# Isoperiodic foliation of the stratum $\mathcal{H}(1,1,-2)$

Gianluca Faraco  Dipartimento di Matematica e Applicazioni U5, Universita‘
degli Studi di Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy
<EMAIL_ADDRESS>,  Guillaume Tahar  Beijing Institute of
Mathematical Sciences and Applications, Huairou District, Beijing, China
<EMAIL_ADDRESS>and  Yongquan Zhang  Institute for Mathematical
Sciences, Stony Brook University, 100 Nicolls Rd, Stony Brook, NY 11794, USA
<EMAIL_ADDRESS>
###### Abstract.

On a Riemann surface, periods of a meromorphic differential along closed loops
define a period character from the absolute homology group into the additive
group of complex numbers. Fixing the period character in strata of meromorphic
differentials defines the isoperiodic foliation where the remaining degrees of
freedom are the relative periods between the zeroes of the differential. In
strata of meromorphic differentials with exactly two zeroes, leaves have a
natural structure of translation surface. In this paper, we give a complete
description of the isoperiodic leaves in marked stratum $\mathcal{H}(1,1,-2)$
of meromorphic $1$-forms with two simple zeroes and a pole of order two on an
elliptic curve. For each character, the corresponding leaf is a connected Loch
Ness Monster. The translation structures of generic leaves feature a ramified
cover of infinite degree over the flat torus defined by the lattice of
absolute periods. By comparison, isoperiodic leaves of the unmarked stratum
are complex disks endowed with a half-translation structure having infinitely
many singular points. Finally, we give a description of the large-scale
conformal geometry of the wall-and-chamber decomposition of the leaves.

###### Key words and phrases:

Translation surface, Elliptic curve, Isoperiodic foliation, Meromorphic
differential, Period character

###### Contents

  1. 1 Introduction
  2. 2 Geometry of translation structures
  3. 3 Period coordinates and isoperiodic foliation
  4. 4 Chambers of cylinder type
  5. 5 Positive leaves
  6. 6 Negative leaves
  7. 7 Non-arithmetic real leaves
  8. 8 Arithmetic real leaves
  9. 9 Conformal geometry of leaves
  10. A Equilateral triangulation, true trees and conformal structure of arithmetic leaves

##  1\. Introduction

Strata of meromorphic $1$-forms with prescribed orders of zeroes and poles are
endowed with the so-called period atlas, see [1, Theorem 2.1] for a a general
construction. Local coordinates of the period atlas are:

  * •

integrals of the differential along closed loops on the surface punctured at
the poles (absolute periods);

  * •

integrals of the differential along paths joining distinct zeroes (relative
periods).

Fixing the absolute periods defines in each stratum the isoperiodic foliation.
In the present paper, we give a complete description of isoperiodic leaves for
stratum $\mathcal{H}(1,1,-2)$ formed by pairs $(X,\omega)$ where $X$ is an
elliptic curve while $\omega$ is a meromorphic $1$-form with two marked simple
zeroes and one double pole.

###  1.1. Main results

Absolute periods are encompassed into the period character. We begin with the
following definition.

######  Definition 1.1.

Any pair $(X,\omega)\in\mathcal{H}(1,1,-2)$ defines a homomorphism $\chi\colon
H_{1}(X^{\ast},\mathbb{Z})\rightarrow\mathbb{C}$ where $X^{\ast}$ is Riemann
surface $X$ punctured at the unique pole of $\omega$. Period character $\chi$
is defined by

(1.1) $\gamma\mapsto\int\limits_{\gamma}\omega.$

We have the identification $H_{1}(X^{\ast},\mathbb{Z})\simeq\mathbb{Z}^{2}$
and changes of basis preserving the standard symplectic structure are
identified with $\textnormal{SL}(2,\mathbb{Z})$. The absolute period group
$\Gamma$ is the image of character $\chi$. It is a subgroup of $\mathbb{C}$
with at most two generators. The first result of the paper is that each
nonzero period character is realized by exactly one (connected) leaf of the
isoperiodic foliation.

######  Theorem 1.2.

In the marked stratum $\mathcal{H}(1,1,-2)$, every non-trivial period
character $\chi$ is realized, up to a symplectic change of basis, by a unique
isoperiodic leaf.

With trivial period character we mean the representation that associates every
element $\gamma\in H_{1}(X^{*},\,\mathbb{C})$ to $0$. Conversely, [9, Theorem
B] states that the trivial period character cannot be realized in stratum
$\mathcal{H}(1,1,-2)$.

For this stratum, isoperiodic leaves are complex curves. The relative period
between the two simple zeroes defines a system of local coordinates that
endows the leaf with a structure of translation surface, see Section 3. Thus,
we can also investigate the geometry of isoperiodic leaves.

######  Theorem 1.3.

In the marked stratum $\mathcal{H}(1,1,-2)$, (the metric completion of) each
isoperiodic leaf is topologically a Loch Ness Monster. Its translation
structure is determined by an Abelian differential with infinitely many double
zeroes, that is conical singularities of angle $6\pi$. Moreover, the leaf has
a natural involution, whose quotient is a half translation surface. This
surface is the corresponding leaf in the unmarked stratum. It is a complex
disk endowed with a quadratic differential having infinitely many simple
zeroes, that is conical singularities of angle $3\pi$. In the case of a
positive leaf or an arithmetic real leaf, the quadratic differential also has
a simple pole, namely a conical singularity of angle $\pi$.

In fact, for a generic unmarked leaf (when $\Gamma$ is a lattice in
$\mathbb{C}$), we construct an explicit biholomorphic map to the Teichmüller
space $\mathcal{T}_{1,1}$, which is conformally a disk. Given a translation
surface $L$ (possibly of infinite type), a homeomorphism $f:L\to L$ is called
_affine_ if it is given by an affine map in each local coordinate defining the
translation structure. The derivative of an affine homeomorphism is a well-
defined element in $\textnormal{GL}^{+}(2,\mathbb{R})$, and the set of all
such elements is the _Veech group_ of $L$. See §2 for an equivalent
definition. We also determine the Veech groups of isoperiodic leaves for
stratum $\mathcal{H}(1,1,-2)$, which are surprisingly large.

######  Theorem 1.4.

In the marked stratum $\mathcal{H}(1,1,-2)$, the Veech group of each
isoperiodic leaf is a conjugate of:

  * •

$\textnormal{SL}(2,\mathbb{Z})$ if $\Gamma$ is not contained in a real line;

  * •

$\left\\{\pm\begin{pmatrix}u&b\\\ 0&a\end{pmatrix}:a,b\in\mathbb{R},a>0,u\in
G\right\\}$ if $\Gamma$ is proportional to a non-discrete subgroup
$\Gamma^{\prime}$ of a real quadratic field $\mathbb{Q}(\sqrt{D})$ so that
$G=\\{u>0:u\Gamma^{\prime}=\Gamma^{\prime}\\}$ is nontrivial;

  * •

$\left\\{\pm\begin{pmatrix}1&b\\\
0&a\end{pmatrix}:a,b\in\mathbb{R},a>0\right\\}$ otherwise.

Each leaf is equipped with a wall-and-chamber decomposition, according to the
combinatorics of the translation surfaces it contains (see §3). The most
prevalent chambers are of _cylinder_ type (see §4). We show that each chamber
of cylinder type is conformally a half plane. In a generic unmarked leaf, the
chambers of cylinder type are parametrized by primitive elements in the period
group $\Gamma$, and via the biholomorphic map to $\mathcal{T}_{1,1}$ and the
identification $\mathcal{T}_{1,1}\cong\mathbb{H}$, we can draw these chambers
as subsets of the upper half plane $\mathbb{H}$. Recall that
${\mathrm{SL}_{2}(\mathbb{Z})}$ acts on $\mathcal{T}_{1,1}$ as mapping
classes, and correspondingly on the unit disk as hyperbolic isometries.

######  Theorem 1.5.

Let $\mathcal{L}$ be a generic unmarked leaf with period group $\Gamma$. Under
a suitable conformal identification
$\mathcal{L}\cong\mathcal{T}_{1,1}\cong\mathbb{H}$, the closure of each
cylinder chamber in $\mathbb{H}\cup\mathbb{R}\cup\\{\infty\\}$ is a rational
number or $\infty$, and every element in $\mathbb{Q}\cup\\{\infty\\}$ arises
this way. Moreover, the actions of ${\mathrm{SL}_{2}(\mathbb{Z})}$ on
$\mathcal{L}\cong\mathbb{H}$ as the Veech group and hyperbolic isometries
agree on the boundary $\mathbb{R}\cup\\{\infty\\}$.

Finally, for each cylinder chamber, it does not contain any horodisk based at
the same point at infinity, nor is it contained in the $K$-neighborhood of a
hyperbolic geodesic ray for any $K>0$.

We give more precise information about the conformal geometry of cylinder
chambers in §9. Figure 1 shows a schematic picture of an unmarked positive
leaf, where some cylinder chambers are shown in red.

Figure 1. A schematic picture of an unmarked positive leaf, drawn in the upper
half plane. For the chamber based at $\infty$, the two ends of its boundary
goes to infinity with asymptotics $\asymp\log x$.

###  1.2. Relations with other works

Under a much bigger perspective, our paper can be seen as a follow-up of a
series of papers and works developed along the years. The period map is
defined as the association

(1.2)
$\text{Per}\colon\Omega\mathcal{S}_{g,n}\longrightarrow\text{Hom}\Big{(}\textnormal{H}_{1}(S_{g,n},\mathbb{Z}),\,\mathbb{C}\,\Big{)}$

where $\Omega\mathcal{S}_{g,n}$ denote the moduli space of homologically
marked translation surfaces. For holomorphic differentials on compact Riemann
surfaces (i.e. $n=0$) the image of the period map has been studied by Haupt in
[18] and subsequently rediscovered by Kapovich in [21] by using Ratner’s
theory. In the same spirit of [21], Le Fils has provided in [17] necessary and
sufficient conditions for a representation $\chi$ to be a character in a given
stratum of holomorphic differentials and, around the same time, Bainbridge-
Johnson-Judge-Park in [5] have provided, with an independent and alternative
approach, necessary and sufficient conditions for a representation to be
realized in a connected component of a prescribed stratum. For meromorphic
differentials on compact Riemann surfaces, equivalently holomorphic
differentials on punctured complex curves (i.e. $n\geq 1$), the image of the
period mapping has been recently determined in [12] (a special version in
which all zeros and poles are at the punctures), in [9] for strata of
meromorphic differentials and subsequently extended by [8] to connected
components of strata.

All previous works concern the study of the period map, and the study of the
fibres is the natural follow-up. For a representation $\chi$, the fibre
$\text{Per}^{-1}(\chi)$ is known as an isoperiodic leaf and it comprises all
translation surfaces with the same absolute periods. Isoperiodic foliation has
already been studied in several recent works. In the case of strata of
holomorphic $1$-forms, ergodicity of the foliation has been obtained in [24]
(in low genus), in [7] (for principal strata111strata of holomorphic $1$-forms
where every zero is simple in the general case) and [16] (with independent
methods). This has recently been generalized to most strata in [30]. In [6],
methods of [7] have been extended to meromorphic differentials with a pair of
simple poles, obtaining a classification result on the closure of leaves. A
study of isoperiodic foliation has been carried out in [22] in the specific
case of meromorphic differentials with an absolute periods group contained in
$\mathbb{R}$ (real-normalized differentials) and a unique double pole. Theorem
1.2 of the present paper confirms the counting of leaves realizing a given
group of absolute periods in the case of genus one [22, Theorem 9].

For strata of meromorphic $1$-forms on the Riemann sphere, isoperiodic
foliation coincides with the isoresidual fibration defined by the map sending
a differential to the vector of its residues at the (labelled) poles, see [14]
for details. In contrast with the results of the present paper, isoresidual
fibers in genus zero have compact closures because they are defined by
algebraic equations (residues at the poles are rational functions of the
locations of zeroes and poles).

In comparison, stratum $\mathcal{H}(1,1,-2)$ is the simplest example of
isoperiodic foliation on a stratum of nonzero genus. Even in this easy case,
isoperiodic leaves are of infinite genus and contain essential singularities.
Wild translation surfaces have already been studied in several works, see [19,
25, 29], mostly with motivations coming from the field of dynamical systems.
Isoperiodic leaves provide a family of examples of wild translation surfaces
having an algebro-geometric origin.

###  1.3. Organization of the paper

In Section §2, we give general background on translation surfaces and their
moduli spaces. In particular, we introduce the notion of core and the walls-
and-chambers structure it defines in strata.

In Section §3, we introduce the isoperiodic foliation, give the classification
of leaves and describe the translation structure of leaves: conical
singularities and decomposition into chambers.

Each leaf contains chambers of "cylinder type" where the core of the
parametrized translation surfaces is a cylinder. The geometric description and
classification of these chambers is given in Section §4.

Proofs of Theorems 1.2, 1.3 and 1.4 are split into the next four sections
depending on the type of the isoperiodic leaf as follows:

  * •

Positive leaves, i.e. $\Gamma$ is a lattice of $\mathbb{C}$ and $\chi$ has
positive volume, are treated in Section §5;

  * •

Negative leaves, i.e. $\Gamma$ is a lattice of $\mathbb{C}$ and $\chi$ has
negative volume, are treated in Section §6;

  * •

Non-arithmetic real leaves, i.e. $\Gamma$ is a dense subgroup of a real line
of $\mathbb{C}$, are treated in Section §7;

  * •

Arithmetic real leaves, i.e. $\Gamma=a\mathbb{Z}$ for some
$a\in\mathbb{C}^{\ast}$, are treated in Section §8.

Finally, in §9 and the appendix, we discuss in greater detail the conformal
geometry of the leaves, describing the large scale asymptotics of the cylinder
chmabers in them in terms of the Teichmüller / hyperbolic metric.

#### Acknowledgements.

The authors would like to thank Dawei Chen, Alexandre Eremenko, Curtis
McMullen and Karl Winsor for valuable remarks and discussions. We also thank
Steffen Rhode for giving an illuminating talk on the paper [20], and Don
Marshall for sharing his program which produces true trees. G.F. and Y.Z.
acknowledge the support of the Max Planck Institute for Mathematics, where
part of the work was done. Finally, G.F. was supported by supported by SSP
2026 DFG Priority Program Geometry at Infinity and he is a member of GNSAGA.

##  2\. Geometry of translation structures

We begin by introducing basic concepts of the main objects of interest.

###  2.1. Elliptic curves and strata of differentials

An elliptic curve $E$ is simply a compact Riemann surface of genus one. By
uniformization Theorem, any elliptic curve is biholomorphic to the quotient
space $\mathbb{C}/\Lambda$ where $\Lambda\cong{\mathbb{Z}}^{2}$ is a rank-two
lattice of $\mathbb{C}$. Two lattices, say $\Lambda$ and $\Lambda^{\prime}$,
define the same complex structure if and only if there exists
$\alpha\in\mathbb{C}^{\ast}$ such that $\Lambda^{\prime}=\alpha\Lambda$. Let
$\mathcal{M}_{1,1}$ denote the moduli space of elliptic curves with a marked
point. This space is well-known to be a one-dimensional complex orbifold with
two singularities given by those elliptic curves admitting non-trivial complex
automorphisms.

We define strata $\mathcal{H}(1,1,-2)$ as the moduli space of pairs
$(X,\omega)$ where $X$ is an elliptic curve and $\omega$ is a meromorphic
$1$-form with two simple zeroes. Two pairs $(X,\omega)$ and
$(X^{\prime},\omega^{\prime})$ are identified if there is a biholomorphic map
$f$ between $X$ and $X^{\prime}$ such that $\omega$ coincides with the
pullback $f^{\ast}\omega^{\prime}$. Stratum $\mathcal{H}(2,-2)$ is defined in
the same way for differentials with a double zero and a double pole.

Assuming the marked point is the double pole, a meromorphic $1$-form with two
simple zeroes and one double pole is characterized by the position of its
zeroes – a principal divisor $A$+$B$ on the elliptic curve – and a scaling
factor $\lambda\in\mathbb{C}^{\ast}$. If the two zeroes coincide, we obtain a
differential with a double pole and Abel-Jacobi relations imply that the
double zero is a $2$-torsion point of the elliptic curve.

A second parametrization uses the fact that any meromorphic function with a
unique double pole at the lattice point is of the form $a\wp+b$ where
$a\in\mathbb{C}^{\ast}$ and $b\in\mathbb{C}$. Consequently, any meromorphic
$1$-form of $\overline{\mathcal{H}}=\mathcal{H}(1,1,-2)\cup\mathcal{H}(2,-2)$
is a linear combination of $\wp dz$ and $dz$. Such a meromorphic $1$-form is
clearly characterized by these coefficients.

###  2.2. Translation structures

On a compact Riemann surface $X$ endowed with a meromorphic $1$-form $\omega$,
we denote by $X^{\ast}$ the complement of the zeroes and poles of $\omega$ in
$X$. Local primitives of $\omega$ are locally injective around each point of
$X^{\ast}$, they provide an atlas of $\mathbb{C}$-valued local coordinates.
Since local primitives of the same differential differ by a constant,
transition maps of this atlas are translations of the complex plane. Vice
versa, such an atlas defines an underlying complex structure $X$ and the
pullbacks of the $1$-form $dz$ on $\mathbb{C}$ via local charts globalize to a
holomorphic differential $\omega$ on $X^{\ast}$.

######  Remark 2.1.

For a translation structure $(X,\omega)$ on a surface $S_{g,n}$, local charts
globalize to a holomorphic mapping
$\text{dev}\colon\widetilde{S}_{g,n}\longrightarrow\mathbb{C}$ called the
developing map, where $\widetilde{S}_{g,n}$ is the universal cover of
$S_{g,n}$. The translation structure on $S_{g,n}$ lifts to a translation
structure $(\widetilde{X},\widetilde{\omega})$ and the developing map turns
out to be locally univalent away from the zeroes of $\widetilde{\omega}$. The
developing map, in particular, satisfies an equivariance property with respect
to a representation
$\chi\colon\text{H}_{1}(X,\mathbb{Z})\longrightarrow\mathbb{C}$ called period
character of the translation structure.

A translation structure locally written as $f(z)dz$ induces a flat metric
$|f(z)|dz$ on the surface and biholomorphisms conjugating the meromorphic form
are isometries for this metric.

###  2.3. Local models of singularities

For a translation structure induced by a meromorphic $1$-form, there are
canonical models for neighborhoods of zeroes and poles, see [4, 9, 26] for
details.

In a neighborhood of a zero of order $a$, a local primitive is a cyclic cover
of degree $a+1$ of the disk totally ramified at the center of the disk. In
other words, the flat metric induced by the $1$-form extends to the zero as a
conical singularity of angle $(2a+2)\pi$.

The present work requires only one kind of poles: double poles with a residue
equal to zero. Notice that this is far from being restrictive. In fact, for
the one-punctured torus, any simple essential loop around the (sole) pole in
$X^{*}$ is the commutator of a pair of simple essential closed curves
generating $\textnormal{H}_{1}(X^{*},\mathbb{Z})$ and commutators always have
trivial period. A neighborhood of such a pole is isometric to a neighborhood
of infinity in the flat plane.

###  2.4. Action of $\text{GL}^{+}(2,\mathbb{R})$

Given a translation surface $(X,\omega)$, elements of
$\text{GL}^{+}(2,\mathbb{R})$ acts by composition with coordinate functions
induced by $\omega$. A more geometric way to see this action is the following.
If $(X,\omega)$ is constructed by identifying parallel sides of a collection
of polygons $P_{i}$ in $\mathbb{C}$, then $g\cdot(X,\omega)$ is obtained by
identifying the corresponding sides of the polygons $g(P_{i})$, viewing $g$ as
a linear function on $\mathbb{C}\cong\mathbb{R}^{2}$.

It is easy to see that this induces an action on each stratum, see [31]. It
follows from definition that period character $\chi$ commutes with the action
of $\text{GL}^{+}(2,\mathbb{R})$.

The $\text{GL}^{+}(2,\mathbb{R})$-action as defined above applies to
translation surfaces of inifinite type as well. Recall that the Veech group of
a translation surface is the set of derivatives of all affine homeomorphisms.
Equivalently, $g\in\text{GL}^{+}(2,\mathbb{R})$ lies in the Veech group of
$(X,\omega)$ if and only if $g\cdot(X,\omega)$ is isomorphic (as translation
surfaces) to $(X,\omega)$.

###  2.5. Volume

For a symplectic basis $\alpha,\beta$ of the absolute homology of a torus,
$\mathfrak{Im}(\overline{\chi(\alpha)}\chi(\beta))$ is called the volume, and
denoted by $Vol_{\chi}$, of character $\chi$. It can be checked directly that
it does not depend on the choice of a specific symplectic basis. The geometry
of an isoperiodic leaf crucially depends on the sign of $Vol_{\chi}$
(positive, negative or zero). See [8] for further details about volume.

###  2.6. Saddle connections

Given a translation surface $(X,\omega)$, a _saddle connection_ is a geodesic
segment, in the flat metric induced by $\omega$ connecting two zeroes of
$\omega$, so that no interior points of the segment are singularities. We
allow the case where the two zeroes are the same; in this case, we shall say
the saddle connection is closed. On a translation surface $(X,\omega)$, every
saddle connection represents a class in the relative homology group
$\textnormal{H}_{1}(X\setminus\\{P\\},\\{Z\\},\mathbb{Z})$ where $\\{Z\\}$ and
$\\{P\\}$ are respectively the set of the poles and the set of the zeroes of
the meromorphic $1$-form.

###  2.7. Core

The notion, first introduced in [15] is used systematically in [13, 26, 27] to
study the geometry of translation structures induced by meromorphic
differentials.

######  Definition 2.2.

A subset $C$ of a translation surface $(X,\omega)$ is said to be _convex_ if,
for any two points $x,y\in C$ joined by a geodesic segment $[x,y]$, the entire
geodesic segment $[x,y]$ is contained in $C$. The _core_ of $X$, denoted by
$\mathcal{C}_{(X,\omega)}$, is the convex hull of the zeroes of $\omega$, i.e.
the smallest closed convex set containing all the zeroes.

The following lemma has been proved in [26, Propositions 4.4 and 4.5]. We
refer to this paper for the proof.

######  Lemma 2.3.

Let $(X,\omega)$ be a translation surface such that $\omega$ has at least one
pole. Then $\mathcal{C}_{(X,\omega)}$ is a topological retract of surface $X$
punctured at the poles of $\omega$. Each connected component of
$X\setminus\mathcal{C}_{(X,\omega)}$ is a topological disk containing exactly
one pole called polar domain. The boundary of each polar domain is formed by
finitely many saddle connections. By convexity, each interior angle of a polar
domain is at least $\pi$.

Differentials of stratum $\mathcal{H}(1,1,-2)$ have a simple enough
topological structure to make possible a topological classification of cores,
see [26, Section §4] for general background.

######  Proposition 2.4.

Let $(X,\omega)$ be a translation surface in the stratum
$\mathcal{H}(1,1,-2)$. Then the interior of the core can be:

  1. (i)

a torus with a slit (we call this a _torus type_);

  2. (ii)

a cylinder (_cylinder type_);

  3. (iii)

empty (_degenerate type_).

###### Proof.

We denote by $\beta$ the number of sides of the unique polar domain. Applying
Gauss-Bonnet formula to the boundary of this polygon, we deduce that the total
angle of interior angles of the domain is $(\beta+2)\pi$ – the local model of
the double pole is the end of an infinite plane. The interior of the core is a
translation surface with boundary $\mathcal{I}$. It has been proved in [28,
Lemma 2.2] that any such surface can be decomposed into finitely flat
triangles glued along saddle connections. The total angle of $\mathcal{I}$ is
$(6-\beta)\pi$ because each of the two simple zeroes is a conical singularity
of angle $4\pi$. It follows that $\mathcal{I}$ is formed by exactly
$t=6-\beta$ triangles. We will investigate possible values of $\beta$. First
note that since the residue at the double pole is zero, its polar domain
cannot be bounded by a unique saddle connection, because its period would be
zero. It follows that $\beta\geq 2$. Case $\beta=6$ is the degenerate case. We
need to consider the remaining cases $t=1,2,3,4$.

When $t=4$, we have $\beta=2$. In this case, the polar domain is a copy of the
complex plane with a slit. The two end points of the slit cannot be
identified, for otherwise the cone angle is strictly bigger than $4\pi$. The
boundary of the core consists of two parallel geodesic segments. If we glue
these two segments together, we obtain a flat torus with two marked points. In
other words, $\mathcal{C}_{(X,\omega)}$ is a torus with slit along a geodesic
segment.

When $t=2$, we have $\beta=4$. In this case, the domain of the pole is complex
plane minus a convex quadrilateral $Q$. The interior of the core is either a
quadrilateral, whose four sides connects to the domain of the pole, or a
cylinder. The former cannot happen, since otherwise the surface is just a
complex plane with four marked points. In the latter case, $Q$ is a
parallelogram. Two opposite sides of $\mathbb{C}\setminus Q$ are identified,
and the other two sides are connected to the ends of the cylinder.

The cases $t=1,3$ cannot happen. Indeed, when $t=1$, we have $\beta=5$. The
polar domain has five sides $A_{1},A_{2},A_{3},A_{4},A_{5}$; two being
identified in the surface. These two identified sides cannot be adjacent
(otherwise one of the two singularities would be a marked point). Without loss
of generality, we assume they are $A_{1}$ and $A_{3}$. Thus, the sum of the
angles corresponding to corners $A_{1}A_{2}$ and $A_{2}A_{3}$ is $3\pi$. These
corners contribute to the same conical singularity $Z_{1}$. Since the two
conical singularities have an angle of $4\pi$, no other corner can contribute
to $Z_{1}$. If the three last corners contribute to the other singularity
$Z_{2}$, then the three vertices of the triangle forming the interior of the
core should contribute to $Z_{2}$. Thus, the two ends of side $A_{2}$ should
contribute to $Z_{2}$. This is a contradiction.

Similarly, when $t=3$, we have $\beta=3$. So the domain of the pole is complex
plane minus a triangle. The interior of the core is a pentagon, two sides of
which are identified. The distribution of 3 corners in the domain of the pole
among the two singularities must be “2+1", so one singularity has contribution
strictly bigger than $3\pi$ from the polar domain. On the other hand, in the
interior, the distribution of the five corners among the two singularities
must be “3+2". In particular, the contribution to the total angle at either
singularity from the core is at least $\pi$. But then one singularity has
angle strictly bigger than $4\pi$, which is impossible. ∎

The volume of period character $\chi$ provides additional constraints on the
shape of the core.

######  Lemma 2.5.

Let $(X,\omega)$ be a translation surface in the stratum $\mathcal{H}(1,1,-2)$
realizing period character $\chi$. If $(X,\omega)$ is of torus type, then
$Vol_{\chi}>0$. If $(X,\omega)$ is of degenerate type, then $Vol_{\chi}\leq
0$. In this latter case, $Vol_{\chi}<0$ if and only if $(X,\omega)$ is
obtained by removing a convex hexagon in an infinite plane and identifying
pairs of opposite sides.

###### Proof.

If $\mathcal{C}_{(X,\omega)}$ is a torus with a slit, we consider a pair
$(\alpha,\beta)$ of loops disjoint from the slit defined a symplectic basis of
$H_{1}(X,\mathbb{Z})$. Gluing the two slides of the slit on each other defines
a flat torus $\mathcal{T}$. Pair $(\alpha,\beta)$ is a symplectic basis of the
homology of $\mathcal{T}$. Since the torus has positive area, it follows that
$Vol_{\chi}>0$.
If the interior of $\mathcal{C}_{(X,\omega)}$ is empty, then $(X,\omega)$ is
obtained by identification of sides in the boundary of the unique polar domain
which is a topological disk. It follows from the computation of Euler
characteristic that the polar domain has six sides and
$\mathcal{C_{(X,\omega)}}$ is formed by three edges.
We obtain $(X,\omega)$ by identifying three pairs of opposite sides (gluing
pattern corresponding to word $abcabc$). Gluing pattern corresponding to words
$aabbcc$ and $aabccb$ imply the existence of more than two conical
singularities. In the last pattern $abcacb$, four angular sectors of the
boundary of the polar domain contribute to one singularity while the two
others contribute to the second one. It follows that the distribution of
angles is $(\pi,\pi,2\pi,\pi,\pi,2\pi)$. This corresponds to a slit in the
infinite plane and some identified sides are then on the same side of the
slit. This is impossible in a translation structure. $(X,\omega)$ is thus
obtained from the complement in $\mathbb{C}$ of a topological hexagon $H$ by
identification of pairs of opposite sides. Since the total angle of each
singularity is $4\pi$ and every angular sector is of magnitude at least $\pi$,
see Lemma 2.3, the magnitude of each angular sector belongs to interval
$[\pi,2\pi]$. Besides, the total angle of three consecutive angular sectors in
the boundary of the polar sector has to be exactly $4\pi$ (by symmetry). It
follows that $H$ is a convex hexagon.
The three vertices corresponding to the same singularity define a triangle of
area $-\frac{Vol_{\chi}}{2}$, see Figure 2. It follows that $Vol_{\chi}\neq
0$. ∎

Figure 2. A surface of degenerate type, with a parallelogram of area
$-Vol_{\chi}$.

##  3\. Period coordinates and isoperiodic foliation

###  3.1. Period atlas on strata

We now introduce a key notion of the present paper. Let $X$ be a compact
Riemann surface of genus $g$. An Abelian differential $\omega\in\Omega(X)$
always determines a representation, say
$\chi\colon\textnormal{H}_{1}(X\setminus
P(\omega),\,\mathbb{Z})\longrightarrow\mathbb{C}$, where $P(\omega)$ denotes
the set of poles of $\omega$ whenever this is a meromorphic differential,
which is defined by the integral; i.e.

(3.1) $\chi(\gamma)=\int_{\gamma}\omega\in\mathbb{C}.$

For a symplectic basis $\\{\alpha_{1},\beta_{1},\dots,\alpha_{g},\beta_{g}\\}$
of $\textnormal{H}_{1}(X,\,\mathbb{Z})\cong\mathbb{Z}^{2g}$, the evaluation of
$\omega$ along this collection of curves provide a system of $2g$ local
coordinates called absolute periods and the residues of $\omega$ – if it is
meromorphic – computed around the poles of $\omega$ provide additional $p-1$
local coordinates, where $p$ is the cardinality of $P(\omega)$. Generally,
absolute periods and residues do not determine a translation surface even
locally. In order to have a local determination of a translation surface one
has to consider more parameters. These correspond to the so called relative
periods. In principle, these periods are not encoded by a period character
$\chi$. For a translation surface $(X,\omega)$ we denote by $Z(\omega)$ the
set of zeroes of $\omega$. If we label the zeroes of $\omega$ as
$\\{z_{1},\dots,z_{n}\\}$ and we choose the latter as a base-point, the
relative periods are then defined as follow

(3.2) $\int_{z_{i}}^{z_{n}}\omega\quad\textnormal{ for }i=1,\dots,n-1.$

In order to keep track of relative periods we need to consider the relative
homology $\textnormal{H}_{1}(X\setminus P(\omega),\,Z(\omega),\,\mathbb{Z})$.
If $Z(\omega)=\\{z_{1},\dots,z_{k}\\}$; let $\gamma_{1},\dots,\gamma_{n-1}$ be
a collection of arcs connecting $z_{1},\dots,z_{k-1}$ with $z_{k}$
respectively. Moreover, if $\omega$ is a meromorphic differential, then
$P=\\{p_{1},\dots,p_{n}\\}\neq\phi$ and denote by $\delta_{i}$ a peripheral
loop around $p_{i}$. The family of curves
$\alpha_{1},\dots,\beta_{g},\gamma_{1},\dots,\gamma_{k-1},\delta_{1},\dots,\delta_{n-1}$
provides a basis for $\textnormal{H}_{1}(X\setminus
P(\omega),\,Z(\omega),\,\mathbb{Z})$. In the case a translation surface
$(X,\omega)$ belongs to a stratum $\mathcal{H}_{g}(\kappa;-\nu)$, the mapping

(3.3)
$\textnormal{Per}\colon\mathcal{H}_{g}(\kappa;-\nu)\longrightarrow\mathbb{C}^{2g+n+k-2}$

given by

(3.4)
$(X,\omega)\longmapsto\left(\,\int_{\alpha_{1}}\omega\,,\dots,\int_{\beta_{g}}\omega\,,\int_{\gamma_{1}}\omega\,,\dots,\int_{\gamma_{k-1}}\omega,\,\int_{\delta_{1}}\omega\,,\dots,\int_{\delta_{n-1}}\omega\,\right)$

provides a system of local coordinates around $(X,\omega)$. There is a natural
mapping
$\textnormal{Abs}\colon\mathcal{H}_{g}(\kappa;-\nu)\longrightarrow\mathbb{C}^{2g}$
that associates to any translation surface $(X,\omega)$ its absolute periods
and forgets relative periods and residues. The image of this mapping has been
completely determined. In fact, in [9, Theorem A] the authors showed that for
every $2g$-tuple the fibres is not-empty. Moreover, the provide necessary and
sufficient condition for a given $2g$-tuple to appear as the absolute periods
of some translation surface with poles in a prescribed stratum. This map is
simply defined as

(3.5)
$(X,\omega)\longmapsto\left(\,\int_{\alpha_{1}}\omega\,,\dots,\int_{\beta_{g}}\omega\,\right).$

The so-called isoperiodic foliation $\mathcal{AP}_{g}$, also known as kernel
foliation or absolute period foliation, is the foliation locally defined by
the fibers of the mapping Abs. Namely, two translation surfaces are in the
same leaf of $\mathcal{AP}_{g}$ if one can be continuously deformed into the
other without changing absolute periods.

###  3.2. Classification of period characters

In the specific case of strata $\mathcal{H}(1,1,-2)$, the underlying
topological surface is of genus one and there is only one pole. Consequently,
the image by the period map of the homology group of the surface punctured at
the pole is a subgroup $\Gamma$ of $\mathbb{C}$ with two generators. We have
the following

######  Lemma 3.1.

The absolute period group $\Gamma$ of a translation surface of
$\mathcal{H}(1,1,-2)$ is either a lattice of $\mathbb{C}$ or a nontrivial
subgroup of a real line of $\mathbb{C}$.

###### Proof.

Since $\Gamma$ is the image of $\mathbb{Z}^{2}$ into $\mathbb{C}$ by a
homomorphism we just have to prove that it cannot be trivial. It follows from
[9, Theorem B], see also [8], where the authors provide necessary and
sufficient conditions to realise the trivial representation in a given
stratum. ∎

######  Remark 3.2.

A different argument for Lemma 3.1 is as follows. Let
$(X,\omega)\in\mathcal{H}(1,1;-2)$ and let $\chi$ be its period character. If
the period of every closed loop is trivial; i.e. $\chi$ is the trivial
representation, then there is no closed saddle connection and the saddle
connections between $B$ and $W$ have the same period. Since every saddle
connection thus belongs to the same direction, there is no flat triangle
inside the core. Consequently, the surface is just formed by the domain of the
pole – a topological disk, see subsection §2.7 – the two conical singularities
and three saddle connections (this number is determined by Euler
characteristic). At each conical singularity, two saddle connections with the
same period are separated by an angular sector which is an integer multiple of
$2\pi$. Here, zeroes are simple so they correspond to conical singularities of
angle $4\pi$. Therefore, the number of saddle connections between them is at
most two.

Based on this lemma, we can thus distinguish four types of isoperiodic leaves:

  1. 1.

Non-arithmetic real leaves if $\Gamma$ is dense in a real line of
$\mathbb{C}$;

  2. 2.

Arithmetic real leaves if $\Gamma=\mathbb{Z}a$ with $a\in\mathbb{C}^{\ast}$
(in this case, we can always find a closed simple loop whose period is zero);

  3. 3.

Positive leaf if $\Gamma$ is a lattice and the image of a symplectic basis of
the homology is a direct basis of $\mathbb{C}$ (positive volume);

  4. 4.

Negative leaf if $\Gamma$ is a lattice and the image of a symplectic basis of
the homology is an indirect basis of $\mathbb{C}$ (negative volume).

###  3.3. Isoperiodic leaves: translation structure

We focus on isoperiodic leaves in $\mathcal{H}(1,1;-2)$ – the stratum we are
interested in this paper. The isoperiodic foliation is defined by fixing
periods of the differential along $2$ loops forming a symplectic basis of the
homology of $X$. Notice that there is only one pole so its residue is
automatically zero. The unique local parameter of deformation of $(X,\omega)$
is the relative period of the differential on a path between the zeroes.
Besides, the zeroes are labelled as Black ($B$) and White ($W$). Notice that
this corresponds to fix two marked points on the underlying complex structure.

######  Remark 3.3.

Since the two zeroes are now labelled, we are technically looking at a marked
stratum, which is a double cover over the one in our definition. In what
follows we shall study isoperiodic leaves in this space.

Since any homotopy class of paths in $X$ punctured at the pole has at least
one geodesic representative, any such class of paths from one conical
singularity to the other has a representative which is a chain of saddle
connections, i.e. geodesic segments, one of which is between the black zero
and the white one. Therefore, there is at least one of these relative homotopy
class represented by a saddle connection.

Around $(X,\omega)$, for any homotopy class $\gamma$ of paths from $B$ to $W$
in $X$ punctured at the poles, the period $z_{\gamma}$ of the differential
along $\gamma$ can serve as a local complex coordinate in the isoperiodic
leaf. For any other class $\gamma^{\prime}$, the difference
$z_{\gamma}-z_{\gamma^{\prime}}$ is the period of the differential along the
closed loop $\gamma\cup-\gamma^{\prime}$. Such an absolute period is fixed all
along the isoperiodic leaf. Therefore, the arbitrary choice of $\gamma$
results in a coordinates system defined up to a constant, up to a translation
in $\mathbb{C}$. These coordinates form a translation atlas in the isoperiodic
leaf.

###  3.4. Isoperiodic leaves: cover structure

In an isoperiodic leaf where the absolute period group $\Gamma$ is a lattice,
the period coordinates of the leaf are defined up to a period of $\Gamma$.
Consequently, we have

######  Proposition 3.4.

Suppose the absolute period group $\Gamma$ of an isoperiodic leaf is a lattice
in $\mathbb{C}$. Then a translation structure on the leaf is induced by a
covering of torus $\mathbb{C}/\Gamma$ ramified over the lattice point.

###### Proof.

Indeed, the conical singularities on the leaf correspond to translation
surfaces obtained by degeneration, i.e. shrinking a saddle connection between
the two zeroes. ∎

Because of the action of $\textnormal{GL}^{+}(2,\mathbb{R})$, every generic
leaf is conjugated to one where $\Gamma$ is normalized to $\mathbb{Z}[\,i\,]$.
In particular, the ramified covering over the square torus endows such
isoperiodic leaves with a structure of square-tiled translation surface.

###  3.5. Isoperiodic leaves: walls-and-chambers structure

Each isoperiodic leaf and stratum can be decomposed into _chambers_ , where
the qualitative shape of the core is the same. The _discriminant_ is the locus
that separates different chambers. It corresponds to a collinerarity condition
between boundary saddle connections of polar domains, see [26, Section §4] for
details.

######  Definition 3.5.

A translation surface $(X,\omega)$ belongs to the _discriminant_ of the leaf
(or stratum) if there exist two consecutive saddle connections on the boundary
of a domain of pole forming an angle of $\pi$. Each connected component of the
complement of the discriminant is a _chamber_ of the leaf.

In each chamber, the topological pair $(X,\partial\mathcal{C}_{(X,\omega)})$
is the same up to homeomorphism. The possible shapes have been classified in
Proposition 2.4. Therefore, we decompose each leaf into:

  * •

chambers of torus type;

  * •

chambers of cylinder type;

  * •

chambers of degenerate type.

#### The stratum $\mathcal{H}(2,-2)$

We now recall the walls-and-chambers structure of the stratum
$\mathcal{H}(2,-2)$. This was extensively discussed in [27]. For this, note
that there is a natural $\mathbb{C}^{\ast}$-action on each stratum by scaling
the meromorphic differential. A _projectivized_ stratum is the quotient of a
stratum by this action. The projectivized stratum
$\mathbb{P}\mathcal{H}(2,-2)$ is biholomorphic to the modular curve
$X_{1}(2)$. The modular curve $X_{1}(2)$ is a complex curve of genus zero with
two cusps and one orbifold point of order $2$. The (projectivized)
discriminant is an arc connecting one cusp to itself, dividing $X_{1}(2)$ into
two chambers:

  * •

One chamber contains the other cusp of $X_{1}(2)$. This chamber consists of
translation surfaces with cylindrical cores, so we call it a chamber of
_cylinder type_. Translation surfaces in this chamber can be constructed by
gluing a cylinder to a copy of the complex plane along a slit. The cusp
contained in this chamber corresponds to an “infinitely long" cylinder.

  * •

The other chamber contains the orbifold point. This chamber consists of
degenerate translation surfaces, so we call it a chamber of _degenerate type_.
Translation surfaces in this chamber can be constructed by removing a
parallelogram from a copy of the complex plane, and then identifying opposite
sides. The orbifold point corresponds to the case where the parallelogram is a
square.

Translation surfaces in the discriminant can be constructed similarly as
degenerate types, by removing a “degenerate parallelogram", whose sides are
all parallel.

###  3.6. Isoperiodic leaves: conical singularities

As already alluded above, each isoperiodic leaf admits a natural translation
atlas that turns the leaf into a translation surface. As such, an isoperiodic
leaf admits conical singularities corresponding to translation surfaces
obtained by means of degeneration. A translation surface degenerates if a
saddle connection shrinks or if its length becomes infinite. Let
$(X,\omega)\in\mathcal{H}(1,1;-2)$ be a translation surface with period
character $\chi$. For a deformation inside an isoperiodic leaf in the stratum
$\mathcal{H}(1,1,-2)$, the lengths of closed saddle connections are fixed.
Therefore, the only saddle connections that can shrink are between the two
conical singularities $B$ and $W$.

If several saddle connections shrink simultaneously, then they have the same
period all along the deformation. Consequently, the angles at the two
singularities between them is an integer multiple of $2\pi$. Since conical
singularities of translation surfaces of $\mathcal{H}(1,1,-2)$ have an angle
of $4\pi$, at most two saddle connections can shrink simultaneously. Notice
that their union is a simple loop, say $\gamma$, of period zero. In the case
$\Gamma$ is not a cyclic group, then this happens only if $\gamma$ is a
peripheral loop around the (unique) pole of the surface. In this case, the
degeneration disconnects the surface into flat plane – the domain of the pole
– and a flat torus. Such a degeneracy occurs only in the chamber of torus
type, see Section §5 for a precise definition. The local model of this
degeneration is just a pointed disk. We compactify by adding a marked point.

For an arithmetic real leaf the picture is quite different from the above one
because there is a closed simple loop $\gamma$ such that
$\Gamma=\chi(\gamma)\,\mathbb{Z}$. If two saddle connection shrink
simultaneously in such isoperiodic leaves, the torus degenerates and we get a
translation surface of genus zero with a pole of order two and thus two marked
points connected by a saddle connection of period $\chi(\gamma)$. The local
model of this degeneration is also a pointed disk (because of the symmetries).
Once again, we compactify it by adding a marked point.

Otherwise, only one saddle connection shrinks. Algebraically, if the two
zeroes of a differential in the stratum $\mathcal{H}(1,1,-2)$ collide, then we
get a differential with only one zero, that is a differential of the minimal
stratum $\mathcal{H}(2,-2)$. Conversely, in a translation surface with a
single zero of order two, we can perform a surgery called breaking up that
splits the zero into two simple zeroes connected by an arbitrarily small
saddle connection, see [4, 11] for details. This surgery consists in replacing
a small disk around the zero of order two, see Figure 3.

Figure 3. Breaking up a zero of order two into two zeroes of order one, with
the courtesy of Corentin Boissy.

By using the period of the saddle connection as a continuous parameter of an
analytic deformation we can see that every translation surface of
$\mathcal{H}(1,1,-2)$ that can be obtained as the deformation of some
translation surface in the stratum $\mathcal{H}(2,-2)$. The surgery is local
and does not affect the absolute periods. What we get is a cover of order
three of the pointed disk as a local model for the degeneration. We compactify
by adding a conical singularity of angle $6\pi$. The position of the conical
singularities of an isoperiodic leaf in $\mathbb{P}\mathcal{H}(2,-2)$ depend
on its type.

######  Proposition 3.6.

In a real leaf, conical singularities belong to the discriminant of
$\mathcal{H}(2,-2)$. In positive leaves, conical singularities belong to the
chamber of cylinder type. Finally, in negative leaves, conical singularities
belong to the chamber of degenerate type.

###### Proof.

The absolute period group $\Gamma$ of a conical singularity of an isoperiodic
leaf is the same as that of the other differentials of the leaf (breaking up
zeroes is a local surgery). For translation surfaces in a stratum with a
single zero, saddle connections are necessarily closed and then their periods
generate $\Gamma$. In $\mathcal{H}(2,-2)$, elements of $\Gamma$ are real
collinear if and only if the translation surface has a degenerate core formed
by a pair of saddle connections with the same direction. In the case of
$\mathcal{H}(2,-2)$, this locus coincides with the discriminant. For an
element of the chamber of cylinder type of $\mathcal{H}(2,-2)$, a symplectic
basis of the homology is a pair of oriented loops, say $\alpha$ and $\beta$
with an intersection number of $+1$. We may choose $\alpha$ as a closed
geodesic of the cylinder and $\beta$ as a loop crossing the cylinder (and
turning around the slit). The image of their periods in $\mathbb{C}$ is
clearly a direct basis. Therefore, such conical singularities can only appear
in positive leaves. Conversely, the neighborhood of any such differential in
$\mathcal{H}(1,1,-2)$ (obtained by breaking up) is a positive leaf. Finally,
for an element of the chamber of degenerate type of $\mathcal{H}(2,-2)$, the
translation surface is a flat plane with a parallelogram removed (it contains
exactly two saddle connections). We consider $\alpha$ and $\beta$ two closed
loops such that each of them crosses one saddle connection and then turns
around the parallelogram removed. The period of each such loop is that of one
of the two closed saddle connections. We check easily that if they intersect
positively, the parallelogram their periods define in $\mathbb{C}$ is
negatively oriented. ∎

###  3.7. Isoperiodic leaves are hyperbolic Riemann surfaces

For the purpose of proving this fact, we exhibit non-constant holomorphic maps
from any isoperiodic leaf to the moduli space $\mathcal{M}_{1,1}$ of elliptic
curves.

######  Lemma 3.7.

For every isoperiodic leaf $L$ in either the marked or unmarked stratum
$\overline{\mathcal{H}}$, there is a non-constant holomorphic map $\phi_{L}$
from $L$ to the moduli space $\mathcal{M}_{1,1}$ of elliptic curves.

###### Proof.

For each translation surface of $\overline{\mathcal{H}}$, we forget the
location of the zeroes of the differential and mark the double pole in the
underlying elliptic curve. In this way, we define a holomorphic map from
$\overline{\mathcal{H}}$ to $\mathcal{M}_{1,1}$. Since the period map defines
holomorphic charts and the isoperiodic foliation is a holomorphic foliation,
for each isoperiodic leaf $L$, we obtain a holomorphic map $\phi_{L}$ from $L$
to $\mathcal{M}_{1,1}$. If $\phi_{L}$ were constant to some complex structure
$\mathbb{C}/\Lambda$, then $L$ would have been contained in the bi-dimensional
vector space $V$ of meromorphic $1$-forms on $\mathbb{C}/\Lambda$ with at most
one double pole at the marked point; generated by $dz$ and $\wp dz$. Since
deformations inside $L$ are isoperiodic, then $L$ is an affine complex line of
$V$ modeled on the subspace of exact $1$-forms in $V$. Since there is no
nontrivial exact $1$-form in $V$, we obtain a contradiction and $\phi_{L}$ is
a non-constant holomorphic map. ∎

######  Corollary 3.8.

For every isoperiodic leaf $L$ in either the marked or unmarked stratum
$\overline{\mathcal{H}}$, $L$ admits a non-constant bounded holomorphic
function.

###### Proof.

Since $\mathcal{M}_{1,1,}$ is biholomorphic to a quotient of the complex disk
$\mathbb{D}$, $\phi_{L}$ lifts to a non-constant holomorphic map to
$\mathcal{D}$. ∎

##  4\. Chambers of cylinder type

We give a detailed description of these chambers appearing in every
isoperiodic leaf of $\mathcal{H}(1,1,-2)$. Following the proof of Proposition
2.4, a translation surface $(X,\omega)$ of cylinder type has a polar domain
with four boundary saddle connections, two of which being glued on the sides
of a cylinder (formed by two flat triangles). The only possible gluing pattern
implies that the two opposite sides are glued on each other identified by a
translation. Consequently, the polar domain is the complement of a
parallelogram $Q$.

A cylinder is constructed by identifying a pair of opposite sides in a
parallelogram $P$. For a translation surface of $\mathcal{H}(1,1,-2)$, each
boundary side is a closed saddle connection. Each of the two conical
singularities belong to exactly one boundary component of the cylinder.
Cylinder is foliated by a $1$-parameter family of closed geodesics, each
representing the same absolute homology class $\gamma$. We choose the
orientation of $\gamma$ in such a way that singularity $B$ belongs to the left
boundary while singularity $W$ belongs to the right boundary, see Figure 4.
Homology class $\gamma$ has to be primitive and its period $u=\chi(\gamma)$
has to be nonzero. We choose the relative period $z$ of the identified sides
of parallelogram $Q$ as coordinate in the chamber. Any diagonal of the
cylinder, oriented from singularity $B$ to singularity $W$, has a period of
the form $z-v$ where $v$ is an absolute period such that pair $(u,v)$ is the
image of symplectic basis by period character $\chi$.

Figure 4. General construction of a surface of cylinder type, with side
identification labeled in red. The two singularities are distinguished by
black or white. For an oriented side signified by an arrow, the letter that
appears nearby denotes the complex number representing that vector.

######  Proposition 4.1.

In stratum $\mathcal{H}(1,1,-2)$, we consider the locus formed by translation
surfaces realizing a nonzero period character $\chi$ (up to a symplectic
change of basis) and a nontrivial absolute period
$u\in\Gamma\setminus\\{{0\\}}$ such that $u$ is the period of a primitive
homology class. Then

  * •

if $Vol_{\chi}\neq 0$ or $\Gamma$ is dense in a real line of $\mathbb{C}$,
there is exactly one chamber $CC_{u}$ formed by translation surfaces of
cylinder type realizing character $\chi$ and such that the appropriately
oriented core geodesic of the cylinder realizes a homology class $\gamma$ of
period $\chi(\gamma)=u$;

  * •

if $\Gamma=a\,\mathbb{Z}$ for some $a\neq 0$ and $u=ka$ for some nonzero
integer $k$, then for each integer $0\leq l<|k|$ with $\gcd(l,k)=1$, there
exists exactly one chamber $CC_{u,v}=CC_{k,l}$ formed by translation surfaces
of cylinder type realizing character $\chi$ and such that the appropriately
oriented core geodesic of the cylinder realizes a homology class $\gamma$ of
period $\chi(\gamma)=u$, and $v=la$ is the period of another homology class
$\eta$, where $(\gamma,\eta)$ forms a symplectic basis of the homology group.

Moreover, in the translation structure of the isoperiodic leaf containing
$CC_{u}$ or $CC_{u,v}$, the chamber is a half-plane defined by inequality
$\mathfrak{Im}(\bar{u}z)<\min(0,Vol_{\chi})$.

###### Proof.

By adopting the notation above, a translation surface of cylinder type is
completely characterized by complex numbers $u,v,z$. Moreover, $v$ is defined
up to elements of $u\,\mathbb{Z}$ and is required to satisfy the equality
$\mathfrak{Im}(\bar{u}v)=Vol_{\chi}$, because $(u,v)$ is the image of a
symplectic basis. This condition fixes $v$ up to $u\,\mathbb{Z}$ when $\Gamma$
is not a cyclic group.

Suppose $\Gamma=a\,\mathbb{Z}$ and let $u=ak$ for some integer $k$. For each
integer $0\leq l<|k|$, the chambers $CC_{u,v}$ with $v=al$, up to
$u\,\mathbb{Z}$, are distinct from each other. Indeed, for a fixed parameter
$z$, the heights of the cylinders determined by parallelograms having sides of
length $u$ and $z-v$ are distinct, even up to an element $u\,\mathbb{Z}$.
Thus, assuming $\chi$ and $u$ are fixed, and so is $v$ in the arithmetic case,
translation surfaces with a cylinder of absolute period $u$ are parameterized
by relative period $z$. The only constraint is that the areas of parallelogram
$P$ and $Q$ should not vanish. These conditions amount to the two
inequalities:

  * •

$\mathfrak{Im}\big{(}\,\bar{u}(z-v)\,\big{)}<0$ or equivalently
$\mathfrak{Im}(\bar{u}z)<Vol_{\chi}$;

  * •

$\mathfrak{Im}(\,\bar{u}z\,)<0$.

These constraints combined together yields the condition
$\mathfrak{Im}(\bar{u}z)<\min(0,Vol_{\chi})$. Hence translation surfaces with
a cylinder of absolute period $u$ form a half-plane in an isoperiodic fiber
realizing character $\chi$. ∎

######  Remark 4.2.

Period $-u$ gives the same chamber in the unmarked stratum
$\mathcal{H}(1,1,-2)$, but a different chamber in the marked double cover.
Also, by the description above, in the arithmetic case where
$\Gamma=a\,\mathbb{Z}$ and $u=ak$ for some nonzero integer $k$, there are
$\varphi(|k|)$ distinct chambers formed by translation surfaces with a
cylinder of absolute period $u$, where $\varphi$ is Euler’s totient function.

Finally, we note that the boundary line of chamber $CC_{u}$ or $CC_{u,v}$ (in
the direction $u\,\mathbb{R}$) corresponds to the degeneration of:

  * •

cylinder $P$ (if $Vol_{\chi}<0$);

  * •

parallelogram $Q$ (if $Vol_{\chi}>0$);

  * •

both of them (if $Vol_{\chi}=0$).

##  5\. Positive leaves

In this section, we give a complete description of positive leaves. Note that
by applying the $\mathrm{GL}^{+}(2,\mathbb{R})$-action, we may assume
$\Gamma=\mathbb{Z}[\,i\,]$.

###  5.1. Chamber of torus type

A surface of torus type is constructed as follows: take a flat torus and a
complex plane, cut parallel slits on both, and then glue them together via
translations; see Figure 5.

Figure 5. A surface of torus type, with a long slit

The holonomy vector $\alpha$ of the slit, oriented from singularity $B$ to
singularity $W$, serves as a coordinate on the chamber. Note that as
$\alpha\to 0$, we obtain a torus and a complex plane glued at a point, as
mentioned in Section §3.6. We add this point to the chamber to obtain a metric
completion of the leaf.

The holonomy vector of the slit cannot be arbitrary. Indeed, the slit cannot
wrap onto itself. In particular, the holonomy vector $\alpha\neq t\gamma$
where $\gamma\in\Gamma$ is a primitive element, and $t\geq 1$. In local
coordinate $\alpha$, the torus chamber is thus a copy of $\mathbb{C}$ minus
slits of the form $\\{t\gamma:t\geq 1\\}$ for all primitive elements
$\gamma\in\Gamma$; see Figure 6.

Figure 6. The torus chamber is a copy of $\mathbb{C}$ minus slits, with the
origin added for metric completion. Blue dots are primitive elements in
$\Gamma$, and give conical singularities of the leaf. The left side of the
slit starting at $\gamma$ is glued to the positive half boundary of the
chamber $CC_{\gamma}$, and the right side is glued to the negative half
boundary of the chamber $CC_{-\gamma}$.

#### Degeneration towards slits

We discuss what happens when moving towards a slit in the torus chamber. Fix a
primitive element $\gamma\in\Gamma$, and $t>1$ – note that the case $t=1$
corresponds to a surface of cylinder type in the stratum $\mathcal{H}(2,-2)$.

We say the holonomy vector $\alpha$ of the slit approaches $t\gamma$ from the
left (resp. right) if $\alpha\to t\gamma$ and the pair $(t\gamma,\alpha)$
remains positively oriented (resp. negatively oriented). We focus on the case
$\alpha$ approaches $t\gamma$ from the left, and the other case is entirely
analogous.

Choose $\delta$ so that $(\gamma,\delta)$ forms a positively oriented
$\mathbb{Z}$-basis for $\Gamma$. Consider $\alpha$ very close to $t\gamma$
from the left. After some cut-and-paste, the core of such a translation
surface looks like a cylinder together with a long and thin strip, see Figure
7. The domain of the pole is a copy of $\mathbb{C}$ with a slit parallel to
$\alpha$. Part of one side of the slit on $\mathbb{C}$ is glued to one side of
the thin strip of the core, and part of the other side of the slit is glued to
the other side of the thin strip.

Figure 7. Cut and paste

As $\alpha\to t\gamma$, the width of the thin strip goes to zero. In the
limit, the core becomes a cylinder, with sides parallel to $\gamma$ and
$\delta-\\{t\\}\gamma$, here $\\{t\\}$ is the fractional part of $t$. Parts of
two sides of the slit on $\mathbb{C}$ originally glued to the thin strip are
thus identified. It is clear that in the limit, the surface also lies on the
boundary of a chamber of cylinder type. It is easy to see that to match the
orientation we introduced for cylinder chambers, when $\alpha\to t\gamma$ from
the left, the corresponding cylinder chamber we approach is the one
corresponding to $\gamma$, and towards the point $(t-1)\gamma$ in the local
coordinate we introduced for this chamber of cylinder type. In particular, as
$t$ ranges from $1$ to $\infty$, we covers a ray $\\{s\gamma:s\geq 0\\}$ in
the boundary of the cylinder chamber.

The geometry of a chamber of torus type is described in Figure 6. It is an
infinite plane with infinitely many slits.

###  5.2. Geometry of a positive leaf

We can now describe the global geometry of a positive leaf. The whole leaf is
obtained by gluing cylinder chambers to the torus chamber.

Indeed, recall that the chamber of torus type is a complex plane with slits of
the form $\\{t\gamma:t\geq 1\\}$ for all primitive elements $\gamma\in\Gamma$.
For the each slit $\\{t\gamma:t\geq 1\\}$, approaching it from the left we end
up on (half of) the boundary of the cylinder chamber corresponding to
$\gamma$, and approaching it from the right we end up on (half of) the
boundary of the cylinder chamber corresponding to $-\gamma$. Similarly, the
other half of these boundaries are glued to the slit $\\{-t\gamma:t\geq 1\\}$.

Every chamber of cylinder type is connected to the chamber of torus type so
there is only one isoperiodic leaf realizing a given period character $\chi$
satisfying $Vol_{\chi}>0$. Besides, there are no chambers of degenerate type
in positive leaves, see Lemma 2.5. This proves Theorem 1.2 in the case of
positive leaves.

After gluing, at the point corresponding to each primitive element $\gamma$,
the leaf has angle $6\pi$. The Abelian differential for the translation
structure on this leaf thus has double zeroes at these points.

It is clear that the leaf supports an involution, given by
$\alpha\mapsto-\alpha$ in local coordinate we gave on the chamber of torus
type, and maps the chamber of cylinder type corresponding to $\gamma$ to the
chamber of cylinder type corresponding to $-\gamma$. In the quotient, the
point $p$ gives a simple pole of the quadratic differential, and other
singularities have cone angle $3\pi$, so they are simple zeros of the
quadratic differential.

We note that the quotient is simply connected. We just have to observe that
the quotient of the chamber of torus type is a half-plane with infinitely many
slits starting at primitive periods, up to a sign. Each side of the same slit
is glued on a half-line bounding the same chamber of cylinder type. In other
words, we insert a half-plane inside each slit. We clearly obtain a surface
homeomorphic to a plane. In terms of complex structures, the surface is either
a complex plane or a complex disk. It follows then from Corollary 3.8 that the
surface is biholomorphic to a disk.

Combined with the following lemma, we conclude that a positive leaf is a Loch
Ness Monster, i.e. a surface of infinite genus with one end. This gives
Theorem 1.3 for positive leaves.

######  Lemma 5.1.

Let $S$ be a topological surface, and let $\mathcal{P}$ denote a topological
surface homeomorphic to the plane $\mathbb{C}$. Suppose there exists a
branched double cover $\pi:S\to\mathcal{P}$, branched over a infinite,
discrete set of points. Then $S$ is a Loch Ness Monster.

###### Proof.

First we show that $S$ has one end. Suppose not. Since $\pi$ is a branched
double cover and $\mathcal{P}$ is one-ended, $S$ has two ends, each covering
the unique end of $\mathcal{P}$. Thus for some large enough simply connected
compact set $K\subset\mathcal{P}$, the preimage of $\mathcal{P}-K$ consists of
two connected components. But $\mathcal{P}-K$ contains ramification points of
$\pi$, so its preimage must be connected, a contradiction.

Next we show that $S$ has infinite genus. Indeed, take a disk $D$ in
$\mathcal{P}$ containing exactly two ramification points. Then $\pi^{-1}(D)$
is connected, and by Riemann-Hurwitz formula, has Euler characteristic $0$. It
has either one or two boundary components. If it has one, then by capping off
the boundary component we obtain a closed surface with Euler characteristic
$1$, which cannot be. So $\pi^{-1}(D)$ has two boundary components, and is
therefore a cylinder. There are infinitely many disjoint pairs of ramification
points, so we obtain infinitely many handles of $S$. By classification of
topological surfaces, $S$ is thus a Loch Ness Monster. ∎

Finally, we can also determine the Veech group of a positive leaf as a
translation surface, Theorem 1.4 for positive leaves.

######  Proposition 5.2.

The Veech group of a positive leaf is a conjugate of
$\textnormal{SL}(2,\,\mathbb{Z})$.

###### Proof.

The action of $g\in\mathrm{GL}^{+}(2,\mathbb{R})$ takes one positive leaf with
absolute period group $\Gamma$ to one with $g\Gamma$. Since the translation
structure of the leaf is characterized by its absolute period group, the Veech
group is exactly the stabilizer of $\Gamma$ in
$\mathrm{GL}^{+}(2,\mathbb{R})$, which is a conjugate of
$\textnormal{SL}(2,\,\mathbb{Z})$. This can also be proved by looking at the
torus chamber: any element in the Veech group must leave the set of primitive
elements in $\Gamma$ invariant. ∎

##  6\. Negative leaves

In this section, we give a complete description of negative leaves. Here,
$\Gamma$ is a lattice and the image of a symplectic basis of the homology is
an indirect basis of $\mathbb{C}$. Once again, up to
$\mathrm{GL}^{+}(2,\mathbb{R})$-action, we may assume
$\Gamma=\mathbb{Z}[\,i\,]$. Chambers of cylinder type are already described in
Section §4 and we have seen that chambers of torus type only exist in positive
leaves, see Lemma 2.5. In the present section we focus on chambers of
degenerate type.

###  6.1. Chamber of degenerate type

We recall in the first place that a translation surface is of degenerate type
if it is formed by identification pairs of opposite sides in the complement of
a convex hexagon in the infinite plane, see Lemma 2.5 and Figure 2. It is
worth mentioning that the sequence of sides is invariant under small
deformations and $\mathrm{GL}^{+}(2,\mathbb{R})$ action.

We denote by $B_{1},W_{1},B_{2},W_{2},B_{3},W_{3}$ the six corners of the
hexagon with the cyclic order corresponding to the clockwise order of the
hexagon (it is equivalent to the standard orientation of the boundary of the
polar domain as a topological disk). Corners $B_{1},B_{2},B_{3}$ correspond to
the Black vertex while $W_{1},W_{2},W_{3}$ correspond to the White vertex.

We introduce the following notations for the three relative periods
corresponding to saddle connections of the boundary of the polar domain:

  * •

$z_{1}$ corresponds to vector $\overrightarrow{B_{1}W_{1}}$;

  * •

$z_{2}$ corresponds to vector $\overrightarrow{B_{2}W_{2}}$;

  * •

$z_{3}$ corresponds to vector $\overrightarrow{B_{3}W_{3}}$.

The also distinguish three absolute periods:

  * •

$u_{1}=z_{1}-z_{3}$, corresponding to vector $\overrightarrow{B_{1}B_{2}}$;

  * •

$u_{2}=z_{2}-z_{1}$, corresponding to vector $\overrightarrow{B_{2}B_{3}}$;

  * •

$u_{3}=z_{3}-z_{2}$, corresponding to vector $\overrightarrow{B_{3}B_{1}}$.

More generally, a characteristic triple is a triple $(u_{1},u_{2},u_{3})$,
defined up to cyclic permutation, of absolute periods of
$\Gamma=\mathbb{Z}[\,i\,]$ such that:

  * •

$(u_{1},u_{2})$, $(u_{2},u_{3})$, $(u_{3},u_{1})$ are periods of a symplectic
basis of the absolute homology;

  * •

$u_{1}+u_{2}+u_{3}=0$.

######  Lemma 6.1.

In the locus of $\mathcal{H}(1,1;-2)$ where $\Gamma$ is the absolute period
group, there is exactly one chamber of degenerate type
$DD_{(u_{1},u_{2},u_{3})}$ for each such triple. For the translation structure
of its isoperiodic leaf, chamber $DD_{(u_{1},u_{2},u_{3})}$ is a triangle
bounded by three sides of vectors $u_{3},u_{2},u_{1}$ (in the trigonometric
order). These boundary segments are shared with chambers of cylinder type
$CC_{u_{3}},CC_{u_{2}},CC_{u_{1}}$ (respectively).

###### Proof.

In the coordinates defined above, a translation surface of degenerate type is
characterized by a triple of absolute periods $(u_{1},u_{2},u_{3})$,
determined by the sides of triangle $B_{1}B_{2}B_{3}$, and three coordinates
$z_{1},z_{2},z_{3}$ – the relative periods. The three coordinates satisfy the
following inequalities:

  * •

$\mathfrak{Im}(z_{1}\overline{u_{1}})>0$;

  * •

$\mathfrak{Im}(z_{2}\overline{u_{2}})>0$;

  * •

$\mathfrak{Im}(z_{3}\overline{u_{3}})>0$.

Conversely, any pair of translation surfaces of degenerate type with the same
triple $(u_{1},u_{2},u_{3})$ satisfying these inequalities can be joined by a
linear path in coordinates $z_{1},z_{2},z_{3}$. It follows that they form a
unique chamber $DD_{(u_{1},u_{2},u_{3})}$.
Coordinates $z_{2}$ and $z_{3}$ differ from $z_{1}$ by a constant. The three
inequalities define a triangle (intersection of three half-planes) in the
translation structure of the isoperiodic leaf. The equality case correspond to
the boundary of the chamber. In each case a white vertex becomes aligned by
its two neighbors, see Figure 8:

  * •

side of period $u_{1}$ corresponds to the alignment of points
$B_{1},W_{1},B_{2}$;

  * •

side of period $u_{2}$ corresponds to the alignment of points
$B_{2},W_{2},B_{3}$;

  * •

side of period $u_{3}$ corresponds to the alignment of points
$B_{3},W_{3},B_{1}$.

Figure 8. A surface of degenerate type, close to the boundary side of period
$u_{1}$ in the chamber.

The hexagon becomes a parallelogram. This locus is the boundary of a chamber
of cylinder type. The period of the cylinder is also the period of the side of
the parallelogram between two black vertices. The identification of the
incident chambers with $CC_{u_{1}},CC_{u_{2}},CC_{u_{3}}$ follows. ∎

###  6.2. Geometry of negative leaves

Similarly to what we have done in Section §5.2, we now describe the global
geometry of a negative leaf. Up to a action of
$\mathrm{GL}^{+}(2,\mathbb{R})$, we can assume that $\Gamma$ coincides with
Gaussian integers.

If a pair $u_{1},u_{2}$ of primitive absolute periods is the image under
character $\chi$ of a symplectic basis of the absolute homology, then chambers
$CC_{u_{1}}$ and $CC_{u_{2}}$ are both incident to chamber
$DD_{(u_{1},u_{2},-u_{1}-u_{2})}$, see Proposition 4.1 for the classification
of chambers of cylinder type. It is then easy to connect each pair of chambers
of cylinder type by a finite sequence of chambers of degenerate type. It
follows that, up to a symplectic change of basis, a unique connected
isoperiodic leaf contains translation surfaces of stratum
$\mathcal{H}(1,1,-2)$ realizing a character $\chi$ of negative volume. This is
the proof of Theorem 1.2 in the case of negative leaves.

Vertices of chambers of degenerate type are translation surfaces of stratum
$\mathcal{H}(2,-2)$. Following results of Section §3.6, they are the
complement of a parallelogram in the infinite plane. Thus, their geometry is
characterized by an ordered pair of absolute periods defined up to the sign.
Consequently, the three vertices of chamber $DD_{u_{1},u_{2},u_{3}}$
correspond to parallelograms of periods:

  * •

$\pm u_{1},\pm u_{2}$;

  * •

$\pm u_{2},\pm u_{3}$;

  * •

$\pm u_{3},\pm u_{1}$.

It follows that in the isoperiodic leaf, the conical singularity corresponding
to parallelogram $\pm u_{1},\pm u_{2}$ is incident to:

  * •

chambers of cylinder type $CC_{u_{1}}$, $CC_{-u_{1}}$, $CC_{u_{2}}$,
$CC_{-u_{2}}$;

  * •

chambers of degenerate type $DD_{(u_{1},u_{2},-u_{1}-u_{2})}$,
$DD_{(-u_{1},-u_{2},u_{1}+u_{2})}$, $DD_{(u_{2},-u_{1},u_{1}-u_{2})}$,
$DD_{(-u_{2},u_{1},u_{2}-u_{1})}$

This proves that conical singularities of the leaf have an angle of $6\pi$ –
there is a contribution of $\pi$ for each chamber of cylinder type and a total
contribution of $2\pi$ for the four others chambers. Under the involution
changing the sign of every period, they become conical singularities of angle
$3\pi$. There are infinitely many of them.

Now, we prove that the quotient $\tilde{\mathcal{L}}$ of the negative leaf
under the canonical involution is homeomorphic to a plane. In every chamber of
degenerate type, we draw three segments between the centroid of the triangle
and the three corners. We denote by $\mathcal{T}$ the graph obtained in
$\tilde{\mathcal{L}}$. Every conical singularity of $\tilde{\mathcal{L}}$ is
incident to exactly two chambers of degenerate type222In the isoperiodic leaf,
each conical singularity is incident to four such chambers. After quotienting
by the involution, we obtain two chambers of degenerate type.. It follows that
$\mathcal{T}$ is a connected graph where every conical singularity is of
valency two while every centroid is of valency three. Choosing an arbitrary
conical singularity as the root, $\mathcal{T}$ becomes an infinite binary
tree. Each connected component of $\tilde{\mathcal{L}}\setminus\mathcal{T}$ is
a chamber of cylinder type, a half-plane, with a third of triangle,
corresponding to a part of a chamber of degenerate type cut out by two
medians, glued on each boundary edge. In other words, each connected component
of $\tilde{\mathcal{L}}\setminus\mathcal{T}$ is a (topological) half-plane. It
follows that $\tilde{\mathcal{L}}$ is homeomorphic to a plane. In terms of
complex structures, the surface is then either a complex plane or a complex
disk. It follows from Corollary 3.8 that the surface is biholomorphic to a
disk.

Applying Lemma 5.1, we conclude that the negative leaf is also a Loch Ness
Monster. This gives Theorem 1.3 for negative leaves.

######  Remark 6.2.

Despite similar descriptions of positive and negative leaves, it is worth
mentioning that positive and negative leaves are generally not isometric to
each other.

The case of negative leaves in Theorem 1.4 is proved with the same argument as
for positive leaves, see Proposition 5.2 above.

##  7\. Non-arithmetic real leaves

Instead of giving a direct description of non-arithmetic real leaves, we
obtain their geometry as a limit of negative leaves for the action of a
subgroup of $\textnormal{GL}^{+}(2,\mathbb{R})$.

###  7.1. Contraction flow

######  Definition 7.1.

For every pair $\theta_{1},\theta_{2}$ of directions in $\mathbb{RP}^{1}$,
contraction flow $\mathcal{C}_{\theta_{1},\theta_{2},t}$ is the subgroup of
$\textnormal{GL}^{+}(2,\mathbb{R})$ conjugated to

(7.1) $\left\\{\begin{pmatrix}e^{-t}&0\\\ 0&1\end{pmatrix}:t>0\right\\}$

contracting exponentially direction $\theta_{1}$ while preserving direction
$\theta_{2}$.

Let $\mathcal{L}_{0}$ be the (unique) negative leaf whose absolute period
group is $\mathbb{Z}[\,i\,]$ while $(\mathcal{C}_{\theta,h,t})_{t\geq 0}$ is
the contraction flow preserving the horizontal direction while contracting
some other direction $\theta$ with $\theta\notin\\{0,\,\pi\\}$. We denote by
$\mathcal{L}_{t}$ the image of $\mathcal{L}_{0}$ under the action of
$\mathcal{C}_{h,t}$. If $\Gamma_{t}$ is the group of absolute periods of
translation surfaces contained in $\mathcal{L}_{t}$, we have
$\Gamma_{t}\longrightarrow\mathbb{Z}+\frac{1}{\tan(\theta)}\mathbb{Z}$; the
action of $\textnormal{GL}^{+}(2,\mathbb{R})$ commutes with the period
character. Contracting directions $\theta$ for which
$\tan(\theta)\notin\mathbb{Q}$ provide any dense subgroup of $\mathbb{R}$ with
two generators.

The action of the contraction flow on the walls-and-chambers structure of
$\mathcal{L}_{0}$, described in Section §6, is the following:

  * •

chambers of cylinder type remain half-planes but the direction of their
boundary line converges to the horizontal line;

  * •

chambers of degenerate type converge to zero area triangles with three
horizontal sides;

  * •

no boundary side of chambers of degenerate type shrinks provided
$\tan(\theta)\notin\mathbb{Q}$.

The family of leaves $(\mathcal{L}_{t})_{t\geq 0}$ converges to limit surface
$\mathcal{L}_{\infty}$ as $t\longrightarrow+\infty$ because all surfaces of
the family keep the discrete singularities property described in [25]. We
recall that:

######  Definition 7.2.

A translation surface has discrete singularities if for each conical
singularity $x$, there is $\epsilon>0$ such that the ball
$\mathbb{B}(x,\epsilon)$ of radius $\epsilon$ centered at $x$ does not contain
any other singularity.

Limit surface $\mathcal{L}_{\infty}$ is a non-arithmetic real leaf with:

  * •

a chamber of cylinder type for each nonzero primitive period of
$\Gamma_{t}\longrightarrow\mathbb{Z}+\frac{1}{\tan(\theta)}\mathbb{Z}$;

  * •

infinitely many conical singularities of angle $6\pi$;

  * •

a gluing pattern of boundary segments of chambers inherited from the gluing
patterns of chambers of degenerate type in leaf $\mathcal{L}_{0}$.

Leaf $\mathcal{L}_{\infty}$ is thus a Loch Ness Monster satisfying the
properties of Theorem 1.3. It contains chamber of cylinder type $CC_{u}$ for
every possible $u$. Besides, in a real periodic leaf, there is no chamber of
degenerate type. Every surface of degenerate type belongs to the boundary of
the chamber of cylinder type. It follows that $\mathcal{L}_{\infty}$ is the
unique leaf realizing character $\chi$. This proves Theorem 1.2 for non-
arithmetic real leaves.

Finally, the quotient of $\mathcal{L}_{\infty}$ by the standard involution is
simply connected for the same reasons as in the case of negative leaves. In
terms of complex structures, the surface is then either a complex plane or a
complex disk. It follows from Corollary 3.8 that the surface is biholomorphic
to a disk.

######  Remark 7.3.

In [22, Theorem 9], it is proved for each dense subgroup
$\Gamma\subseteq\mathbb{R}$ with two generators, there are exactly two
connected leaves for which the absolute period group coincides with $\Gamma$.
There is no contradiction with Theorem 1.2 because $\Gamma$ is the image of
exactly two period characters, up to change of basis.

###  7.2. Veech group

Finally, we determine the Veech group of a non-arithmetic real leaf as a
translation surface. We first need a number-theoretic lemma. To prepare for
the statement of the lemma, recall that given a square-free integer $D\geq 2$,
the ring of integers of the quadratic field $\mathbb{Q}(\sqrt{D})$ is given by
$\mathbb{Z}[\gamma]$, where

$\gamma=\begin{cases}\sqrt{D}&\text{if }D\equiv 2,3\mod 4\\\
\frac{1+\sqrt{D}}{2}&\text{if }D\equiv 1\mod 4\end{cases}$

is the standard generator of the ring. The conjugate of any element
$s+t\sqrt{D}$ with $s,t\in\mathbb{Q}$ is given by $\bar{x}=s-t\sqrt{D}$, and
the norm of $x$ is defined as $N(x)=x\bar{x}$. The norm is multiplicative,
i.e. $N(xy)=N(x)N(y)$.

Note that $x\in\mathbb{Z}[\gamma]$ if and only if $N(x)$ is an integer, and
$x$ is a unit in $\mathbb{Z}[\gamma]$ if and only if $N(x)=\pm 1$. The units
of $\mathbb{Z}[\gamma]$ form a group, which is generated by $-1$ and a
positive fundamental unit $u$.

Let $\tau=(t,l,m)$ be a triple of integers so that $t,m\neq 0$ and
$\gcd(t,l,m)=1$. Consider the $\mathbb{Z}$-module defined as
$\Gamma=t\,\mathbb{Z}+(l+m\gamma)\,\mathbb{Z}\subseteq\mathbb{Q}(\sqrt{D})$.
For any positive real number $a$, if $a\Gamma\subseteq\Gamma$, then $a$ acts
as a matrix in the basis $\\{t,l+m\gamma\\}$. We use $\det(a)$ to denote the
determinant of this matrix. We have

######  Lemma 7.4.

Consider the group

$G_{D,\tau}:=\\{a\in\mathbb{R}_{+}:a\Gamma=\Gamma,\det(a)=1\\}.$

We have

  1. (1)

$G_{D,\tau}$ is a subgroup of units in $\mathbb{Z}[\gamma]$;

  2. (2)

$G_{D,\tau}$ contains a nontrivial element $\eta=\alpha+\beta\gamma$ with
$N(\eta)=1$ if and only if $m$ divides $\beta$, and $t$ divides
$(\beta/m)N(l+m\gamma)$;

  3. (3)

Let $k$ be the smallest non-negative integer so that $u^{k}\in G_{D,q}$. Then
$G_{D,\tau}=\langle u^{k}\rangle$.

###### Proof.

Set $N(\Gamma):=\min\\{|{N(x)}|:x\in\Gamma,x\neq 0\\}$. Clearly $N(\Gamma)>0$,
and $N(a\Gamma)=|N(a)|N(\Gamma)$. This immediately implies that every element
in $G_{D,\tau}$ is a unit.

For the second part, first note that

$\eta\cdot t=\alpha t+\beta t\gamma=\left(\alpha-\frac{\beta
l}{m}\right)t+\frac{\beta t}{m}(l+m\gamma).$

For this to be in $\Gamma$, we must have $m$ divides $\beta t$, and $m$
divides $\beta l$. If $\gcd(m,\beta)\neq|m|$, then $t$ and $l$ share a
nontrivial factor with $m$, contradicting the assumption $\gcd(t,l,m)=1$. So
we must have $m$ divides $\beta$. Set $\beta=\beta_{0}m$ for some integer
$\beta_{0}$. Next we can calculate

$\begin{split}\eta\cdot(l+m\gamma)&=\alpha l+\beta m\gamma^{2}+(\beta l+\alpha
m)\gamma=\alpha l+\beta
mD-(\beta_{0}l+\alpha)l+(\beta_{0}l+\alpha)(l+m\gamma)\\\
&=\beta_{0}m^{2}D-\beta_{0}l^{2}+(\beta_{0}l+\alpha)(l+m\gamma)\\\
&=-\beta_{0}N(l+m\gamma)+(\beta_{0}l+\alpha)(l+m\gamma)\end{split}$

when $\gamma=\sqrt{D}$, and

$\begin{split}\eta\cdot(l+m\gamma)&=\alpha l+\beta
m\left(\gamma+\frac{D-1}{4}\right)+(\beta_{0}l+\alpha)m\gamma\\\ &=\alpha
l+\beta_{0}m^{2}\frac{D-1}{4}-(\beta_{0}l+\alpha+\beta)l+(\beta_{0}l+\alpha+\beta)(l+m\gamma)\\\
&=\beta_{0}\left(m^{2}\frac{D-1}{4}-l^{2}-ml\right)+(\beta_{0}l+\alpha+\beta)(l+m\gamma)\\\
&=-\beta_{0}N(l+m\gamma)+(\beta_{0}l+\alpha+\beta)(l+m\gamma)\end{split}$

when $\gamma=\frac{1+\sqrt{D}}{2}$. In either case, for this to be in
$\Gamma$, we must have $t$ divides $\beta_{0}N(l+m\gamma)$.

On the other hand, we have when $\gamma=\sqrt{D}$,

$\begin{split}\det(\eta)&=(\alpha-\beta_{0}l)(\beta_{0}l+\alpha)+\beta_{0}^{2}N(l+m\gamma)=\alpha^{2}-\beta_{0}^{2}l^{2}+\beta_{0}^{2}N(l+m\gamma)\\\
&=\alpha^{2}-\beta^{2}D+\beta_{0}^{2}(m^{2}D-l^{2})+\beta_{0}^{2}N(l+m\gamma)\\\
&=N(\eta)-\beta_{0}^{2}N(l+m\gamma)+\beta_{0}^{2}N(l+m\gamma)=1,\end{split}$

and when $\gamma=\frac{1+\sqrt{D}}{2}$,

$\begin{split}\det(\eta)&=(\alpha-\beta_{0}l)(\beta_{0}l+\alpha+\beta)+\beta_{0}^{2}N(l+m\gamma)\\\
&=\alpha^{2}-\beta_{0}^{2}l^{2}+\alpha\beta-\beta\beta_{0}l+\beta_{0}^{2}N(l+m\gamma)\\\
&=\alpha^{2}+\alpha\beta+\beta^{2}\frac{1-D}{4}-\beta_{0}^{2}\left(m^{2}\frac{1-D}{4}+ml-l^{2}\right)+\beta_{0}^{2}N(l+m\gamma)\\\
&=N(\eta)-\beta_{0}^{2}N(l+m\gamma)+\beta_{0}^{2}N(l+m\gamma)=1.\end{split}$

Thus it is easy to see that when the divisibility conditions are met, we
indeed have $\eta\in G_{D,\tau}$. Finally, the third part follows from the
fact that any positive unit is a power of the fundamental unit $u$. ∎

Note that if $G_{D,\tau}$ contains a nontrivial element $\eta$, $\eta^{2}$ is
always of norm $1$, so the lemma above covers all cases.

######  Proposition 7.5.

The Veech group of a non-arithmetic real leaf is a conjugate of one of the
following groups:

  * •

$V=\left\\{\pm\begin{pmatrix}1&b\\\
0&a\end{pmatrix}:a,b\in\mathbb{R},a>0\right\\}$ if the ratios of periods do
not all belong to a fixed quadratic number field;

  * •

$V_{D,\tau}=\left\\{\pm\begin{pmatrix}u&b\\\
0&a\end{pmatrix}:a,b\in\mathbb{R},a>0,u\in G_{D,\tau}\right\\}$ if up to
scaling $\Gamma=t\mathbb{Z}+(l+m\gamma)\mathbb{Z}$ for some triple of integers
$\tau=(t,l,m)$ with $m\neq 0$ and $\gcd(t,l,m)=1$, and $\gamma$ is the
standard generator of the ring of integers in $\mathbb{Q}(\sqrt{D})$ for some
square free integer $D\geq 2$.

###### Proof.

Indeed, any element in the Veech group for the leaf with absolute period group
contained in $\mathbb{R}$ must preserve the real line. Such an element is of
the form $\begin{pmatrix}a&b\\\ 0&c\end{pmatrix}$ where
$a,c\in\mathbb{R}^{\ast}$ and $b\in\mathbb{R}$. Besides, elements of the Veech
group have to map $\mathbb{Z}+\frac{1}{\tan(\theta)}\mathbb{Z}$ to itself. In
other words, scaling by $a$ preserve group $\Gamma$. Unless
$\frac{1}{\tan(\theta)}$ satisfies a quadratic equation with integer
coefficients, this forces $a$ to be either $1$ or $-1$. We obtain precisely
the group $V$. Conversely, any element in $V$ leaves each chamber invariant
and its boundary fixed, so is indeed in the Veech group. In the remaining
cases, ratios of absolute periods belong to a quadratic number field
$\mathbb{Q}(\sqrt{D})$ where $D\geq 2$ is a square-free integer. We can
consider, up to a scaling of $\chi$, that $\Gamma$ is of the form
$\Gamma=t\,\mathbb{Z}+(l+m\gamma)\,\mathbb{Z}$. It follows from Lemma 7.4 that
the coefficient $a$ belongs to $G_{D,\tau}$; provided $a>0$. Conversely, any
element in $V_{D,\tau}$ simply permute the chambers. ∎

######  Remark 7.6.

Note that it is possible $G_{D,\tau}=\\{1\\}$, e.g. we may choose $D=m=3$, in
which case $\beta$ is never a multiple of $3$, so the second group can be the
same as the first for some parameters. On the other hand, Lemma 7.4 gives a
criterion to construct parameters where this group is nontrivial.

##  8\. Arithmetic real leaves

In this section, we give a complete description of arithmetic real leaves. In
this case, the absolute period group $\Gamma=\mathbb{Z}a$ for some nonzero
complex number $a$. Up to the $\mathrm{GL}^{+}(2,\mathbb{R})$-action, we may
assume $\Gamma=\mathbb{Z}$.

We have discussed chambers of cylinder type in an arithmetic real leaf in §4,
and it is easy to see that chambers of degenerate type do not exist in such a
leaf. So it remains to understand when and how chambers of cylinder type are
glued together.

###  8.1. Degeneration towards boundary

We first discuss how surfaces in a chamber of cylinder type degenerate as we
approach the boundary of the chamber. For each positive integer $k$ and any
integer $l$ so that $0\leq l<k$ and $\gcd(k,l)=1$, there are two chambers of
cylinder type in the marked stratum, each is a half plane bounded by
$\mathbb{R}$. One of them, denoted by $CC^{+}_{k,l}$, corresponds to the
parameters $u=k$ and $v=l$, while the other, denoted by $CC^{-}_{k,l}$,
corresponds to $u=-k$ and $v=-l$. In the coordinate $z$, $CC^{+}_{k,l}$ is the
lower half plane, while $CC^{-}_{k,l}$ is the upper half plane. They are
images of each other under $z\mapsto-z$, so we focus on $CC^{+}_{k,l}$.

To simplify our discussion, let $S(l_{1},l_{2},l_{3})$, where
$l_{1},l_{2},l_{3}>0$, be the translation surface of degenerate type obtained
as follows. Cut a complex plane along a horizontal slit of length
$l_{1}+l_{2}+l_{3}$. The upper slit is divided into three segments from left
to right: segment $A$ of length $l_{1}$, segment $B$ of length $l_{2}$, and
segment $C$ of length $l_{3}$. The lower slit is similarly divided into three
segments from left to right: segment $C^{\prime}$ of length $l_{3}$, segment
$B^{\prime}$ of length $l_{2}$, and segment $A$ of length $l_{1}$. Identify
$A$ with $A^{\prime}$, $B$ with $B^{\prime}$, and $C$ with $C^{\prime}$ by
translations we obtain a translation surface in $\mathcal{H}(1,1,-2)$.

Notice that if we allow one of $l_{i}$’s to be zero, we obtain a surface in
$\mathcal{H}(2,-2)$.

#### The case $\bm{l=0}$

Then $k=1$. Consider the surface with coordinate $z=n+t+i\epsilon$, where
$n\geq 1$ is an integer, $0\leq t<1$, and $\epsilon>0$. Letting $\epsilon\to
0$, we obtain a surface of degenerate type, and it is easy to see that it is
precisely $S(t,1-t,n+t)$.

The point $z=0$ does not lie in the leaf it self, but its metric completion.
It corresponds to a pinched torus: the surface is obtained by identifying two
points on a complex plane. This gives the nodal curve in the Deligne-Mumford
compactification of $\mathcal{M}_{1,1}$.

Translation surfaces corresponding to coordinates with negative real part can
be described similarly – they are, in fact reflections of those with positive
real part. In particular, the nonzero integers on $\mathbb{R}$ correspond to
points in $\mathcal{H}(2,-2)$.

#### The case $\bm{l\neq 0}$

Consider the surface with coordinate $z=t+i\epsilon$, so the diagonal of the
cylinder is given by $z-v=t-l+i\epsilon$. We consider four cases.

  * •

$l-k<t<0$ In this case, $-k<\mathfrak{Re}(z-v)<-l$. Taking $\epsilon\to 0$,
the cylinder degenerates, and we obtain the surface $S(|t|,k-l-|t|,l+|t|)$.

  * •

$0<t<l$ In this case, $-l<\mathfrak{Re}(z-v)<0$. Letting $\epsilon\to 0$, we
get the surface of degenerate type $S(k+t-l,l-t,t)$.

  * •

$-(n+1)k+l<t<-nk+l$ for some integer $n\geq 1$ In this case,
$-(n+1)k<\mathfrak{Re}(z-v)<-nk$. Taking $\epsilon\to 0$, we get the surface
of degenerate type $S(|t|,(n+1)k-l-|t|,l+|t|-nk)$.

  * •

$nk+l<t<(n+1)k+l$ for some integer $n\geq 0$ In this case,
$nk<\mathfrak{Re}(z-v)<(n+1)k$. Taking $\epsilon\to 0$, we get the surface of
degenerate type $S(t-l-nk,(n+1)k+l-t,t)$.

In particular, the points $nk+l$ for any integer $n$ together with $0$ on the
real line correspond to surfaces in $\mathcal{H}(2,-2)$.

The description for the boundary of the chamber $CC^{-}_{k,l}$ can be obtained
by taking $w\mapsto-w$. Note that for surfaces of degenerate type on the
boundary, taking $w\mapsto-w$ yields the same surface but with a different
marking of its zeros.

###  8.2. Gluing of chambers

We now describe how different chambers are glued together. Fix a pair of
integers $(k,l)$ with $0\leq l<k$ and $\gcd(k,l)=1$. As discussed above, the
corresponding cylinder chamber $CC^{+}_{k,l}$ is the lower half plane bounded
by the real line, which is divided by the points $nk+l$ for $n\in\mathbb{Z}$
and $0$ into segments.

First consider the segment $[l-k,0]$ on the boundary of $CC^{+}_{k,l}$. Let
$k^{\prime}=k-l$, $n^{\prime}$ be the unique integer satisfying
$\frac{2l-k}{k-l}<n^{\prime}\leq\frac{l}{k-l},$ and
$l^{\prime}=(n^{\prime}+1)l-n^{\prime}k$. Then the chamber
$CC^{-}_{k^{\prime},l^{\prime}}$ has a boundary segment
$[-(n^{\prime}+1)k^{\prime}-l^{\prime},-n^{\prime}k^{\prime}-l^{\prime}]=[l,k]$.
One can easily check from our discussion in the last section that the
corresponding surfaces on this segment match those on $[l-k,0]$ of
$CC^{+}_{k,l}$.

Next consider the segment $[-(n+1)k+l,-nk+l]$ for some natural number $n\geq
1$. Set $k^{\prime}=(n+1)k-l$ and $l^{\prime}=k$. The chamber
$CC^{-}_{k^{\prime},l^{\prime}}$ has a boundary segment
$[-l^{\prime},0]=[-k,0]$, where the corresponding flat surfaces match those on
$[-(n+1)k+l,-nk+l]$ of $CC^{+}_{k,l}$.

For the segment $[0,l]$ (which does not exist for $l=0$), set $k^{\prime}=l$,
$l^{\prime}=(n^{\prime}+1)l-k$ where $n^{\prime}$ is the unique integer $\geq
k/l-1$ and $<k/l$. It is easy to see that flat surfaces on the segment
$[n^{\prime}k^{\prime}-l^{\prime},(n^{\prime}+1)k^{\prime}-l^{\prime}]=[k-l,k]$
on the boundary of $CC^{-}_{k^{\prime},l^{\prime}}$ match those on $[0,l]$ of
$CC^{+}_{k,l}$. In fact, it is the “inverse" of the case above.

For the segment $[nk+l,(n+1)k+l]$, where $n$ is a nonnegative integer, set
$k^{\prime}=(n+1)k+l$ and $l^{\prime}=nk+l$. Then it is easy to see that flat
surfaces on the segment $[0,k^{\prime}-l^{\prime}]=[0,k]$ on the boundary of
$\mathcal{C}^{-}_{k^{\prime},l^{\prime}}$ match those on $[nk+l,(n+1)k+l]$ of
$\mathcal{C}^{+}_{k,l}$. In fact, it is the “inverse" of the first case.

This description of gluing implies the following:

######  Lemma 8.1.

For any period character $\chi$ such that $\Gamma=\mathbb{Z}$, the locus in
marked stratum $\mathcal{H}(1,1,-2)$ realizing character $\chi$, up to a
symplectic change of basis, is connected.

###### Proof.

Since $CC^{+}_{1,0}$ is glued to $CC^{-}_{1,0}$ along the segment $[-1,1]$, we
only need to show that via gluing we can reach any pair of admissible indices
$(k,l)$ from $(1,0)$. We note that from $(k,l)$, we can then reach
$((n+1)k-l,k)$ for any natural number $n$.

We prove this by induction on $l$. Since we can reach $(n+1,1)$ for any
natural number $n$ from $(1,0)$, the statement holds for $l=1$.

Suppose the statement holds for any $l\leq l_{0}$. For $l=l_{0}+1$, take
$t\leq l_{0}$ with $\gcd(t,l)=1$. By inductive hypothesis, $(l,t)$ can be
reached by gluing from $(1,0)$. Then we can also reach $(nl+(l-t),l)$ for any
natural number $n$. This in fact gives all admissible pairs with second index
$l$. So the statement also holds for $l_{0}+1$. By induction, any admissible
pair of index can be reached by gluing, as desired. ∎

###  8.3. Geometry of arithmetic real leaves

We start with the following:

######  Lemma 8.2.

The singularities of any arithmetic real leaf in the marked stratum
$\mathcal{H}(1,1,-2)$ all have cone angle $6\pi$.

###### Proof.

Note that by the description of gluing, every singularity comes from the
points $nk+l$ and $0$ on the boundary of $CC^{+}_{k,l}$, or $nk-l$ and $0$ on
the boundary of $CC^{-}_{k,l}$. Moreover, it is easy to see that every such
point is glued to $0$ on the boundary of some chamber. So it suffices to show
that $6$ chambers are glued at $0$ for each $CC^{+}_{k,l}$.

It is then easy to check the following, based on our description of gluing:

  * •

When $2l-k>0$, the point $0$ on $\partial CC^{+}_{k,l}$ is glued to the point
$-l$ on $\partial{CC}^{-}_{k^{\prime},l^{\prime}}$ where $k^{\prime}=k-l$ and
$l^{\prime}=(n^{\prime}+1)l-n^{\prime}k$ for the unique integer
$n^{\prime}\in(\frac{2l-k}{k-l},\frac{l}{k-l}]$, as well as the point $-k+l$
on $\partial{CC}^{+}_{l,2l-k}$, the point $0$ on $\partial{CC}^{-}_{k,l}$, $l$
on $\partial{CC}^{+}_{k^{\prime},l^{\prime}}$, and $k-l$ on
$\partial{CC}^{-}_{l,2l-k}$. So $6$ chambers are glued here.

  * •

When $2l-k<0$, the point $0$ on $\partial{CC}^{+}_{k,l}$ is glued to the point
$-l$ on $\partial{CC}^{-}_{k-l,l}$, as well as the point $-k+l$ on
$\partial{CC}^{+}_{l,l^{\prime}}$ where $l^{\prime}=(n^{\prime}+2)l-k$ for the
unique integer $n^{\prime}\in[k/l-2,k/l-1)$, the point $0$ on
$\partial{CC}^{-}_{k,l}$, $l$ on $\partial{CC}^{+}_{k-l,l}$, and $k-l$ on
$\partial{CC}^{+}_{l,l^{\prime}}$. So $6$ chambers are glued here.

  * •

When $k=2l$, the only possible pair is $(k,l)=(2,1)$. In this case, $0$ on
$\partial{CC}^{+}_{2,1}$ is glued to both $-1$ and $1$ on
$\partial{CC}^{-}_{1,0}$, both $-1$ and $1$ on $\partial{CC}^{+}_{1,0}$, and
$0$ on $\partial{CC}^{-}_{2,1}$. Counting the chambers ${CC}^{\pm}_{1,0}$
twice, we still have $6$ chambers glued here.

We have dealt with all possibilities, so we are done. ∎

Recall that the point $0$ on the boundary of ${CC}^{\pm}_{1,0}$ corresponds a
pinched torus, so it does not lie on the leaf, but in its metric completion.
In this completion, it does not produce a singularity, since the two chambers
$CC^{\pm}_{1,0}$ are glued along the segment $[-1,1]$. We refer to the
corresponding point on the completion of the leaf as its _center_.

As in the case of positive and negative leaves, a real arithmetic leaf
supports an involution $\iota$, given by identifying ${CC}^{\pm}_{k,l}$ via
$w\mapsto-w$. We have

######  Lemma 8.3.

The quotient of the completion of the leaf with absolute period group
$\mathbb{Z}$ by the involution $\iota$ is a complex disk. The projection is
ramified at the cone points and the center.

###### Proof.

The only statement that requires a proof is the topology of the quotient. The
others follow immediately from the description of gluing and the definition of
$\iota$.

Under $\iota$, $CC_{k,l}^{+}$ is identified with $CC_{k,l}^{-}$, so we will
use $CC_{k,l}$ to denote the corresponding chamber in the quotient. Starting
with $CC_{1,0}$, we can obtain the whole leaf inductively as follows.

  * •

We start by gluing the segment $[0,1]$ to $[-1,0]$ on $\partial CC_{1,0}$.
This creates a cone angle of $\pi$ at the center. Let the resulting surface be
$\mathcal{L}_{0}$. The boundary of $\mathcal{L}_{0}$ is topologically a line,
being images of $[1,\infty)$ and $(-\infty,-1]$ on $\partial CC_{1,0}$, where
$\pm 1$ maps to the same point $V^{0}$.

  * •

The segment $[-1,1]$ on $\partial\mathcal{CC}_{2,1}$ is glued to
$\mathcal{L}_{0}$, so that $0$ on $\partial\mathcal{CC}_{2,1}$ is glued to
$V^{0}$. Let this new surface be $\mathcal{L}_{1}$. It has two boundary
components, and each boundary component consists of two rays coming from the
boundary of two different chambers meeting at a point. Let this point be
$V^{1}_{1}$ and $V^{1}_{2}$ for the two components.

  * •

Inductively, the surface $\mathcal{L}_{n}$ has $2^{n}$ boundary components,
each with a distinguished point $V^{n}_{i}$. The next surface
$\mathcal{L}_{n+1}$ is obtained from $\mathcal{L}_{n}$ by gluing the origin of
same chamber $CC_{k_{i},l_{i}}$ to $V^{n}_{i}$ for each $i$. This creates
$2^{n+1}$ boundary components, each with a distinguished point $V^{n+1}_{j}$.

It is easy to see that the resulting surface
$\mathcal{L}_{\infty}=\bigcup_{n\geq 0}\mathcal{L}_{n}$ is thus topologically
a plane. In fact, the set of surfaces of degenerate type in the quotient form
a binary tree rooted at the center; see Figure 9. The branching points of the
tree are precisely the zeros of the quadratic differential. If we identify the
ends of the tree with a Cantor set, the chambers can be identified with the
intervals complementary to the Cantor set.

In terms of complex structures, the completion of the leaf is either a complex
plane or a complex disk. It follows from Corollary 3.8 that the leaf itself
must be a punctured disk, so the completion is biholomorphic to a disk. ∎

Figure 9. A portion of the arithmetic leaf in the unmarked stratum. Numbers in
read are (relative) lengths of the segments. The wall is a binary tree,
consisting of surfaces of degenerate type. The root and branching points of
the binary tree are the pole and the zeros, respectively, of the quadratic
differential defining the flat structure of the leaf

The gluing described above implies that real arithmetic leaves satisfy the
properties of Theorem 1.3. Indeed, applying Lemma 5.1 again, we conclude that
the leaf is a Loch Ness Monster. Moreover, in the quotient by involution, the
simple pole of the quadratic differential defining the flat structure comes
from the center of the leaf.

Again, we remark that while this gives the same topology as positive and
negative leaves, the translation structure is very different.

###  8.4. Veech group

Finally, we determine the Veech group of an arithmetic real leaf as a
translation surface, i.e. we conclude the proof of Theorem 1.4.

######  Proposition 8.4.

The Veech group of an arithmetic real leaf is a conjugate of the following
group:

$V=\left\\{\pm\begin{pmatrix}1&b\\\
0&a\end{pmatrix}:a,b\in\mathbb{R},a>0\right\\}.$

###### Proof.

Indeed, any element in the Veech group must preserve the real line, and maps
$\mathbb{Z}$ to itself. This gives precisely the group $V$. Conversely, any
element in $V$ leaves each chamber invariant and its boundary fixed, so is
indeed in the Veech group. ∎

##  9\. Conformal geometry of leaves

In this section, we discuss the conformal geometry of leaves in the unmarked
stratum. This is written as a separate section to include more details without
diverging too much from the main focus in previous sections.

###  9.1. Biholomorphic maps to the Teichmüller space

#### Generic case

In the generic case, we construct an explicit map from each leaf to the
Teichmüller space $\mathcal{T}_{1,1}$, which is a lift of the map to
$\mathcal{M}_{1,1}$.

Suppose first that the period group $\Gamma$ is a lattice of $\mathbb{C}$. For
simplicity, we may assume $\Gamma=\mathbb{Z}[i]$. Given a point $(X,\omega)$
on the leaf, we have an isomorphism $H_{1}(X,\mathbb{Z})\cong\mathbb{Z}[i]$
via integration of $\omega$. Moreover, if we stay on a positive (resp.
negative) leaf, this isomorphism also preserves (reverses) the symplectic
pairing. Thus each point is a _homologically marked_ translation surface. In
genus one, the space of homologically marked elliptic curve is the same as the
Teichmüller space, as the Torelli group is trivial.

Clearly the map defined above is injective: if $(X_{1},\omega_{1})$ and
$(X_{2},\omega_{2})$ are two translation surfaces giving the same marked
surface, then $X_{1}=X_{2}$, and $\omega_{1}-\omega_{2}$ is an exact
holomorphic 1-form. Thus $\omega_{1}=\omega_{2}$ as well. The map is also
surjective. Indeed, given any elliptic curve
$\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ where $\tau\in\mathbb{H}^{2}$, a
holomorphic differential with a double pole is given by
$\omega=(a+b\wp_{\tau})dz$, where $\wp_{\tau}$ is the Weierstrass
$\wp$-function. Solving for $a,b$ in

(9.1)
$\int_{0}^{1}(a+b\wp_{\tau})dz=1\qquad\textnormal{and}\qquad\int_{0}^{\tau}(a+b\wp_{\tau})dz=\pm
i$

we obtain a differential with period group $\mathbb{Z}[i]$.

#### Non-arithmetic real case

Essentially the same argument works for the non-arithmetic real case as well:
without loss of generality we may assume $\Gamma=\mathbb{Z}+\omega\mathbb{Z}$,
where $\omega$ is a irrational number. Replacing $\pm i$ with $\omega$ in the
arguments above, we obtain the desired map.

#### Arithmetic real case

Next consider the case of arithmetic real leaves. Again, for simplicity, we
may assume $\Gamma=\mathbb{Z}$. Recall that the leaf $\mathcal{L}$ with period
group $\Gamma$ is biholomorphic to a punctured disk. We can construct a
covering map from $\mathcal{T}_{1,1}$ to $\mathcal{L}$ as follows. Indeed, for
any $\tau\in\mathbb{H}^{2}$, we can solve for $a,b$ in

$\int_{0}^{1}(a+b\wp_{\tau})dz=0\qquad\textnormal{and}\qquad\int_{0}^{\tau}(a+b\wp_{\tau})dz=1$

as above. This determines a point $(X,\omega)$ on the leaf with
$X\cong\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ and $\omega=a+b\wp_{\tau}$.
However, since the horizontal period is zero, it is easy to see that the
translation surface we obtain for $\tau$ is the same as that for $\tau+1$. In
particular, the map described above from
$\mathcal{T}_{1,1}\cong\mathbb{H}^{2}$ to $\mathcal{L}$ descends to a map from
$\mathbb{H}^{2}/\langle z\mapsto z+1\rangle$ to $\mathcal{L}$.

We claim this latter map is bijective. Indeed, an inverse can be constructed
as follows. Given $(X,\omega)$, we can choose a symplectic basis
$\alpha,\beta$ for $H_{1}(X,\mathbb{Z})$ so that $\int_{\alpha}\omega=0$ and
$\int_{\beta}\omega=1$. This gives a marking on $H_{1}(X,\mathbb{Z})$. Given
$\alpha$, the choice of $\beta$ is not unique, since we can always replace
$\beta$ by $\beta+n\alpha$ for any integer $n$. So we obtain a map from
$\mathcal{L}$ to $\mathbb{H}/\langle z\mapsto z+1\rangle$, which is clearly
the inverse of the map defined in the previous paragraph.

###  9.2. The Teichmüller geometry of cylinder chambers

Recall that the Teichmüller space $T(S)$ is endowed with a mapping class group
invariant complete metric $d_{T}$ called the Teichmüller metric. Given two
points in Teichmüller space $T(S)$ represented by $f:S\to X$ and $g:S\to Y$,
the Teichmüller distance is given by

(9.2) $d_{T}([f],[g]):=\frac{1}{2}\log\inf\\{K:h:X\to Y\text{ is
$K$-quasiconformal, and homotopic to }g\circ f^{-1}\\}.$

For $S=S_{1,1}$, the Teichmüller space
$\mathcal{T}_{1,1}=\mathcal{T}(S_{1,1})$ is bi-holomorphic to
$\mathbb{H}^{2}$, and the Teichmüller metric there agrees with the hyperbolic
metric (up to a constant multiple).

Recall that every cylinder chamber $\mathcal{C}$ is a half plane, whose
boundary is parallel to $\mathbb{R}u$ for some complex number $u$. Denote by
$X_{t}$ the point in Teichmüller space $\mathcal{T}_{1,1}$ corresponding to
$tu$ on the boundary of $\mathcal{C}$. Since the each leaf is biholomorphic to
(the quotient) of $\mathcal{T}_{1,1}\cong\mathbb{H}^{2}$, we may map
$\mathcal{C}$ conformally (and hence isometrically) to a region in the upper
half plane $\mathbb{H}^{2}$ so that the limit point at infinity of the chamber
is $\infty$, and $X_{0}$ corresponds to the point $i$. Let
$\sigma_{\mathcal{C}}(t)$ denotes the point $X_{t}$ corresponds to.

The main goal of this section is to prove the following:

######  Theorem 9.1.

For any cylinder chamber $\mathcal{C}$, we have $\sigma_{\mathcal{C}}(t)\asymp
t+i\log t$. More precisely, there exist constants $D,T$ depending only on the
leaf containing $\mathcal{C}$ so that
$d_{\mathbb{H}^{2}}(\sigma_{\mathcal{C}}(t),t+i\log t)\leq D$ when $|t|\geq
T$.

In particular, $\mathcal{C}$ does not contain any horodisks, and is also not
contained in a $D$-neighborhood of some complete Teichmüller geodesic for any
$D>0$.

As before, for simplicity, we assume the period group $\Gamma=\mathbb{Z}[i]$.
The result for other leaves can be obtained via the
$\mathrm{GL}^{+}(2,\mathbb{R})$-action.

First consider the torus obtained as follows. Let $P$ be the unit square
$[0,1]\times[0,1]$ and $Q$ the rectangle $[0,1]\times[0,s]$ for some $s>0$. As
in §4, we obtain a torus $T_{s}$ by gluing the vertical sides of $P$, the
vertical sides of $\hat{\mathbb{C}}-Q$, and the corresponding horizontal sides
of $P$ and $\hat{\mathbb{C}}-Q$. By horizontal and vertical reflection
symmetry, it is easy to see that $T_{s}$ corresponds to a point $i\tau(s)$ on
the imaginary axis of $\mathcal{T}_{1,1}\cong\mathbb{H}^{2}$. We can estimate
$\tau(s)$ as follows.

######  Lemma 9.2.

The function $\tau(s)$ satisfies $\tau(s)\asymp\log(s)$.

The lower bound can be obtained from the following result in [10]:

######  Theorem 9.3.

The modulus of the quadrilateral $Q^{c}=\hat{\mathbb{C}}-Q$, defined as the
extremal length of the path family in $\hat{\mathbb{C}}-Q$ connecting top and
bottom sides of $Q$, satisfies
$\operatorname{Mod}(Q^{c})\sim\frac{1}{\pi}\log(s)$.

Here $f(s)\sim g(s)$ means $\lim f(s)/g(s)=1$, so the $\frac{1}{\pi}\log(s)$
gives the exact asymptotics of $\operatorname{Mod}(Q^{c})$ as $s\to\infty$.

###### Proof of Lemma 9.2..

For the lower bound, note that $\tau(s)$ is equal to the extremal length of
the path family homotopic to the curve obtained by concatenation of a vertical
side of $P$ and a vertical side of $Q^{c}$. By additive property of extremal
length, we conclude that $\tau(s)$ is bounded below by the sum of the moduli
of $P$ and $Q^{c}$. Since the modulus of $P$ is $1$, the theorem quoted above
implies $\tau(s)\geq 1+\frac{1}{\pi}\log(s)$.

For the upper bound, we construct a quasiconformal map
$f:Q^{c}\to\mathbb{C}-B((s+1/2)i,1/2)-B(-1/2i,1/2)$ as follows. The
quadrilateral $Q^{c}$ is divided into several regions. Outside the red and
blue regions in Figure 10, the map $f$ is identity / translation. In the blue
regions, $f$ is a scaling in the $x$-direction.

Figure 10. The quasiconformal map

It remains to define $f$ in the two red regions. Over every boundary segment
of the upper red region except the top side of $Q$, $f$ is either already
defined, or can be defined easily as a translation. It also maps the top side
of $Q$ to the circle of radius $1/2$ centered at $(s+1/2)i$, via a map to be
determined below.

Consider the map $g(z)=\frac{\sqrt{s^{2}+s}+iz}{\sqrt{s^{2}+s}-iz}$. This maps
the disk $B((s+1/2)i,1/2)$ to $B(0,r(s))$, and the disk $B(-(s+1/2)i,1/2)$ to
the complement of $B(0,1/r(s))$, where $r(s)=2s+1-2\sqrt{s^{2}+s}$. Now we can
define $f$ on $[0,1]\times\\{s\\}$ as follows

$f(t)=g^{-1}(r(s)e^{2\pi i(1-t)}).$

It is easy to check that $f$ has uniformly bounded derivative on the boundary
of the red region, and hence can be extended to the red region as a
quasiconformal map with uniformly bounded dilation $K$. It can be similarly
defined on the lower red region.

Now $f$ is constructed so that $\frac{i}{2\pi}\log(g\circ f)$ maps $Q^{c}$ to
a rectangle of height $\frac{1}{\pi}\log(r(s))$ and width $1$, which maps the
top and bottom sides of $Q^{c}$ isometrically to the bottom and top sides of
this rectangle. Now we can glue a copy of $P$ to both rectangles, and obtain a
quasiconformal map from $T_{s}$ to a torus with modulus
$\frac{1}{\pi}\log(1/r(s))+1$. By property of quasiconformal maps, we then
have the modulus of $T_{s}$, given by $\tau(s)$ is bounded above by
$\frac{K}{\pi}\log(1/r(s))+K\asymp\log(s)$. This gives the upper bound. ∎

Of course, the same argument above also gives a lower bound, but by quoting
Theorem 9.3 we obtain the lower bound without an undetermined multiplicative
factor.

First suppose $\mathcal{C}$ is contained in a positive leaf. Recall that its
boundary is parallel to a primitive element $u\in\mathbb{Z}[i]$. Let $X_{t}$
be the torus corresponding to the point $tu$ on the boundary of the chamber
$\mathcal{C}$. Recall that this is constructed from the complement of a slit
of length $(|t|+1)|u|$ in $\hat{\mathbb{C}}$, and a parallelogram $P_{t}$ with
vertices at $0$, $u$, $tu+v$ and $(t+1)u+v$, where $(u,v)$ form a positive
basis for $\mathbb{Z}[i]$. Note that we may choose so that the projection of
$v$ onto $u$ has length shorter than $|u|$. Suppose the corresponding point in
the Teichmüller space $\mathcal{T}_{1,1}\cong\mathbb{H}^{2}$ is $\sigma(t)$.
We claim

######  Lemma 9.4.

There exists a constant $D>0$ independent of the chamber so that
$d_{\mathbb{H}^{2}}(\sigma(t),t+i\log|t|)\leq D$ for all $|t|$ large enough.

###### Proof.

It suffices to construct a quasiconformal map between $X_{t}$ and the torus
corresponding to $t+i\log(t)$.

For simplicity, assume $t>0$. The case $t<0$ is entirely similar. First note
that we can construct a quasiconformal map $h_{t}$ from the complement of the
slit to the complement of $Q=[0,1]\times[0,t]$; see Figure 11. We divide the
complement of the slit into pieces by rays perpendicular to or forming $45$
degree angles with the slit, as in Figure 11.

Figure 11. A piecewise affine quasiconformal map

The complement of $Q$ can be divided into pieces similarly. Note that we can
map each piece to one with the same color by an affine map. The only pieces
varying with $t$ are green ones, but we can map each green piece to a
corresponding one by a conformal map (rotation plus scaling). So the dilation
of $h_{t}$ is independent of $t$.

Following the same construction as before, $Q^{c}$ is quasiconformal to a
rectangle with modulus $\asymp\log(t)$. Thus the torus $X_{t}$ is
quasiconformal to the torus constructed by gluing $P_{t}$ to this rectangle;
see Figure 12. It is easy to see that up to a rotation and scaling by $1/|u|$,
the parallelogram $P_{t}$ has lengths labelled as in the figure. This new
torus corresponds to a point within bounded distance from $t+i\log(t)$, as
desired.

Figure 12. $X_{t}$ is quasiconformal to this torus

∎

Theorem 9.1 for cylinder chambers in a positive leaf then follows, since
$\mathrm{GL}^{+}(2,\mathbb{R})$ acts as quasiconformal maps between leaves, so
each group element only causes bounded distortion on Teichmüller distance.

For the negative leaf with period group $\mathbb{Z}[i]$, recall that each
cylinder chamber is again parametrized by a primitive element $u$ in
$\mathbb{Z}[i]$, whose boundary is parallel to $\mathbb{R}u$. The surface
$X_{t}$ corresponding to $tu$ is given by the complement of a parallelogram
$Q$ with vertices at $0,u,tu+v,(t+1)u+v$ and identification specified on the
left of Figure 13. Here $(u,v)$ again form a positive basis for
$\mathbb{Z}[i]$ where $v$ is chosen so that the projection of $v$ onto $u$ has
length shorter than $|u|$.

Figure 13. Illustration for negative leaves

As in the case of positive leaves, we can construct an explicit quasiconformal
map from $X_{t}$ to a surface obtained from a rectangle of width $1$ and
height $\asymp\log t$, with side identification specified on the right of
Figure 13. The dilation of the quasiconformal map is independent of $t$. The
closed curve with absolute period $v$ in $X_{t}$ is mapped to a curve
homotopic to the curve in the rectangle with slope $\log(t)/t$. Hence the
corresponding point in the Teichmüller space is of bounded distance from
$t+i\log(t)$, as in the positive case. This concludes Theorem 9.1 for cylinder
chambers in negative leaves.

Finally, the same construction works for arithmetic and non-arithmetic real
leaves as well.

###  9.3. Conjugacy of action of ${\mathrm{PSL}_{2}(\mathbb{Z})}$

Recall that ${\mathrm{SL}_{2}(\mathbb{Z})}$ acts as the Veech group of each
generic leaf. On the other hand, ${\mathrm{SL}_{2}(\mathbb{Z})}$ acts on
$\mathcal{T}_{1,1}$ by isometries.

######  Proposition 9.5.

The natural map defined in the previous section from a generic (unmarked) leaf
to $\mathcal{T}_{1,1}$ conjugates these two actions of
${\mathrm{SL}_{2}(\mathbb{Z})}$ on the boundary at infinity.

For simplicity, we focus on positive leaves. The case for negative leaves is
entirely similar. Without loss of generality, we may also assume the period
group is $\mathbb{Z}[i]$.

Recall that a cylinder chamber is parametrized by a primitive element
$u=p+qi\in Z[i]$, up to multiplication by $-1$. Here $p,q$ are relatively
prime integers. An element $\gamma\in{\mathrm{SL}_{2}(\mathbb{Z})}$ maps the
chamber $\mathcal{C}_{u}$ associated with $u$ to that associated with
$\gamma\cdot u=\gamma\cdot\begin{pmatrix}p\\\ q\end{pmatrix}$.

On the other hand, fixing a base surface $X\in\mathcal{T}_{1,1}$ and a
symplectic basis $(\alpha,\beta)$ for $H^{1}(X,\mathbb{Z})$, each point
$p/q\in\mathbb{Q}\cup\\{\infty\\}$ on $\partial\mathbb{H}^{2}$ corresponds to
a pinched hyperbolic surface where the simple closed curve $p\alpha+q\beta$ is
pinched. Moreover, an element $\gamma\in{\mathrm{SL}_{2}(\mathbb{Z})}$ maps
$p/q$ to $\gamma\cdot p/q=\gamma\cdot\begin{pmatrix}p\\\ q\end{pmatrix}$.

Thus to prove Proposition 9.5, it suffices to show that the point at infinity
of $\mathcal{C}_{u}$ is precisely $p/q$, noting the density of
$\mathbb{Q}\cup\\{\infty\\}$ in
$\partial\mathbb{H}^{2}=\mathbb{R}\cup\\{\infty\\}$ and the continuity of
actions. But the discussions in the previous section imply that the point at
infinity of $\mathcal{C}_{u}$ corresponds to pinching the curve with period
$u$. If we choose $\alpha,\beta$ to be the curves with periods $1$ and $i$
respectively, then the curve with period $u$ is given by $p\alpha+q\beta$. We
then have the desired result by the discussions above.

##  Appendix A Equilateral triangulation, true trees and conformal structure
of arithmetic leaves

In this appendix, we describe another perspective of the conformal structure
of arithmetic leaves, inspired by recent work of Ivrii, Lin, Rohde, and Sygal
[20].

#### Equilateral triangulation and true trees

A _Belyi function_ is a holomorphic map $f$ from a Riemann surface $X$ to
$\hat{\mathbb{C}}$ with only $3$ critical values (usually taken to be
$0,1,\infty$). Equivalently, $X$ admits an _equilateral triangulation_ , by
taking the inverse images under $f$ of the upper and lower half planes in
$\mathbb{C}$. Conformally, we may reconstruct $X$ by gluing identical pieces
of equilateral triangles according to the combinatorics given by the
triangulation.

Note that not every Riemann surface admits an equilateral triangulation. A
classical result of Belyi states that a compact Riemann surfaces admits an
equilateral triangulation iff it is algebraic [2]. On the other hand, by a
recent result of Bishop and Rempe, every noncompact Riemann surface admits an
equilateral triangulation [3].

If $X=\hat{\mathbb{C}}$ and the Belyi function $f$ is a polynomial, then it is
also called a _Shabat polynomial_. It is easy to see that in this case
$f^{-1}([0,1])$ is a tree. Such a tree is called a _true tree_ , and up to
Möbius transformations, every finite tree in the plane is equivalent to a
unique true tree.

True trees are _conformally balanced_ , in the sense that the harmonic measure
seen from infinity of every edge on either side is the same. Equivalently, the
Riemann mapping from the complement of a true tree in $\hat{\mathbb{C}}$ to
$\hat{\mathbb{C}}-\mathbb{D}$ fixing $\infty$ maps each side of the edges to
an circular arc of equal length.

Let $\mathcal{L}$ be the (unmarked) arithmetic real leaf with period group
$\mathbb{Z}$. Then $\mathcal{L}$ admits an equilateral triangulation. Indeed,
each chamber is identified with the upper half plane, which can be tiled by
vertical half strips bounded by vertical rays starting at integers. Mapping
the vertex at infinity of each triangle to $\infty$, we get the corresponding
Belyi function $f:\mathcal{L}\to\mathbb{C}$. The preimage $f^{-1}([0,1])$
gives precisely the walls of the chambers, shown in Figure 9.

#### Approximation by finite trees

Let $\mathcal{T}_{\infty}$ be the trivalent infinite tree with a distinguished
vertex $v$, and $\mathcal{T}_{n}$ be the subtree induced by all vertices of
distance $n$ from $v$. Denote also by $\mathcal{T}_{n}$ the conformally
balanced form of the tree, normalized so that the Riemann mapping
$\varphi_{n}:\hat{\mathbb{C}}-\mathbb{D}\to\hat{\mathbb{C}}-\mathcal{T}_{n}$
has expansion $z\mapsto z+O(1/z)$ at infinity. Then the main results in [20]
state:

  * •

The trees $\mathcal{T}_{n}$ converge in the Hausdorff topology to an infinite
trivalent tree union a Jordan curve $\mathcal{T}_{\infty}\cup\partial\Omega$;

  * •

The corresponding Shabat polynomials $p_{n}$ converge to a Belyi function
$F:\Omega\to\hat{\mathbb{C}}$, so that $\mathcal{T}_{\infty}=F^{-1}([0,1])$;

  * •

Each connected component $\mathcal{C}$ of $\Omega-\mathcal{T}_{\infty}$ is a
disk, whose boundary meets $\partial\Omega$ at a unique point and otherwise
consists of edges of $\mathcal{T}_{\infty}$.

  * •

More precisely, the domain $\Omega$ is the so-called _developed deltoid_ ,
studied in [23].

In particular, $\Omega$ is conformally equivalent to the surface obtained by
gluing half planes along the infinite tree $\mathcal{T}_{\infty}$, in the
spirit of Figure 9.

Inspired by this, we consider the following way of producing a conformal image
of the arithmetic leaf $\mathcal{L}$. Instead of looking at the rooted binary
tree in Figure 9, we look at a degree $3$ cover of it ramified over the root
vertex $v$. This tree is combinatorially the same as $\mathcal{T}_{\infty}$,
but each edge is assigned a weight according to its length in the period
coordinates, as marked in Figure 9. We still denote this tree by
$\mathcal{T}_{\infty}$. Similarly we have the level $n$ subtree
$\mathcal{T}_{n}$, and the corresponding weight function on its edges.

Let $\mathcal{T}_{n}^{*}$ be the tree obtained from $\mathcal{T}_{n}$ by
adding $t-1$ interior vertices to an edge with weight $t$. In this way, in the
corresponding true form, the harmonic measure of an arc between branching
points viewed from infinity is proportional to the weight of the corresponding
edge in $\mathcal{T}_{n}$. We similarly obtain $\mathcal{T}_{\infty}^{*}$.
Again, we use the same symbols $\mathcal{T}_{n}^{*},\mathcal{T}_{\infty}^{*}$
to denote the corresponding true trees, normalized so that the Riemann mapping
$\varphi_{n}^{*}:\hat{\mathbb{C}}-\mathbb{D}\to\hat{\mathbb{C}}-\mathcal{T}^{*}_{n}$
has expansion $z\mapsto z+O(1/z)$ at infinity. The estimates in [20] can be
adapted to prove the following:

  * •

A subsequence of the trees $\mathcal{T}_{n}^{*}$ converge in the Hausdorff
topology to $\mathcal{T}_{\infty}^{*}$ union a Jordan curve
$\partial\Omega^{*}$;

  * •

The corresponding Shabat polynomials $p_{n}^{*}$ converge to a Belyi function
$F^{*}:\Omega^{*}\to\hat{\mathbb{C}}$, so that
$\mathcal{T}_{\infty}^{*}={F^{*}}^{-1}([0,1])$;

  * •

Each connected component $\mathcal{C}$ of $\Omega-\mathcal{T}_{\infty}$ is a
disk, whose boundary meets $\partial\Omega^{*}$ at a unique point, and
otherwise consists of edges of $\mathcal{T}_{\infty}^{*}$.

  * •

In fact $\Omega^{*}$ is conformally equivalent to a degree $3$ cover of the
leaf $\mathcal{L}$, with the cusp at the center filled in.

See Figure 14 for the picture of a true tree approximating $\Omega^{*}$. The
proof of these statements follow the same strategy as [20], although due to a
break in symmetry, many estimates need to be adapted here. We leave the
details to an upcoming sequel.

Figure 14. True trees approximate the conformal structure of $\mathcal{L}$

In [20], as mentioned above, the domain $\Omega$ is identified with the
developed deltoid, which arises as the mating of the dynamics of the ideal
triangle group (an index 2 subgroup of ${\mathrm{PSL}_{2}(\mathbb{Z})}$) in
the interior and the anti-holomorphic map $z\mapsto\bar{z}^{2}$ in the
exterior (see [23]). They also prove the uniqueness of Hausdorff limit by
showing that every subsequential limit gives the same mating. Moreover,
because of the action of the ideal triangle group, the Belyi function
$F:\Omega\to\hat{\mathbb{C}}$ composed with the Riemann mapping
$R:\mathbb{D}\to\Omega$ is a modular function invariant under the group.

In our case, however, the different lengths of the edges break the symmetry of
the trivalent tree, and there is no natural ${\mathrm{PSL}_{2}(\mathbb{Z})}$
action (recall that the Veech group of the arithmetic leaf is not
${\mathrm{PSL}_{2}(\mathbb{Z})}$!). We are not sure what the exterior
conformal structure could be, and hence do not claim uniqueness of the
Hausdorff limit. On the other hand, it is clear that Figure 14 is homeomorphic
to the picture produced in [20]; see their Figure 1. Could $\Omega^{*}$ also
arise dynamically (e.g. as the mating of a Fuchsian group and an anti-
holomorphic map)? We hope to further investigate these questions in the
sequel.

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