2}$ with (elliptic) local monodromy given by
$r$, while

$P_{3}=X\cdot P_{4},\quad P_{2}\,=\,Y\cdot X\cdot P_{4}\quad\text{ and }\quad
P_{1}\,=\,X^{-1}\cdot Y\cdot X\cdot P_{4}.$

Consider now in the hyperbolic plane $\mathbb{H}^{2}$ the unique hyperbolic
geodesic $c$ perpendicular to both geodesics generated by $P_{1}P_{2}$ and
$P_{4}P_{3}$. Due to its uniqueness, this geodesic is fixed by $Y$ (it is the
axis of the loxodromic isometry $Y$). Moreover, since two distinct geodesics
in $\mathbb{H}^{2}$ intersect at most once, $c$ does not contain any of the
vertices $P_{j}$, for all $j=1,\,...,\,4$. Hence $c$ is a simply closed
geodesic representing the free homotopy class of $\gamma_{y}$ and $c$ does not
contain the conical point. The same argument shows the existence of a unique
hyperbolic geodesic $\widetilde{c}$ perpendicular to both geodesics generated
by $P_{1}P_{4}$ and $P_{2}P_{3}$ representing the free homotopy class of
$\gamma_{x}$ that does not contain the conical point.

Therefore, grafting of conical hyperbolic structures can be performed along
both geodesics representing the free homotopy class of $\gamma_{x}$ and
$\gamma_{y}$.

######  Lemma 4.5.

The conical hyperbolic structure on the torus is rectangular if and only if

$\operatorname{tr}(YX)=\operatorname{tr}(YX^{-1}).$

###### Proof.

The trace condition $\operatorname{tr}(YX)=\operatorname{tr}(YX^{-1})$ implies
that the reflexion across the unique geodesic $c$, which sends the edge
$P_{2}P_{3}$ to the edge $P_{1}P_{4},$ induces a real symmetry of the conical
hyperbolic structure. The fix point set consists of two disjoint circles – the
geodesic $c$ and the edge $P_{2}P_{3}$ (which is identified with the edge
$P_{1}P_{4}$). Hence the corresponding torus is rectangular (and not rhombic)
and the reflexion across $c$ coincides with the reflexion across the edge
$P_{1}P_{4}$ (this reflexion corresponds to the involution $i\eta$ described
in Section 3.4). ∎

Orbifold grafting is well-defined on the one-punctured torus and the grafted
Riemann surface remains rectangular:

######  Lemma 4.6.

Grafting the $1$-punctured rectangular torus $T^{2}_{\tau}=\mathbb{C}/\Gamma$,
$\Gamma=\mathbb{Z}\oplus i\tau\mathbb{Z},$ along $\gamma_{x}$ or $\gamma_{y}$,
respectively, yields another $1$-punctured and rectangular torus
$T^{2}_{\tau^{G}}$ with a different real projective structure.

###### Proof.

Let $Y\in\operatorname{SL}(2,\mathbb{R})$ denotes the monodromy along
$\gamma_{y}$. Since $y=\operatorname{tr}(Y)$ satisfies $|y|>2$, we get that
$Y$ is a hyperbolic element in $\operatorname{SL}(2,\mathbb{R}).$ Therefore,
grafting glues the annulus $|T_{Y}|$ to $T^{2}_{\tau}$ along $\gamma_{y}$.
This adds a rectangle to the fundamental domain of the torus $T^{2}_{\tau}$
along the edge $\gamma_{y}$. Therefore, the resulting torus remains
rectangular and $\tau$ decreases. With the same argument grafting along
$\gamma_{x}$ add a rectangle along the edge $\gamma_{y}$ and the conformal
type of $\tau$ increases. ∎

On the grafted punctured torus we fix (without loss of generality) the real
involution, which we again denote by $\eta,$ whose fix point set lies in the
homology class $2[\gamma_{x}]$ and contains the singular point. Let

$\mathcal{M}_{\mathbb{R}}^{r}=\mathcal{M}_{1,1}^{r,\eta}\cap\mathcal{M}_{1,1}^{r}(\mathbb{R})\subset\mathcal{M}_{1,1}^{r}$

be the subset of $\eta$-symmetric $\text{SL}(2,\mathbb{R})$-representations on
the one-punctured torus with local monodromy determined by the parabolic
weight $r$. From (3.8) and Lemma 3.17 (see also Remark 3.18 and Figure 1) the
space $\mathcal{M}_{\mathbb{R}}^{r}$ has four connected components, each of
them is a non-compact manifold of real dimension one.

######  Lemma 4.7.

Let $\rho\in\mathcal{M}_{\mathbb{R}}^{r}$. Then there is a unique
$\tau\in\mathbb{R}^{>0}$ such that the uniformization representation
$\rho^{U}(\tau)$ of the 1-punctured torus $T^{2}_{\tau}$ satisfies
$\rho=\rho^{U}(\tau)$.

###### Proof.

The proof of [Go03, Theorem 3.4.1] which identifies each connected component
of $\mathcal{M}_{1,1}^{r}(\mathbb{R})$ with conical hyperbolic structures also
implies that the one-dimensional submanifold of $\eta$-symmetric
representations $\mathcal{M}_{\mathbb{R}}^{r}$ is given by those hyperbolic
structures compatible with the rectangular conformal structure of
$T^{2}_{\tau}$, i.e., $\tau\in\mathbb{R}^{>0}$. ∎

The following Lemma is analogous to the main result of [Tan].

######  Lemma 4.8.

For each $r\in(0,\tfrac{1}{2})$, simple-grafting (along $\gamma_{x}$ or
$\gamma_{y}$) induces a homeomorphism of the Teichmüller space
$\mathbb{R}_{>0}$ of 1-punctured rectangular tori to itself.

###### Proof.

Without loss of generality we only consider grafting along $\gamma_{y}.$ The
map $\tau\mapsto\tau^{G}$, see Lemma 4.6, is smooth and moreover,
$\tau^{G}\,<\,\tau$. Therefore, $\tau^{G}\rightarrow 0$ for $\tau\rightarrow
0.$ For surjectivity we use the intermediate value theorem: it remains to show
$\tau^{G}\rightarrow\infty$ for $\tau\rightarrow\infty$. This follows by
observing that the conformal type $\tau_{Y}$ of the Hopf torus $T_{Y}$
satisfies $\tau_{Y}\to\infty$ when $\tau\to\infty$ and
$\tau^{G}=(\tfrac{1}{\tau}+\tfrac{1}{\tau_{Y}})^{-1}$. Injectivity follows
from the fact that the conformal type of the Hopf tori $T_{Y}$ and
$T_{\widetilde{Y}}$ satisfies $\tau_{Y}>\tau_{\widetilde{Y}}$ for
$\tau>\widetilde{\tau}$. ∎

###  4.6. Spin structures and projective structures on the 1-punctured torus

If the underlying holomorphic line bundle $L$ of a logarithmic
SL$(2,\mathbb{C})$-connection on the 1-punctured torus in (3.12) is not a spin
bundle, then the corresponding $\gamma_{\chi}^{\pm}$ must have a zero. Thus an
oper $\nabla$ on a 1-punctured torus gives rise to a spin bundle on the
compact torus through the special form of the connection in (3.14) and (3.15).

Moreover, the corresponding quadratic form (see Section 4.4) $Q$ on
$H_{1}(T^{2}_{\tau}\setminus\\{o\\},\,\mathbb{Z}_{2})$ is well defined on
$H_{1}(T^{2}_{\tau},\,\mathbb{Z}_{2})$, as the local conjugacy class around
the puncture is trivial in $\mathbb{Z}_{2}$, see also [John, Bob]. Hence,
using the same arguments as in the compact case, an suitably adjusted version
of Lemma 4.3 holds on the one-punctured torus, i.e., grafting changes the
induced spin structure.

The grafted connection $\nabla^{G}$ has by construction
SL$(2,\mathbb{R})$-monodromy lying in the same connected component as the
orbifold uniformization connection $\nabla^{U}$ of $T^{2}_{\tau}.$ Moreover,
the Hitchin section based at $T^{2}_{\tau}$ maps $q=0$ to $\nabla^{U}$ and
there exist a $q_{G}\in\mathbb{R}_{*}$ with $\nabla^{q_{G}}=\nabla^{G}$ by
$\eta$-invariance.

Recall that we have chosen the spin structure of $\nabla^{U}$ to be trivial,
i.e., it is represented by $\chi(\nabla^{U})=0\in$ Jac$(T^{2}_{\tau}).$ For
the grafted connection $\nabla^{G}$ it turns out, using abelianization, that
the underlying line bundle $L$ must be a particular spin bundle, i.e.,
$\chi(\nabla^{G})$ is a half lattice point of Jac$(T^{2}_{\tau})$.

######  Lemma 4.9.

Via abelianization the spin structure of simple-grafting along $\gamma_{y}$ is
given by $\chi(\nabla^{G,y})=i\tfrac{\pi}{2}$ (up to lattice points and sign),
and the spin structure of simple-grafting along $\gamma_{x}$ is given by
$\chi(\nabla^{G,x})=\tfrac{\pi}{2\tau}$ (up to lattice points and sign),

###### Proof.

The unitary line bundle connection on the induced spin bundle for
$\chi(\nabla^{G,y})$ is

$d+i\tfrac{\pi}{2}d\overline{w}+i\tfrac{\pi}{2}dw$

which has monodromy $-1$ along $\gamma_{x}$ and monodromy $1$ along
$\gamma_{y}$. For $\chi(\nabla^{G,x})$ it is

$d+\tfrac{\pi}{2\tau}d\overline{w}-\tfrac{\pi}{2\tau}dw$

and has monodromy $-1$ along $\gamma_{y}$ and monodromy $1$ along
$\gamma_{x}$. Thus, the statement follows by the same arguments as Lemma 4.3,
since after grafting along $\gamma_{y}$ the spin structure change by
multiplication with $1$ along $\gamma_{y}$ (meaning unchanged) and by
multiplication with $-1$ along $\gamma_{x}$. It is the other way around when
grafting along $\gamma_{x}.$ ∎

##  5\. Holomorphic connections with
$\operatorname{SL}(2,\mathbb{R})$-monodromy

Consider the natural 2-fold covering map $\pi\colon
Jac(T^{2}_{\tau})\to\mathbb{C}P^{1}$ branched at the four spin bundles. The
real subspace provided by Lemma 3.10 is mapped to
$\mathbb{R}\subset\mathbb{C}$. In fact, as the corresponding elliptic curve
$T^{2}_{\tau}$ being rectangular, its associated $\wp$-function maps the four
spin bundles in the Jacobian (which identifies with the half periods and the
critical points of $\wp$) to the real axis.

######  Lemma 5.1.

The map
$h\colon\mathbb{R}\rightarrow\mathbb{R}\subset\mathbb{C}P^{1}=\pi(\text{Jac}(T^{2}_{\tau}))$
given by $q\mapsto\pi(\chi(\nabla^{q}))$ is continuous. Moreover, there exists
$q_{H}\in\mathbb{R}\setminus\\{0\\}$ satisfying either
$\chi(\nabla^{q_{H}})=\frac{\pi}{4\tau}$ or
$\chi(\nabla^{q_{H}})=i\frac{\pi}{4}$ (up to sign and adding lattice points).

###### Proof.

The holomorphic structure $\chi(\nabla^{q})$ as a map into Jac$(T^{2}_{\tau})$
is only well-defined up to sign. The projection to $\mathbb{C}P^{1}$ removes
this multivaluedness, and $h$ is well-defined and continuous. It should be
noted that this is, up to normalisation, the Tu-invariant of $\nabla^{q}$ in
[Lo]. By $\eta$-invariance, i.e., Lemma 3.19 and Lemma 3.10, and using the
fact that $\pi$ maps exactly the (real) $\eta$-invariant points of the
Jacobian (i.e., $\chi$ lying in the lines given by Lemma 3.10) to real points
(including $z=\infty$) in $\mathbb{C}P^{1}$, we obtain that
$h\colon\mathbb{R}\to\mathbb{R}P^{1}\subset\mathbb{C}P^{1}.$

Grafting along $\gamma_{y}$ (or $\gamma_{x}$) changes the spin structure by
Lemma 4.9. The map $h$ sends the point $q=0$ to $0\in$ Jac$(T_{\tau}^{2})$ and
the point $q=q_{G}$ to $i\tfrac{\pi}{2}\in$Jac$(T_{\tau}^{2})$ (or to
$\tfrac{\pi}{2\tau}\in$Jac$(T_{\tau}^{2})$).

By continuity the image of $h$ contains either the interval
$\pi([0,i\frac{\pi}{2}])\subset\mathbb{R}P^{1},$ where

$[0,i\tfrac{\pi}{2}]:=\\{ti\tfrac{\pi}{2}\mid 0\leq t\leq 1\\}$

and analogously for the subsequent intervals, or the closure of its complement
(inside $\mathbb{R}P^{1}$) given by

$\mathbb{R}P^{1}\setminus\pi((0,i\tfrac{\pi}{2}))=\pi([0,\tfrac{\pi}{2\tau}])\cup\pi([\tfrac{\pi}{2\tau},\tfrac{\pi}{2\tau}+i\tfrac{\pi}{2}])\cup\pi([\tfrac{\pi}{2\tau}+i\tfrac{\pi}{2},i]).$

Consequently, there exists $q_{H}$ (and a second $\widetilde{q}_{H}$ for
grafting along $\gamma_{x}$) in the interval between $q=0$ and $q=q_{G}$ such
that $\chi(\nabla^{q_{H}})=\frac{\pi}{4\tau}$ or
$\chi(\nabla^{q_{H}})=i\frac{\pi}{4}$ (up to sign and adding lattice points).
∎

######  Remark 5.2.

The Lemma shows that the holomorphic structure $H$ (see Definition 3.21) which
lifts to the trivial holomorphic structure on the associated Riemann surface
$\Sigma$ is attained whenever the weight $r$ is rational. Since the map $h$
depends continuously in $\tau$, the bundle type $H$ does not change for a
continuous deformation of $\tau.$

Lemma 3.20 together with Remark 3.22 gives the following theorem.

######  Theorem 5.3.

For every rational weight
$\widetilde{r}=\tfrac{l}{k}\in(\frac{1}{4},\frac{1}{2})$ and every
$\tau\in\mathbb{R}_{>0}$, the Riemann surface $\Sigma(\tau)$ given by the
$g+1$-fold covering (3.20) or (3.21) (with $g=k-1$ for k odd and $g=k/2-1$ for
$k$ even) of the four punctured sphere $\mathcal{S}_{4}(\tau)$ (as in (3.10))
admits a holomorphic connection on the trivial bundle
$\underline{\mathbb{C}}^{2}\rightarrow\Sigma(\tau)$ with
SL$(2,\mathbb{R})$-monodromy.

######  Remark 5.4.

Restricting to rational weights $r=\tfrac{k-1}{2k}$ we obtain a new proof of
the main theorem of [BDHH], without applying the WKB analysis. In particular,
we obtain holomorphic systems on compact Riemann surfaces with Fuchsian
representations that are not far out in the Betti moduli space of real
representations.

######  Corollary 5.5.

The triple $\nabla^{U},$ $\nabla^{H}$ and $\nabla^{G}$ is ordered in the
character variety, i.e., either

$y(\nabla^{U})<y(\nabla^{H})<y(\nabla^{G})\quad\text{ or }\quad
y(\nabla^{U})>y(\nabla^{H})>y(\nabla^{G}).$

###### Proof.

From (3.22), each connected component of $\mathcal{M}_{\mathbb{R}}^{r}$ admits
$y=\operatorname{tr}(Y)$ as a global coordinate, where $Y$ is defined as the
monodromy along $\gamma_{y}$ (see (3.6)). Moreover, the image of $y$ is either
$(2,\infty)$ or $(-\infty,-2).$

Hence, for any choice of spin structure on the torus the map

$q\in\mathbb{R}\rightarrow y(\nabla^{q})\in\mathbb{R}$

is a diffeomorphism onto the corresponding connected component given by
$y\in(2,\infty)$ or $y\in(-\infty,-2).$ Therefore, the map $y(\nabla^{q})$ is
strictly monotonic in $q\in\mathbb{R}$ and the Corollary follows from $q_{H}$
lying between $q=0$ and $q=q_{G}.$ ∎

######  Remark 5.6.

With the above notations, since $q\in\mathbb{R}$ (by Hitchin-Kobayashi
correspondence) and $y\in(2,\infty)$ (the trace) are both global coordinates
on every connected component of $\mathcal{M}^{r}_{\mathbb{R}},$ the space of
$\eta$-invariant conical hyperbolic structures, we have that $y(\nabla^{q})$
is either strictly monotonically increasing or decreasing. Moreover, the
conformal type for grafting $n$-times along $\gamma_{x}$ and $n$-times along
$\gamma_{y}$ degenerate for $n\rightarrow\infty$ to the two different ends of
$\mathcal{M}^{r}_{\mathbb{R}}$. Therefore, either

$y(\nabla^{q_{G,x}})<y(\nabla^{q=0})<y(\nabla^{q_{G,y}})\quad\text{ or }\quad
y(\nabla^{q_{G,x}})>y(\nabla^{q=0})>y(\nabla^{q_{G,y}}).$

Since the $y$-coordinate of connection $\nabla^{q_{H}}$ lies between
$y(\nabla^{q_{G,x}})$ and $y(\nabla^{q=0})$, or respectively,
$y(\nabla^{q_{H}})$ lies between $y(\nabla^{q_{G,y}})$ and $y(\nabla^{q=0})$,
we reverse the ordering in Corollary 5.5 by grafting along $\gamma_{x}$
instead of $\gamma_{y}$.

######  Lemma 5.7.

Consider the family of 1-punctured rectangular tori $T^{2}_{\tau}$, for
$\tau\in\mathbb{R}_{>0}$. Then the map defined using the abelianization
coordinates

$\mathfrak{M}\colon\mathbb{R}^{2}\ni(a,\tau)\rightarrow\mathcal{M}_{1,1}^{r,\eta},\quad(a,\tau)\mapsto\rho(\nabla^{a,\chi_{0}}(\tau))$

is a local diffeomorphism for fixed $\chi_{0}=\tfrac{\pi}{4\tau}$ away from
$a_{0}=-\tfrac{\pi}{4\tau}.$

######  Remark 5.8.

An analogue Lemma also holds when choosing $\chi_{0}=i\tfrac{\pi}{4}$ and
$a\in i\mathbb{R}$ away from $a_{0}=i\tfrac{\pi}{4}.$

###### Proof.

Consider the map

$\widetilde{\mathfrak{M}_{\mathbb{C}}}\colon\mathbb{C}^{3}\ni(a,\chi,\tau)\rightarrow\mathcal{M}_{1,1}^{r},\quad(a,\chi,\tau)\mapsto\rho(\nabla^{a,\chi}(\tau))$

from $\mathbb{C}^{3}$ to the complex 2-dimensional space
$\mathcal{M}^{r}_{1,1}.$ By the Riemann-Hilbert correspondence and the fact
that, for fixed conformal type $\tau$, the variables $(a,\chi)$ are
coordinates of $\mathcal{M}_{1,1}^{r}$ (away from half lattice points of
$\chi$), the kernel of the differential is only $1$-dimensional, and can be
computed using the differential of isomonodromic deformations.

To show that $\mathfrak{M}$ is a local diffeomorphism, it suffices to prove
that

$\mathfrak{M}_{\mathbb{C}}\colon\mathbb{C}^{2}\ni(a,\tau)\rightarrow\mathcal{M}_{1,1}^{r},\quad(a,\tau)\mapsto\rho(\nabla^{a,\chi_{0}}(\tau))$

is an immersion at all points $(a,\tau)\in\mathbb{C}^{2}$, with
$a\neq-\tfrac{\pi}{4\tau}$ (i.e., without restricting to the $\eta$-invariant
subspaces). Indeed, Lemma 3.10 then implies that the image of
$\mathfrak{M}_{\mathbb{C}}$ restricted to real points
$(a,\tau)\in\mathbb{R}^{2}$ lies in the real 2-dimensional submanifold
$\mathcal{M}_{1,1}^{r,\eta}\subset M_{1,1}^{r}$.

Since $\mathfrak{M}_{\mathbb{C}}$ is obtained from
$\widetilde{\mathfrak{M}_{\mathbb{C}}}$ by fixing the $\chi$-coordinate, it
suffices to show the kernel of $d\widetilde{\mathfrak{M}_{\mathbb{C}}}$ is
transversal to the slice $\chi=\chi_{0}.$ The proof uses a result by Loray
[Lo]. Be aware that the parameter $\tau$ used here is the conformal type of
the torus $T^{2}_{\tau},$ while the parameter $t\in\mathbb{C}P^{1}$ in [Lo] is
the conformal type of the $4$-punctured sphere with punctures
$\\{0,1,\infty,t\\}$. The transformation from $\tau$ to $t\in\mathbb{C}P^{1}$
is a local diffeomorphism since the conformal type of the torus is non-
degenerated.

First note that, for every fixed $\tau$, the parameters
$\chi_{0}(\tau)=\tfrac{\pi}{4\tau}$ and $a_{0}(\tau)=-\tfrac{\pi}{4\tau}$
define a reducible connection on the 4-punctured torus
$\Sigma_{2}(\tau)=\mathbb{C}/(2\mathbb{Z}\oplus 2\tau i\mathbb{Z})$
corresponding to a specific fixed irreducible representation of the
1-punctured torus $T^{2}_{\tau}$. This reducible connection is the tensor
product of the pull-back of $D$ from the 4-punctured sphere to the four-
punctured torus with a $\mathbb{Z}_{2}$ line bundle connection. Therefore,
when varying $\tau$, these reducible connections define an isomonodromic
deformation.

For isomonodromic deformations, the bundle type $\widetilde{q}(t)$ (see[Lo,
Corollary 10 and Section 5.8.4]), determined by $\pi(\chi)$ (referred to as
the Tu-invariant in [Lo]) solves the Painlevé VI equation in $t$ away from
four values of $\widetilde{q}(t)$ 666The Painlevé VI equation is a second
order ODE, and the four exceptional points corresponds to $\chi$ being a half
lattice point of Jac$(T^{2}_{\tau})$. Therefore the flat connections (as a
family in $t$) are determined by the initial value $\widetilde{q}(t_{0})$ and
the initial direction $\tfrac{d}{dt}|_{t=t_{0}}\widetilde{q}$ (provided by
$p$; see [Lo, Theorem 8]).

Hence the $t$-family of connections corresponding to
$\chi_{0}(\tau)=\tfrac{\pi}{4\tau}$ and $a_{0}(\tau)=-\tfrac{\pi}{4\tau}$
determines a solution $\widetilde{q}_{H}(t)$ of the second order Painlevé VI
equation. Note that though this family of connections are reducible on the
4-punctured torus, they are irreducible on the 1-punctured $T^{2}_{\tau}$ on
which the Theorem of Loray holds. Let $\widetilde{q}(t)$ be another solution
for a different isomonodromic deformation at $t_{0}$ (where $t_{0}$
corresponds to $\tau_{0}$) with initial value
$\widetilde{q}(t_{0})\,=\,\widetilde{q}_{H}(t_{0}).$ Then
$\widetilde{Q}=\widetilde{q}-\widetilde{q}_{H}$ must have a simple zero at
$t_{0},$ i.e., $\tfrac{d}{dt}|_{t=t_{0}}\widetilde{Q}\neq 0,$ since the
connections are determined by $\widetilde{q}(t_{0})$ and the initial direction
$\tfrac{d}{dt}|_{t=t_{0}}\widetilde{q}$ [Lo, Theorem 8].

Since $\widetilde{q}\,=\,\pi(\chi)$, we have that $\chi-\chi_{0}$
corresponding to the isomonodromic deformation obtained by $\widetilde{q}$ has
simple zero at $\tau_{0}$, implying transversality away from
$a=-\tfrac{\pi}{4\tau}$. ∎

###  5.1. Proof of the main Theorem

Let $\rho$ be the given RSR-representation. It defines a point
$\widetilde{\rho}$ (in fact there are 4 preimages) in the character variety of
$\eta$-symmetric and real representations $\mathcal{M}_{\mathbb{R}}^{r}$ on
the 1-punctured torus. The aim is to show that there exist a
$\tau\in\mathbb{R}_{>0}$ such that $\widetilde{\rho}$ can be realized as the
monodromy representation of a logarithmic connection $\nabla^{H}(\tau)$ on the
parabolic bundle over $T^{2}_{\tau}$ determined by $H,$ see Definition 3.21,
corresponding to the trivial holomorphic bundle on the covering $\Sigma$.

Recall that the trace $y=\operatorname{tr}(Y)$ of the monodromy $Y$ along
$\gamma_{y}$ is a global coordinate on each connected component of
$\mathcal{M}_{\mathbb{R}}^{r}$. Consider the coordinate $y(\widetilde{\rho})$
of our given element. Without loss of generality we restrict in the following
to the connected component with $y(\rho)>2$ and $x(\rho)>2$.

Choose a representation $\rho_{0}\in\mathcal{M}_{\mathbb{R}}^{r}$ with
$y(\rho_{0})<y(\widetilde{\rho})$. By Section 4.5, there is a (rectangular)
conformal type $\tau^{uG}_{0}>0$ of the 1-punctured torus such that the
orbifold uniformization connection $\nabla^{U}(\tau^{uG}_{0})$ has monodromy
representation $\rho_{0}.$ Grafting $\nabla^{U}(\tau^{uG}_{0})$ once along
$\gamma_{y}$ yields a new projective structure given by a connection
$\nabla^{G}(\tau_{0}),$ where $\tau_{0}>0$ is the rectangular conformal type
obtained from grafting $T^{2}_{\tau^{uG}_{0}}$.

Assume further without loss of generality that we have
$y(\nabla^{U}(\tau_{0}))<y(\nabla^{G}(\tau_{0}))$ at $\tau_{0}$. If this
assumption does not hold for simple-grafting along $\gamma_{y}$, it holds for
simple-grafting along $\gamma_{x}$, and the remainder of the proof works with
the same arguments, see Remark 5.6. Then there exists a
$\nabla^{H}(\tau_{0})=\nabla^{q_{H}}(\tau_{0}),$ see Lemma 5.1, on the
parabolic bundle determined by $H\in$ Jac$(T^{2}_{\tau_{0}})$ with

$y(\nabla^{U}(\tau_{0}))\,<\,y(\nabla^{H}(\tau_{0}))\;<\,y(\nabla^{G}(\tau_{0}))\,=\,y(\rho_{0})\,<\,y(\widetilde{\rho}).$
(5.1)

Since $y$ is a global coordinate on each connected component of
$\mathcal{M}_{\mathbb{R}}^{r},$ the above inequality remains true for
continuous deformations of the connections induced by a continuous deformation
of the conformal type $\tau$. Since $\nabla^{U}(\tau)$ and $\nabla^{G}(\tau)$
depends continuously on $\tau$ by Lemma 4.7 and Lemma 4.8, the inequality
holds for a deformation in $t\mapsto\tau(t)$, if there exist a corresponding
continuous deformation

$t\mapsto\nabla^{H}(a(t),\tau(t))$

with prescribed underlying parabolic structure $H(\tau(t))$, see the
definition of $\mathfrak{M}$ in Lemma 5.7. Locally such a deformation of
$\nabla^{H}(\tau_{0})$ exists, since $\mathcal{M}^{r}_{\mathbb{R}}$ is a real
1-dimensional submanifold of $\mathcal{M}_{1,1}^{r,\eta},$ on which $(a,\tau)$
are local coordinates by Lemma 5.7.

In the following, we use

$t=y$

as deformation parameter, i.e., the family $t\mapsto\nabla^{H}(a(t),\tau(t))$
of logarithmic connections on the prescribed parabolic bundle H over the
1-punctured torus given by $\tau(t)$ satisfies

$y(\nabla^{H}(a(t),\tau(t)))=t.$

We call such a continuous (and therefore smooth) family admissible if
additionally the following condition hold

$\tau(t_{0})=\tau_{0}\quad\text{and}\quad\nabla^{H}(a(t_{0}),\tau(t_{0}))=\nabla^{H}(\tau_{0}),$

for $t_{0}=y_{0}:=y(\nabla^{H}(\tau_{0})).$ The aim is to show that

$y_{max}:=\text{sup}\left\\{y\in\mathbb{R}_{\geq y_{0}}\mid\exists\text{ an
admissible family }t\mapsto\nabla^{H}(a(t),\tau(t))\text{ with
}t\in[y_{0},y)\right\\}$

satisfies

$y_{max}=\infty>y(\widetilde{\rho}).$

By construction we therefore have

$y(\nabla^{U}(\tau(t)))<y(\nabla^{H}(t,\tau(t)))\;<y(\nabla^{G}(\tau(t))).$
(5.2)

for all $t\in[y_{0},y_{max}).$ Let us assume that $y_{max}<\infty.$ Since
$\mathcal{M}_{\mathbb{R}}^{r}$ satisfies (3.22) we obtain
$x_{max}\in(2,\infty)$ is finite as well. Moreover, the corresponding
conformal type

$\tau_{sup}:=\limsup_{t\to y_{max},t<y_{max}}\tau(t)\in\mathbb{R}^{>0}<\infty$

is finite by (5.2) using Lemma 4.8).

Next, by definition of $\nabla^{H}(a(t),\tau(t))$ together with the fact that
$(a,\chi)$ are coordinates on the moduli space, there is either a real or
purely imaginary function $a(t),$ see Lemma 3.10, such that

$\nabla^{H}(t,\tau(t))=\nabla^{a(t),H,r}$

on the 1-punctured torus $T^{2}_{\tau(t)}$ is real and $\eta$-invariant.
Assume that

$a_{sup}:=\sup_{t\to y_{max},t<y_{max}}|a(t)|=\infty.$

Using the WKB analysis of Mochizuki in [BDHH, Appendix] along $\gamma_{x}$ if
$a(t)$ is real, or along $\gamma_{y}$ if $a(t)$ is purely imaginary, with
respect to the diagonal Higgs field

$\Phi_{t}=a(t)\begin{pmatrix}dw&0\\\ 0&-dw\end{pmatrix}$

we obtain that, up to taking a suitable subsequence for which the conformal
type converges to $\tau_{sup}$, either

$x(\nabla^{H}(a(t),\tau(t)))\rightarrow\infty\quad\text{ or }\quad
y(\nabla^{H}(a(t),\tau(t)))\rightarrow\infty$

which is a contradiction. Therefore $y_{max}=\infty$ concluding the proof. ∎

##  6\. The dodecahedral example

It is well-know that there exists exactly 4 compact, regular, and space-
filling tessellations of the hyperbolic 3-space. They were first described by
Coxeter in [Co1, Co2]. Here we are interested in the order-4 dodecahedral
honeycomb with Schlaefli symbol $\\{5,3,4\\}$. The fundamental domain of this
tessellation is the regular dodecahedron with dihedral angle $\pi/2$. There
are four dodecahedra around each edge and eight dodecahedra around each vertex
in an octahedral arrangement. Note that, in contrary to the case of dihedral
angle $2\pi/5$ which leads to the construction of a Seifert-Weber hyperbolic
3-manifold, the cocompact discrete subgroup of $\text{PSL}(2,\mathbb{C})$,
constructed by identification of opposite pairs of faces of the regular
dodecahedron with dihedral angle $\pi/2$, admits nontrivial torsion. Though
the $\pi/2$-dodecahedron is not a fundamental domain for any discrete torsion-
free subgroup in $\text{PSL}(2,\mathbb{C})$, there are discrete groups with
torsion in $\text{PSL}(2,\mathbb{C})$ which admit the $\pi/2$-dodecahedron as
its fundamental domain. Groups containing torsion-free subgroups of finite
(small) index can be determined using the Reidemeister-Schreier method [Be].

A barycentric subdivision of the regular dodecahedron cuts it into $120$
copies of its characteristic cell. In our case where the dihedral angle is
$\pi/2$, the characteristic cell is the tetrahedron $\mathbb{T}$ with dihedral
angles

$\tfrac{\pi}{2},\tfrac{\pi}{2},\tfrac{\pi}{2},\tfrac{\pi}{5},\tfrac{\pi}{3},\tfrac{\pi}{4}.$

Figure 2. The fundamental $(5,3,4)$ tetrahedron.

The aim is to use $\mathbb{T}$ to explicitly writing down a RSR-representation
$\rho$ that is compatible with the cocompact lattice given by the dodecahedral
tiling of $\mathbb{H}^{3}.$ For computational convenience, we use the
hyperboloid model of the hyperbolic 3-space. To fix notations, consider the
Lorentzian space

$\mathbb{R}^{3,1}=\left\\{h=\begin{pmatrix}x_{0}+x_{1}&x_{2}+i\,x_{3}\\\
x_{2}-i\,x_{3}&x_{0}-x_{1}\end{pmatrix}\,\mid\,\,\overline{h}^{T}\,=\,h\right\\}$

equipped with its canonical indefinite inner product $(.,.)$ of signature
$(1,3).$ The quadratic form of $(.,.)$ is the negative of the determinant.

Then the hyperbolic 3-space is given by

$\begin{split}\mathbb{H}^{3}&=\\{(x_{0},x_{1},x_{2},x_{3})\mid-
x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-1,\;x_{0}>0\\}\\\
&=\left\\{h=\begin{pmatrix}x_{0}+x_{1}&x_{2}+i\,x_{3}\\\
x_{2}-i\,x_{3}&x_{0}-x_{1}\end{pmatrix}\mid\;\text{det}(h)=1;\,\overline{h}^{T}=h;\,\text{Tr}(h)>0\right\\}\end{split}$
(6.1)

endowed with the Riemannian metric induced by the quadratic form $q.$ The
submanifold

$\mathbb{H}^{2}\,=\\{(x_{0},x_{1},x_{2},0)\mid-
x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-1,\;x_{0}>0\\}$

endowed with the induced Riemannian metric is totally geodesic and a copy of
the hyperbolic 2-plane.

Recall that there exist a natural double covering of the connected component
of the identity of the isometry group of $\mathbb{H}^{3}$ by
SL$(2,\mathbb{C})$

$\mathrm{SL}(2,\mathbb{C})\to\mathrm{SO}(3,1)\subset\mathrm{Isom}(\mathbb{H}^{3})$

given by the right action

$(h,g)\in\mathbb{H}^{3}\times\mathrm{SL}(2,\mathbb{C})\longmapsto h\cdot
g=\bar{g}^{T}hg\in\mathbb{H}^{3}.$

Restricting this map to the real subgroup
$\mathrm{SL}(2,\mathbb{R})\subset\mathrm{SL}(2,\mathbb{C})$ preserves
$\mathbb{H}^{2}\subset\mathbb{H}^{3}$ and defines a double covering

$\mathrm{SL}(2,\mathbb{R})\to\mathrm{SO}(2,1)\subset\mathrm{Isom}(\mathbb{H}^{2}).$

Consider the hyperbolic tetrahedron in $\mathbb{H}^{3}$ defined by the
following 4 vertices 777We choose $\frac{1}{5^{1/4}}\in\mathbb{R}^{>0}$, and
also $\sqrt{x}>0$ for all $x>0$.

$\begin{split}P_{0}&=(1,0,0,0)\\\
P_{1}&=\left(\sqrt{1+\tfrac{2}{\sqrt{5}}},-\frac{1}{5^{1/4}},-\frac{1}{5^{1/4}},0\right)\\\
P_{2}&=\left(\sqrt{\tfrac{1}{2}(3+\sqrt{5})},-\sqrt{\tfrac{2}{-1+\sqrt{5}}},0,0\right)\\\
P_{3}&=\left(\;\tfrac{1}{2}\sqrt{7+3\sqrt{5}},-\tfrac{1}{2}\sqrt{1+\sqrt{5}},-\tfrac{1}{2}\sqrt{1+\sqrt{5}},\tfrac{1}{2}\sqrt{1+\sqrt{5}}\;\right)\end{split}$

with (geodesic triangle) faces

$\begin{split}&T_{0}\subset\text{span}\\{P_{1},P_{2},P_{3}\\}\cap\mathbb{H}^{3}\\\
&T_{1}\subset\text{span}\\{P_{0},P_{2},P_{3}\\}\cap\mathbb{H}^{3}\\\
&T_{2}\subset\text{span}\\{P_{0},P_{1},P_{3}\\}\cap\mathbb{H}^{3}\\\
&T_{3}\subset\text{span}\\{P_{0},P_{1},P_{2}\\}\cap\mathbb{H}^{3}\subset\mathbb{H}^{2}.\\\
\end{split}$

Each face has a unit length normal, unique up to sign, given by

$\begin{split}L_{0}&=\left(\,\tfrac{1}{\sqrt{1+\sqrt{5}}},-\tfrac{1}{2}\sqrt{3+\sqrt{5}},0,0\,\right)\\\
L_{1}&=\left(\,0,0,\tfrac{1}{\sqrt{2}},\tfrac{1}{\sqrt{2}}\,\right)\\\
L_{2}&=\left(\,0,-\tfrac{1}{\sqrt{2}},\tfrac{1}{\sqrt{2}},0\,\right)\\\
L_{3}&=(\,0,0,0,1\,).\\\ \end{split}$

Computing the angles between $L_{k},$ $k=0,\,...,\,3$ the tetrahedron
constructed here is in fact $\mathbb{T}$, see Figure 2. The corresponding
Coexter group is generated by the reflections $R_{k}$ across the faces $T_{k}$
for $k\,=\,0,...,\,3,$ i.e.,

$R_{k}(v)=v-2(v,L_{k})L_{k},\quad v\in\mathbb{R}^{3,1}.$

Consider its order 2 subgroup $\mathfrak{G}$ consisting of orientation
preserving transformations generated by

$G_{m,n}:=R_{m}R_{n},\quad 0\leq m<n\leq 3.$

We denote by $\Gamma$ the subgroup of SL$(2,\mathbb{C})$ given by the preimage
of $\mathfrak{G}$ through the above double cover. Its generators are
determined by choosing lifts $g_{k,l}$ of $G_{k,l}$

$\begin{split}g_{0,1}&:=\begin{pmatrix}0&-\tfrac{1+i}{4}\left(1+\sqrt{5}+\sqrt{2(-1+\sqrt{5})}\right)\\\
\frac{(2-2i)}{(1+\sqrt{5}+\sqrt{2(-1+\sqrt{5})}}&0\end{pmatrix}\\\ &\\\
g_{0,2}&:=\begin{pmatrix}-\tfrac{1}{2}\sqrt{1+\sqrt{5}-\sqrt{2(1+\sqrt{5})}}&\left(-\sqrt{1+\sqrt{5}-\sqrt{2(1+\sqrt{5})}}\right)^{-1}\\\
\tfrac{1}{2}\sqrt{1+\sqrt{5}-\sqrt{2(1+\sqrt{5})}}&\left(-\sqrt{1+\sqrt{5}-\sqrt{2(1+\sqrt{5})}}\right)^{-1}\end{pmatrix}\\\
&\\\
g_{0,3}&:=\begin{pmatrix}0&-i\sqrt{\tfrac{1}{2}(1+\sqrt{5}+\sqrt{2(1+\sqrt{5})})}\\\
\frac{-i}{\sqrt{\tfrac{1}{2}(1+\sqrt{5}+\sqrt{2(1+\sqrt{5})})}}&0\end{pmatrix}\\\
&\\\ g_{1,2}&:=\frac{1}{2}\begin{pmatrix}-1+i&1+i\\\ -1+i&-1-i\end{pmatrix}\\\
&\\\ g_{1,3}&:=\frac{1}{\sqrt{2}}\begin{pmatrix}-1-i&0\\\
0&-1+i\end{pmatrix}\\\ &\\\ g_{2,3}&:=\frac{-i}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}.\\\ \end{split}$ (6.2)

Note that all $g_{k,l}$ are of finite order, e.g., $g_{0,2}$ is of order $5$.
Consider

$j_{0}:=g_{1,2}g_{1,3}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&-1\\\
1&1\end{pmatrix}\in\text{SL}(2,\mathbb{R})$

and

$J_{1}:=-(g_{0,2})^{2}\in\text{SL}(2,\mathbb{R}),$

and define

$\begin{split}J_{2}&:=j_{0}J_{1}j_{0}^{-1}\\\ J_{3}&:=j_{0}J_{2}j_{0}^{-1}\\\
J_{4}&:=j_{0}J_{3}j_{0}^{-1}.\\\ \end{split}$ (6.3)

Note that $j_{0}$ is of order 8 and $J_{1},\dots,J_{4}$ are all of order 10. A
direct computation then shows

$J_{4}J_{3}J_{2}J_{1}=\text{Id}.$

######  Theorem 6.1.

The representation
$\rho\,\colon\,\pi_{1}(S_{4},\,s_{0})\,\longrightarrow\,\text{SL}(2,\mathbb{C})$
of the 4-punctured sphere $S_{4}$ given by

$\rho(\gamma_{k})\,=\,J_{k}$

is a genus $4$ RSR-representation compatible with the cocompact lattice
$\Gamma\subset\mathrm{SL}(2,\mathbb{C}).$

###### Proof.

The group $\Gamma$ (the preimage of $\mathfrak{G}$) is a cocompact lattice of
$\mathrm{SL}(2,\mathbb{C})$, since it corresponds (up to a reflection) to the
tessellation of $\mathbb{H}^{3}$ by copies of the tetrahedron $\mathbb{T}$.
Since $J_{k}$ lies in SL$(2,\mathbb{R})$ for all $k\,=\,1,\,\cdots,\,4$, and

$\text{tr}(J_{2}J_{1})=\text{tr}(J_{3}J_{2})=-1-\sqrt{5}<-2,\quad\text{tr}(J_{3}J_{1})=-\tfrac{3}{2}(1+\sqrt{5})<-2,$

the image of the representation $\rho$ defines a real lattice
$\widehat{\Gamma}$ in $\mathrm{SL}(2,\mathbb{R})$. Moreover,
$\widehat{\Gamma}$ is symmetric with
$\widetilde{r}=\tfrac{3}{10}\in(\tfrac{1}{4},\tfrac{1}{2})$, since

$\text{tr}(J_{k})=\frac{1}{2}(1-\sqrt{5})=2\cos\left(2\pi\tfrac{3}{10}\right)\quad\text{for}\quad
k=1,...,4,$

and rectangular by

$\text{tr}(J_{3}J_{1})\,=\,\text{tr}(J_{4}J_{2})\,=\,-\tfrac{3}{2}(1+\sqrt{5}).$

Since the order of the $J_{j}$ is 10, the genus of the representation is $4$
by Definition 2.2. ∎

Applying Theorem 2.4 we obtain:

######  Corollary 6.2.

There exist a cocompact lattice $\Gamma$ in $\mathrm{SL}(2,\mathbb{C})$ and a
compact Riemann surface $\Sigma$ of genus $g=4$ admitting a holomorphic map
$f\colon\Sigma\to\mathrm{SL}(2,\mathbb{C})/\Gamma$ which does not factor
through a curve of lower genus.

###  6.1. Outlook: CMC 1 surfaces in hyperbolic 3-manifolds

Surfaces with constant mean curvature $H$ in Riemannian 3-manifolds are the
critical points of the area functional with fixed enclosed volume. By the
Lawson correspondence [La], CMC 1 surfaces (i.e., surfaces with $H=1$) in
hyperbolic 3-space are (locally) in one-to-one correspondence with minimal
surfaces in $\mathbb{R}^{3}.$ The latter can be explicitly parametrized in
terms of holomorphic data by the classical Weierstrass representation.

Figure 3. An equivariant CMC 1 surface compatible with the dodecahedron
tesselation of $\mathbb{H}^{3}$. The shown lines are the trajectories of a
holomorphic quadratic differential of the underlying compact Riemann surface
of genus 3. Image by Nick Schmitt.

As first noticed by Bryant [Bry], see also [UY], there exists a local
Weierstrass representation of CMC 1 surfaces in $\mathbb{H}^{3}$ as well.
Consider for this a nilpotent nowhere vanishing
$\mathfrak{sl}(2,\mathbb{C})$-valued holomorphic 1-form $\Phi$ on a Riemann
surface $\Sigma.$ The frame $F$ is a solution of the ordinary differential
equation $dF=\Phi F$. Then the conformal immersion $f$ (defined on the
universal covering of $\Sigma$) given by $f=\bar{F}^{T}F$ has constant mean
curvature $1$ in the hyperbolic 3-space. In invariant terms, every CMC 1
surface is given by a flat unitary connection $\nabla$ together with a
nilpotent Higgs field $\Phi,$ see [Pi]. The frame $F$ is then a parallel frame
with respect to the flat connection $\nabla-\Phi.$ The unitary connection
$\nabla$ gives rise to an associated flat unitary bundle $V\rightarrow\Sigma$
whose transition functions are determined by the local unitary frames of
$\nabla.$ Since the representation formula is not sensitive with respect to
appropriate changes of the parallel unitary frames, the choice of the unitary
frame does not alter the CMC 1 immersion. We say a CMC 1 surface in a
hyperbolic manifold admits a simple Weierstrass representation, if the
corresponding flat unitary bundle is trivial. The following theorem holds.

######  Theorem 6.3.

A CMC 1 surface $f\colon\Sigma\to\mathbb{H}^{3}/\Gamma$ into a hyperbolic
3-manifold $\mathbb{H}^{3}/\Gamma$ admits a simple Weierstrass representation
if and only if it is the projection of a holomorphic curve
$F\colon\Sigma\to\mathrm{SL}(2,\mathbb{C})/\Gamma.$

Even though we have shown the existence of holomorphic curves into
$\mathrm{SL}(2,\mathbb{C})/\Gamma$ where $\Gamma$ is the dodecahedron
tesselation group, and there exists compact CMC 1 surfaces in
$\mathbb{H}^{3}/\Gamma$, see Figure 3, it is unclear whether there exist
compact CMC 1 surfaces in $\mathbb{H}^{3}/\Gamma$ with simple Weierstrass
representation, since the Higgs fields of the holomorphic curves we
constructed here are not nilpotent.

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