# Computing Veech groups

Brandon Edwards  Intel Corporation JF2
2111 NE 25th Ave.
Hillsboro, OR 97124<EMAIL_ADDRESS>,  Slade Sanderson  Oregon
State University
Corvallis, OR 97331<EMAIL_ADDRESS>and  Thomas A. Schmidt  Oregon
State University
Corvallis, OR 97331<EMAIL_ADDRESS>
(Date: 22 December 2020)

###### Abstract.

We present an algorithm, based on work of the first named author, for
computing the Veech group, $\mathrm{SL}(X,\omega)$, of a translation surface
$(X,\omega)$. This presentation is informed by an implementation of the
algorithm by the second named author.

For each stratum of the space of translation surfaces, we introduce an
infinite translation surface containing in an appropriate manner a copy of
every translation surface of the stratum. We show that a matrix is in
$\mathrm{SL}(X,\omega)$ if and only if an associated affine automorphism of
the infinite surface sends each of a finite set, the “marked” Voronoi staples,
arising from orientation-paired segments appropriately perpendicular to
Voronoi 1-cells, to another pair of orientation-paired “marked” segments.
(“Segments” are also known as “saddle connections”.)

We prove a result that is of independent interest. For each real
$a\geq\sqrt{2}$ there is an explicit hyperbolic ball such that for any
Fuchsian group trivially stabilizing $i$, the Dirichlet domain centered at $i$
of the group already agrees within the ball with the intersection of the
hyperbolic half-planes determined by the group elements whose Frobenius norm
is at most $a$. When $\mathrm{SL}(X,\omega)$ is a lattice we use this to give
a condition guaranteeing that the full group $\mathrm{SL}(X,\omega)$ has been
computed.

###### Key words and phrases:

Veech group, Fuchsian group, Dirchlet domain

###### 2010 Mathematics Subject Classification:

37F30 (30F60 32G15 37D40 52C20)

###### Contents

  1. 1 Introduction
    1. 1.1 Veech groups
    2. 1.2 Computing Veech groups
  2. 2 Background
    1. 2.1 Translation surfaces
      1. 2.1.1 Three views of a translation surface.
      2. 2.1.2 Affine diffeomorphisms and the Veech group
      3. 2.1.3 Saddle connections, strata, matrix group action
      4. 2.1.4 Voronoi decomposition
  3. 3 A model surface, Voronoi staples and a Veech group membership criterion
    1. 3.1 Model translation surface, $\mathcal{O}$
    2. 3.2 Closed Voronoi 2-cells as convex bodies on $\mathcal{O}$
    3. 3.3 Voronoi staples determine $(X,\omega)$
    4. 3.4 Membership criterion
  4. 4 Nesting hyperbolic polygons agreeing on ever larger balls
    1. 4.1 Bounded Frobenius norm elements give Dirichlet domain within bounded ball
    2. 4.2 Recognizing a generating set
  5. 5 From theory to algorithm
    1. 5.1 Basic steps of algorithm
    2. 5.2 Some technical details
  6. 6 Verified calculations
    1. 6.1 An “L”-surface
    2. 6.2 Investigating an infinitely generated Veech group

##  1\. Introduction

Any non-zero holomorphic 1-form $\omega$ on a Riemann surface $X$ gives local
coordinates by integration, the result is a translation surface, see say [M].
There is a natural action by $\text{SL}_{2}(\mathbb{R})$ on the set of
translation surfaces, the stabilizer of a translation surface is its Veech
group. (See below for a definition in terms of intrinsic properties of the
surface.) We give a new algorithm for the computation of the Veech group of a
translation surface.

###  1.1. Veech groups

Various dynamical properties of the linear flow on a translation surface are
determined by its Veech group, [V]. Every Veech group is a non-cocompact
Fuchsian group [V]. By way of Teichmüller theory, when the group is a lattice,
there is a corresponding algebraic curve in Riemann moduli space,
$\mathcal{M}_{g}$ where $g$ is the genus of $X$, which is embedded with
respect to the Teichmüller metric [V]; Veech named such curves Teichmüller
curves, the translation surface is said to be a lattice surface. Veech also
showed that certain triangle groups arise as what we call Veech groups, Bouw-
Möller [BM] showed that up to finite index every non-cocompact Fuchsian
triangle group so arises. On the other hand, almost every translation surface
has a trivial Veech group [Mö]; and, whereas any polygon whose vertex angles
are rational multiples of $\pi$ can be “unfolded” to achieve a translation
surface, only three non-isosceles acute Euclidean triangles give lattice
surfaces [KS, P]. Although constraints on which non-cocompact Fuchsian groups
can be Veech groups, such as equality of trace field with invariant trace
field if the group contains a hyperbolic element (essentially a result of
[KS]) are known, a complete list of Fuchsian groups realized as Veech groups
remains to be determined.

###  1.2. Computing Veech groups

We give a new algorithm for computing Veech groups of translation surfaces
(compact and without boundary). For a lattice surface, our algorithm
completely computes the group. Our approach is different from the general
algorithms in the literature [Bow] (see also [V2]), [BrJ, Mu, SmW] (of which
it seems only the algorithm of Mukamel [Mu] has been implemented), as well as
from the special case algorithms [Sch, F]. As with most of these others, this
algorithm has two parts: determining group elements and building a fundamental
domain for the action of the group on the hyperbolic plane.

To determine elements, we view a translation surface as being assembled from
the disjoint union of its Voronoi 2-cells by identifying shared edges. We
introduce a model surface for each stratum into which the open Voronoi 2-cells
inject. There is a finite set of saddle connections whose images on the model
surface lead to a reconstruction of the original surface by identifying the
two edges lying on the perpendicular bisector of the image of a saddle
connection and that of its orientation-reversed saddle connection. We call
these pairs of saddle connections Voronoi staples, and show that an element of
$\mathrm{SL}_{2}(\mathbb{R})$ is in the Veech group exactly if it sends the
Voronoi staples on the model surface to copies of orientation-paired saddle
connections.

While we build a fundamental domain in a standard manner, we insist on finding
elements in increasing Frobenius norm. We show a result of independent
interest: For any Fuchsian group (trivially stabilizing $z=i$), the
intersection of the half-planes determined by the elements of any explicitly
bounded Frobenius norm agrees with the Dirichlet domain based at $z=i$ of the
group within a corresponding explicit hyperbolic ball.

Recall that the Dirichlet domain is the nested (appropriately decreasing)
limit of the convex bodies defined by intersecting the half-planes
appropriately defined by finite sets of elements. Thus, the new result is that
by taking the Frobenius norm ordering, there are also common domains which
converge (appropriately increasing) to the interior of the Dirichlet domain.
With this, we give a test for when the elements of a bounded Frobenius norm
generate the group. This test thus gives a stopping condition for our
algorithm, one that can only be fulfilled if the group is a lattice.

We have confirmed the computation of various Veech groups using the algorithm
[Sa]. Here we show use a simple example to illustrate matters (see Example 20,
Figure 6, and Subsection 6.1). We also compute elements within an infinitely
generated Veech group, finding that the first tens of thousands of elements in
order of Frobenius norm are all contained in a subgroup generated by three
elements (see Subsection 6.2 ).

This paper is based on the Ph.D. dissertation [E] of the first named author,
as well as the implementation by the second named author [Sa]. That
implementation was built upon Delecroix-Hooper’s “flat surface” computer code
package [DH] written in SageMath [S].

##  2\. Background

We introduce basic notation as well as remind the reader of some standard
results.

###  2.1. Translation surfaces

See, say, [M, Mö, W] for much of the following.

####  2.1.1. Three views of a translation surface.

A translation surface is a real surface $X$ such that on the complement of a
finite set of points $X\setminus\Sigma$ there is an atlas whose transition
functions are all translations, and such that the flat structure on this
complement extends to $X$ to have cone singularities of angles integral
multiples of $2\pi$. Equivalently, a translation surface is a Riemann surface
$X$ with a non-zero holomorphic differential (also called an abelian
differential) $\omega$ where $\Sigma$ is the set of zeros of $\omega$; due to
this view we often use the notation $(X,\omega)$ for a translation surface. A
third equivalent form is as a collection of polygons in the Euclidean plane,
with equal length parallel sides identified by translation (with the result an
oriented surface); singularities occur only at points arising from vertices.
We say $(X,\omega)$ in this form is polygonally presented.

It is common to change without warning from one to another of these
perspectives when discussing any given translation surface.

Note that it is sometimes useful to allow the inclusion of removable
singularities, thus points of cone angle $2\pi$, in $\Sigma$.

####  2.1.2. Affine diffeomorphisms and the Veech group

The group of translations of the plane is
$\mathrm{Trans}(\mathbb{R}^{2})\cong\mathbb{R}^{2}$. Recall that the affine
group of the plane is
$\mathrm{Trans}(\mathbb{R}^{2})\rtimes\mathrm{GL}_{2}(\mathbb{R})$. An affine
diffeomorphism of a translation surface is a homeomorphism $f:X\to X$ which
sends $\Sigma$ to itself, while inducing an affine diffeomorphism on
$X\setminus\Sigma$; that is, the maps on local coordinates (as induced in the
usual manner) are elements of the affine group of the plane. Under
composition, the set of all of these forms a group, $\mathrm{Aff}(X,\omega)$.
Because of the normality of translations in the affine group of the plane, the
linear part of each affine diffeomorphism $f$ is constant, independent of
choice of local coordinates, and thus there is a group homomorphism
$\mathrm{der}:\mathrm{Aff}(X,\omega)\to\mathrm{GL}_{2}(\mathbb{R})$. The image
of the subgroup of orientation-preserving affine diffeomorphisms,
$\mathrm{Aff}^{+}(X,\omega)$, is called the Veech group of $(X,\omega)$. We
denote the kernel of $\mathrm{der}$ by $\mathrm{Trans}(X,\omega)$.

Convention. Unless otherwise stated, the notation $(X,\omega)$ will denote a
closed translation surface: compact without boundary.

The flat structure on any translation surface allows Lebesgue measure on the
plane to induce a measure on the surface. When the translation surface is
compact this is a finite measure, and it is an observation of Veech [V] that
the Veech group is then a subgroup of $\mathrm{SL}_{2}(\mathbb{R})$. To
emphasize this, one denotes the Veech group by $\mathrm{SL}(X,\omega)$. Veech
[V] also showed that $\mathrm{SL}(X,\omega)$ is a discrete subgroup of
$\mathrm{SL}_{2}(\mathbb{R})$.

####  2.1.3. Saddle connections, strata, matrix group action

It is easy to see that the foliation of $\mathbb{R}^{2}$ by horizontal lines
induces a foliation on $X\setminus\Sigma$. The lines of this foliation meet at
points of $\Sigma$ of cone angle greater than $2\pi$ to give saddle
singularities.

Any geodesic ray emanating from a singularity is called a separatrix. A
separatrix joining two singularities (with no other singularity in its
interior) is called a saddle connection, or simply as in [KS] a segment.
Integration along a segment results in the corresponding element of
$\mathbb{C}$, called a holonomy vector.

The collection of all (closed) translation surfaces $(X,\omega)$ with the
holomorphic 1-form $\omega$ having zeros of orders $d_{1},\dots,d_{s}$ (with
repetition allowed) is called a stratum and denoted
$\mathcal{H}(d_{1},\dots,d_{s})$. Note that the cone angle at a zero of order
$d$ is $2(d+1)\pi$. By either the Riemann-Roch Theorem or that of Gauss-
Bonnet, all translation surfaces of a fixed stratum have the same genus.

The group $\mathrm{GL}_{2}(\mathbb{R})$ acts on the collection of all
translation surfaces by way of post-composition with local coordinate
functions. This action preserves each stratum. In the setting of closed
translation surfaces, Veech showed that the
$\mathrm{SL}_{2}(\mathbb{R})$-stabilizer of $(X,\omega)$ under this action is
isomorphic to $\mathrm{SL}(X,\omega)$. Furthermore, for any
$M\in\mathrm{SL}_{2}(\mathbb{R})$, one has
$M^{-1}\mathrm{SL}(X,\omega)M=\mathrm{SL}(M\cdot(X,\omega)\,)$.

####  2.1.4. Voronoi decomposition

Each translation surface has a flat metric with conical singularities on it.
Masur-Smillie [MS] sketch the theory of Voronoi decompositions subordinate to
the set of singularities. A Voronoi 2-cell is an open, connected, set of
points which are closer to some singularity than to any other. The boundary of
each Voronoi 2-cell is a union of Voronoi 1-cells consisting of open geodesics
(each of whose points have exactly two distinct shortest paths to $\Sigma$),
meeting in Voronoi 1-cells (single points each having at least three distinct
shortest paths to $\Sigma$). Each compact $(X,\omega)$ is the union of
finitely many open Voronoi 2-cells (exactly one per each element of $\Sigma$),
finitely many Voronoi 1-cells and finitely many Voronoi 0-cells. We will refer
to the union of a Voronoi 1-cell and its two endpoint 0-cells simply as an
edge.

##  3\. A model surface, Voronoi staples and a Veech group membership
criterion

###  3.1. Model translation surface, $\mathcal{O}$

Given a stratum $\mathcal{H}(d_{1},\dots,d_{s})$, for each subscript $i$ let
$\mathcal{O}_{i}$ be the infinite translation surface
$(\mathbb{C},z^{d_{i}}\,dz)$ and let
$\mathcal{O}=\mathcal{O}(d_{1},\dots,d_{s})$ be the disjoint union of the
$\mathcal{O}_{i}$. Partition the index set $\\{1,\dots,s\\}$ corresponding to
repeated values of the $d_{i}$, with partition elements of cardinalities say
$n_{1},\dots,n_{t}$.

######  Lemma 1.

Each $\mathrm{Trans}(\mathcal{O}_{i})$ is a cyclic group of order $1+d_{i}$,
whose generator acts as a rotation of angle $2\pi$. The group
$\mathrm{Trans}(\mathcal{O})$ is generated by the product of permutation
groups $S_{n_{1}}\times\cdots\times S_{n_{t}}$ acting by change of index,
along with the direct product of the various
$\mathrm{Trans}(\mathcal{O}_{i})$.

###### Proof.

Any element of $\mathrm{Trans}(\mathcal{O}_{i})$ maps the singularity to
itself and off of this induces translations in local coordinates. These local
coordinates arise from the ramified covering $\mathcal{O}_{i}\to\mathbb{C}$
and thus any induced translation is in fact the identity. In other words,
$\mathrm{Trans}(\mathcal{O}_{i})$ restricts to
$\mathcal{O}_{i}\setminus\\{0\\}$ to be the deck transformation group of the
covering of the punctured complex plane. Thus,
$\mathrm{Trans}(\mathcal{O}_{i})$ is indeed a cyclic group of order $1+d_{i}$,
generated by a rotation of $2\pi$.

Acting by permutation of indices, $S_{n_{1}}\times\cdots\times S_{n_{t}}$
certainly gives a subgroup of $\mathrm{Trans}(\mathcal{O})$. Furthermore, the
trivial extension to $\mathcal{O}$ of the action of
$\mathrm{Trans}(\mathcal{O}_{i})$, acting as the identity on all other
$\mathcal{O}_{j}$, gives an injection of $\mathrm{Trans}(\mathcal{O}_{i})$
into $\mathrm{Trans}(\mathcal{O})$. Any affine diffeomorphism $\varphi$ of
$\mathcal{O}$ must preserve cone angles, and hence composition with some
element of $S_{n_{1}}\times\cdots\times S_{n_{t}}$ gives an affine
diffeomorphism that sends each $\mathcal{O}_{i}$ to itself. If furthermore
$\varphi\in\mathrm{Trans}(\mathcal{O})$ then so is this composition, and hence
for each $i$ it restricts to give an element of
$\mathrm{Trans}(\mathcal{O}_{i})$. Thus, the result holds. ∎

######  Lemma 2.

For each $A\in\mathrm{GL}_{2}\mathbb{R}$ there is an affine diffeomorphism
$f_{A}\in\mathrm{Aff}(\mathcal{O})$ such that $\mathrm{der}(f_{A})=A$.

###### Proof.

We first briefly sketch the result in the setting of $(\mathbb{C},dz)$, see
Figure 1. The surface $A\cdot(\mathbb{C},dz)$ is formed by post-composing the
chart maps of $(\mathbb{C},dz)$ with $A$, using the standard action of
$\mathrm{GL}_{2}\mathbb{R}$ on $\mathbb{C}\cong\mathbb{R}^{2}$. The identity
map on $\mathbb{C}$ gives $\mathrm{Id_{\mathbb{C}}}:(\mathbb{C},dz)\to
A\cdot(\mathbb{C},dz)$, of Jacobian equal to $A$. Similarly, the map sending
points of $\mathbb{C}\cong\mathbb{R}^{2}$ to their corresponding image under
$A$ induces a diffeomorphism $F_{A}:A\cdot(\mathbb{C},dz)\to(\mathbb{C},dz)$
of Jacobian equal to $I_{2}$, the identity matrix. Thus the composition
$F_{A}\circ\mathrm{Id_{\mathbb{C}}}$ is an affine diffeomorphism whose
Jacobian equals $A$.

${(\mathbb{C},dz)}$${A\cdot(\mathbb{C},dz)}$${(\mathbb{C},dz)}$${\cup}$${\cup}$${\cup}$${U}$${U}$${A(U)}$${\mathbb{C}\supset
U}$${A(U)}$${A(U)}$$\scriptstyle{\mathrm{Id}_{\mathbb{C}}}$$\scriptstyle{\mathrm{F}_{A}}$$\scriptstyle{\varphi_{U}=\mathrm{Id}_{U}}$$\scriptstyle{A\circ\varphi_{U}}$$\scriptstyle{\mathrm{Id}}$
Figure 1. Local coordinate calculation shows that
$F_{A}\circ\mathrm{Id_{\mathbb{C}}}$ has Jacobian $A$. The similarly defined
map $f_{A}:\mathcal{O}\to\mathcal{O}$ is also of linear part $A$.

The above shows that whenever $\mathcal{O}_{i}$ corresponds to a removable
singularity, there is an affine diffeomorphism of it, say $f_{i}$, whose
linear part is $A$. Now, for any other index $i$, we have
$\mathrm{Id_{\mathcal{O}_{i}}}:\mathcal{O}_{i}\to A\cdot\mathcal{O}_{i}$ of
linear part $A$. Furthermore, similar to $F_{A}$ above, we have
$\widehat{F}_{A,i}:A\cdot\mathcal{O}_{i}\to\mathcal{O}_{i}$ of trivial linear
part. Letting $\widehat{F}_{A}:A\cdot\mathcal{O}\to\mathcal{O}$ be the map
which restricts to $\widehat{F}_{A,i}$ on each $A\cdot\mathcal{O}_{i}$, one
finds that $f_{A}=\widehat{F}_{A}\circ\mathrm{Id_{\mathcal{O}}}$ is a
homeomorphism taking singularities to singularities and whose linear part is
$A$. ∎

The locally length-minimizing paths on $\mathcal{O}$ are easily described.

######  Lemma 3.

For each $\mathcal{O}_{i}$, any two distinct points are connected by a unique
geodesic. Each such geodesic is either a straight line segment, or a connected
sequence of such segments meeting at the singularity with (minimum) angle at
least $\pi$.

###### Proof.

The description of the geodesics is a special case of Strebel’s result for an
arbitrary (half-)translation surface $(Y,\alpha)$, see Theorem 8.1 of [Str].
Since $\mathcal{O}_{i}$ is simply connected, any two points of
$\mathcal{O}_{i}$ are connected by a unique geodesic (see [Str] Theorem 14.2.2
and Corollary 18.2). ∎

######  Lemma 4.

Suppose that $(X,\omega)$ is an element of $\mathcal{H}(d_{1},\dots,d_{s})$.
Then there is an isometric injection of the disjoint union of the open Voronoi
2-cells of $(X,\omega)$ into $\mathcal{O}$. This map is unique, up to the
action of $\mathrm{Trans}(\mathcal{O})$ on the image.

###### Proof.

Enumerate the elements of $\Sigma$ so that for each $i$, $\sigma_{i}\in\Sigma$
has cone angle $2(1+d_{i})\pi$. Fix $i$ and for typographic ease let
$\sigma=\sigma_{i}$, and let $\mathcal{C}_{\sigma}$ be the Voronoi 2-cell
centered at $\sigma$.

By definition, the open set $\mathcal{C}_{\sigma}$ contains no other
singularity, hence it contains some singular coordinate patch about $\sigma$.
This coordinate patch is a totally ramified covering of some disk centered at
the origin of $\mathbb{C}$, of the same degree as the totally ramified
covering from $\mathcal{O}_{i}$ to $\mathbb{C}$. Restricted to a common disk,
the ramified coverings have the same complex structure, by (ramified) covering
space theory; see say [Mi]. Moreover, each of the ramified coverings respects
the translation surface structure in the sense that the horizontal foliations
of the regular local coordinate patches are respected. In particular,
emanating from the origin there is an open ray along the positive real axis
whose preimage under each map is a “star” of $1+d_{i}$-edges. Thus, isometry
between the two is unique up to a rotation given by the action of an element
of $\mathrm{Trans}(\mathcal{O}_{i})$. Indeed, in a compatible manner, each can
be given an extended system of polar coordinates (singular in the standard
manner at its central point), with central angle from zero (at our chosen lift
of the positive real axis) to $2(1+d_{i})\pi$. For ease, we refer to these
simply as polar coordinate systems.

The polar coordinate system extends to all of $\mathcal{O}_{i}$. Although this
is not true for $(X,\omega)$, since each point of $\mathcal{C}_{\sigma}$ is
connected to $\sigma$ by a unique shortest path and these paths vary
continuously in $\mathcal{C}_{\sigma}$, we can extend the polar coordinates to
all of $\mathcal{C}_{\sigma}$. The map
$\iota_{\sigma}:\mathcal{C}_{\sigma}\to\mathcal{O}_{i}$ sending a point to the
corresponding point with the same polar coordinates is thus a local isometry.

Putting these isometries together, we have an isometry from the disjoint union
of the Voronoi 2-cells into $\mathcal{O}$. Given another such isometry, we can
compose with an element of $S_{n_{1}}\times\cdots\times S_{n_{t}}$ to ensure
that both send the Voronoi 2-cell of $\sigma_{i}$ to $\mathcal{O}_{i}$ for
each $i$. The only remaining freedom is accounted for by the action of each
$\mathrm{Trans}(\mathcal{O}_{i})$. The result thus holds. ∎

We fix some notation.

######  Definition 5.

We will assume throughout that some isometric embedding $\iota$ as above of
the union of the Voronoi 2-cells of $(X,\omega)$ into $\mathcal{O}$ has been
fixed. For each singularity $\sigma$ we let
$\mathcal{O}_{\sigma}=\mathcal{O}_{i}$ where $\iota$ sends $\sigma$ to the
origin of $\mathcal{O}_{i}$. As in the previous proof, we also let
$\mathcal{C}_{\sigma}$ be the Voronoi 2-cell centered at $\sigma$ and
$\iota_{\sigma}:\mathcal{C}_{\sigma}\to\mathcal{O}_{\sigma}$ the corresponding
isometry. Unless otherwise stated, $0$ will denote the singularity of
$\mathcal{O}_{\sigma}$.

Recall that $(X,\omega)$ is isometric to the quotient space of the disjoint
union of the closure of its Voronoi cells under the relation defined by the
identification of shared edges. Similarly, each $\iota(\mathcal{C}_{\sigma})$
has a collection of line segments comprising its closure which can be
identified so as to recover $(X,\omega)$. Note that for each $\sigma$, we have
$\overline{\iota(\mathcal{C}_{\sigma})}=\overline{\iota_{\sigma}(\mathcal{C}_{\sigma})}$.

######  Proposition 6.

Let $Y$ be the disjoint union of the various
$\overline{\iota(\mathcal{C}_{\sigma})}$. For each $\sigma$ any edge of
$\mathcal{C}_{\sigma}$ can be uniquely identified with a line segment in
$\overline{\iota(\mathcal{C}_{\sigma})}$ so that the two line segments
corresponding to a common edge of Voronoi 2-cells are translation equivalent
and such that the resulting equivalence relation on $Y$ results in a quotient
metric space that is isometric to $(X,\omega)$.

###### Proof.

Fix $\sigma$ and choose any edge $e$ of $\mathcal{C}_{\sigma}$ as well as a
point $p$ in the interior of $e$. The polar coordinates of
$\mathcal{C}_{\sigma}$ extend to give local coordinates in some sufficiently
small neighborhood of this regular point. Since the inverse function $j$
taking $\iota(\mathcal{C}_{\sigma})$ to $\mathcal{C}_{\sigma}$ is given by
identifying points of the same polar coordinates, it extends linearly to
include all of this neighborhood in its image. Suppose that $e$ is shared with
$\mathcal{C}_{\sigma^{\prime}}$ and let $j^{\prime}$ be the inverse function
in this setting. Polar coordinates here also give local coordinates in some
sufficiently small common neighborhood $U$ of $p$. These two local coordinates
on $U\subset(X,\omega)$ differ by a translation, from which it follows that
$j^{-1}(U)$ is a translation equivalent copy of $(j^{\prime})^{-1}(U)$ in
$\mathcal{O}_{\sigma}$.

Figure 2 is related to the following construction. Let $\hat{U}$ be the set
taken by $j$ to $U$ and let $\hat{p}\in\hat{U}$ be the preimage of $p$. There
is a unique shortest path from $\sigma$ to the regular point on
$p\in(X,\omega)$. Correspondingly there is a radial line from $0$ to
$\hat{p}$. As we vary $p\in U$ (while remaining in the interior of $e$), the
radial lines vary continuously. We have the analogous situation with respect
to the shortest paths from $\sigma^{\prime}$. The translation equivalence can
thus be extended to the set comprised of the radial lines in
$\mathcal{C}_{\sigma^{\prime}}$ (the lines translating ‘rigidly’ with their
endpoints). Note that $\sigma^{\prime}$ is sent to some regular point, say
$w\in\mathcal{O}_{\sigma}$. As well, each of our $\hat{p}$ has equal length
straight line segments to $0$ and to $w$. It follows that the set of these
$\hat{p}$ is a line segment on the perpendicular bisector $L$ of the line
segment joining $0$ and $w$.

Varying $p$ shows that $j$ can be extended so that all of the interior of $e$
has a well defined $j$-preimage, and that this preimage lies on $L$ and by
continuity we find that $j$ sends a line segment $\hat{e}$ on the boundary of
of $\overline{\iota(\mathcal{C}_{\sigma})}$ to $e$, and that all of $\hat{e}$
lies on $L$. It also follows that $\hat{e}$ and the subset of the closure of
$\iota_{\sigma^{\prime}}(\mathcal{C}_{\sigma^{\prime}})$ corresponding to $e$
are translation equivalent. Thus, the announced equivalence relation exists,
and the quotient space of $Y$ is indeed isometric to $(X,\omega)$. ∎

When $\sigma$ is fixed we will continue to use $\hat{p}$ and $\hat{e}$ to
denote the preimage of a point $p$ and of an edge $e$ respectively, under the
extension $j$ of the inverse of $\iota_{\sigma}$. We also introduce another
bit of notation.

$\hat{e}$$\hat{U}$$\hat{p}$$w$0

Figure 2. Construction for Proposition 6, $\iota$-images of a common edge of
two Voronoi 2-cells can be identified by a translation equivalence. Here
$\hat{p}$ is an interior point of a boundary edge $\hat{e}$ of
$\iota(\mathcal{C}_{\sigma})$, $0$ denotes the origin of
$\mathcal{O}_{\sigma}$, $\hat{U}$ is a neighborhood of $\hat{p}$, $w$ is the
reflection of $0$ through the line of $\hat{e}$. Black lines outline a
triangle in the closure of $\iota(\mathcal{C}_{\sigma})$. Blue dotted lines
give the equal length segments joining $\hat{p}$ to $0$ to $w$

######  Definition 7.

For $z\in\mathcal{O}_{\sigma}\setminus\\{0\\}$, let $D(z)$ be the open disk of
center $z$ and radius equal to the distance from $0$ to $z$.

$\hat{e}$$T$$T^{\prime}$$w_{1}$$w_{2}$$w$0$\hat{q}$

Figure 3. Voronoi 0-cell $q$ corresponds to an endpoint $\hat{q}$ of a
boundary line segment $\hat{e}$ of
$\overline{\iota(\mathcal{C}_{\sigma})}\subset\mathcal{O}_{\sigma}$. The proof
of Proposition 6 shows that a reflection of the triangle $T$ through the line
of $\hat{e}$ is translation equivalent to a triangle within a paired Voronoi
2-cell. Proposition 8 shows that a neighborhood of $\hat{q}$ is covered by
translation equivalent copies of triangles in the $n+2$ (counted with possible
repetition) Voronoi 2-cells on whose boundaries $q$ lies; pictured: $n=2$. The
points $w,w_{1},\dots,w_{n}$ are the resulting images of the singularities.
The dotted blue arc lies on the circle of center $\hat{q}$ and radius equal to
the length of the shortest path of $q$ to $\Sigma$. By repeating this
construction with respect to the“lower” endpoints of edges in the role of $q$,
one finds that the sector of the open disk bounded by this arc lies in the
union of translation equivalent copies of triangles.

######  Proposition 8.

Suppose that $e$ is an edge of $\mathcal{C}_{\sigma}$ and that $p\in e$. Then
$D(\hat{p})$ is contained in the union of a finite number of triangles in
$\mathcal{O}_{\sigma}$, each of which is translation equivalent to a triangle
whose vertices are the singularity and endpoints of one edge of the 2-cell of
some Voronoi 2-cell of $(X,\omega)$. In particular, the inverse $j$ to
$\iota_{\sigma}$ extends so as to take $D(\hat{p})$ to an open set of
$(X,\omega)\setminus\Sigma$.

###### Proof.

We begin by showing that the initial result holds for the endpoints of $e$.
Normalize so that $\hat{e}$ is vertical and denote its lowest point by
$\hat{q}$, see Figure 3. Now, $\iota(\mathcal{C}_{\sigma})$ includes (the
interior of) the triangle $T$ whose vertices are $0$ and the endpoints of
$\hat{e}$. The proof of Proposition 6 shows that the reflection of $T$ through
$\hat{e}$, say $T^{\prime}$, is translation equivalent to a triangle inside
$\iota(\mathcal{C}_{\sigma^{\prime}})$. As in that proof, let $w$ be the
reflection of $0$ through the line of $\hat{e}$. Let $S$ be the sector of
$D(\hat{q})$ bounded by the radial lines ending at $0$ and $w$. We aim to show
that $S$ is contained in an appropriate union of triangles.

Corresponding to $\hat{q}$ is an endpoint of $e$ say, $q$. There is a finite
number of Voronoi 2-cells, say
$\mathcal{C}_{\sigma_{1}},\dots,\mathcal{C}_{\sigma_{n}}$ (repetition allowed)
such that $q$ is an endpoint of an edge of each; enumerate so that the
corresponding edges are $e_{1},\dots,e_{n+1}$ where $e_{1}$ is shared by
$\mathcal{C}_{\sigma}$ and $\mathcal{C}_{\sigma_{1}}$, $e_{n+1}$ is a shared
edge of $\mathcal{C}_{\sigma_{n}},\mathcal{C}_{\sigma^{\prime}}$ and any other
$e_{i}$ is shared by $\mathcal{C}_{\sigma_{i}},\mathcal{C}_{\sigma_{i+1}}$.
Since $q$ is a regular point, the polar coordinates of $\mathcal{O}_{\sigma}$
allow the extension of the inverse map $j$ to include a neighborhood of
$\hat{q}$. As well, the directions of the various edges as given by geodesic
segments emanating from $q$ are well defined. This allows us to repeatedly use
the construction of the proof of Proposition 6. Thus, for each $i$, we can
translate the $\iota$-copy of a triangle within
$\iota(\mathcal{C}_{\sigma_{i}})$ so that the vertices of the closure on
$\mathcal{O}_{\sigma}$ are $\hat{q}$, the second endpoint of a line segment
$\hat{e}_{i}$, and a point $w_{i}$ on the boundary arc of $S$. In particular,
there is an open neighborhood of $\hat{q}$ contained in the union of all of
the triangles on $\mathcal{O}_{\sigma}$.

If $S$ is not already contained in the union of these triangles, then let
$\hat{q}_{i}$ be the lower endpoints of the $\hat{e}_{i}$. These also
correspond to regular points, and thus there are translates of closures of
$\iota$-images of triangles which meet at them. Note that the reflections
involved place the images of the corresponding singularities outside of $S$.
(One can also show that this images must be external to $S$, as if not then
there is a straight path from $\hat{q}$ to such an image that gives rise to a
path from $q$ to $\Sigma$ that is shorter than the radius of $S$, a
contradiction.) We can now continue this process, finding translate images of
triangles so as to share edges, with singularities of images always external
to $S$. By the compactness of $(X,\omega)$ there are finitely many edges of
Voronoi 2-cells, and thus the triangles must have area bounded below by a
positive constant. Therefore, allowing for ever more generations, we must
eventually have $S$ included in the union of a finite number of our triangles
on $\mathcal{O}_{\sigma}$.

We can now argue with the other endpoint of $e$ instead of $q$. We find that
the sector $S^{+}$ of the open disk whose center is this second point and
whose radius is $R$ the distance from it to $0$ bounded by the radial lines to
$0$ and to $w$ is also covered by such triangles. Now, $r\leq R+|\hat{e}|$ and
$R\leq r+|\hat{e}|$ must both hold (where $|\hat{e}|$ denotes its length), as
for example the definition of $r$ shows that the path from $\hat{p}$ along
$\hat{e}$ and then to $0$ has image under the map $j$ that joins $q$ to
$\sigma$ (passing through only regular points). That is, $D(\hat{q})$ is
contained in the union of $S$ and $S^{+}$. For the same reason this is true
for the disk of the other endpoint. But, similar comparisons of lengths of
paths show that $D(\hat{p})\subset S\cup S^{+}$ also holds for points
$\hat{p}$ lying in the interior of $\hat{e}$.

Finally, since none of the images of the singularities of the various
triangles lies in $S\cup S^{+}$, given $\hat{p}$ the map $j$ extends so as to
send $D(\hat{p})$ to an open subset of $(X,\omega)\setminus\Sigma$. ∎

###  3.2. Closed Voronoi 2-cells as convex bodies on $\mathcal{O}$

We define key notions for the following.

######  Definition 9.

Any separatrix $\rho$ emanating from $\sigma$ is such that the map $\iota$
sends $\rho\cap\mathcal{C}_{\sigma}$ to a line segment emanating from the
origin of $\mathcal{O}_{\sigma}$. Let $\breve{\rho}$ be the open ray formed by
stopping $\rho$ at the first singularity it encounters thereafter, if any such
exists. The linear continuation of $\iota(\,\rho\cap\mathcal{C}_{\sigma}\,)$
whose length agrees with that of $\breve{\rho}$ then gives the unique
isometric embedding of $\breve{\rho}$ into $\mathcal{O}$ agreeing with $\iota$
restricted to $\mathcal{C}_{\sigma}$. We denote this image by
$\hat{\iota}(\rho)$.

Given a saddle connection $s$, we call $\hat{\iota}(s)$ a marked segment for
$(X,\omega)$.

######  Definition 10.

Given a saddle connection $s$, let $\sigma$ be the initial endpoint of $s$. We
define $\hat{s}$ to be the terminal endpoint of $\hat{\iota}(s)$. We also
define the half-space of $s$ to be

$H_{\iota}(s)=\\{z\in\mathcal{O}_{\sigma}\mid d(0,z)\leq d(z,\hat{s})\\},$

where $d(\cdot,\cdot)$ denotes the distance on $\mathcal{O}_{\sigma}$ and (for
simplicity’s sake) $0$ denotes its origin, $\iota(\sigma)$.

######  Proposition 11.

Given a saddle connection $s$ emanating from a singularity $\sigma$, let
$s^{\perp}$ denote the perpendicular bisector of $\hat{\iota}(s)$. Then
$s^{\perp}$ is the boundary of $H_{\iota}(s)$.

###### Proof.

Away from its origin, $\mathcal{O}_{\sigma}$ has a Euclidean metric, and thus
$s^{\perp}$ is indeed well-defined. Similarly, in the case that $\sigma$ is a
removable singularity the result is well known from elementary geometry.

In all other cases, we invoke Strebel’s description of geodesics, Lemma 3
above. If neither point is the origin and the radial segments having these
points as endpoints are of angle less than $\pi$, then the geodesic between
them is simply the straight line segment connecting them in a single
coordinate chart. If their angle is greater than $\pi$, then the geodesic is
the union of two radial line segments.

Now, if $\hat{\iota}(s)$ makes an angle greater than or equal to $\pi$ with
the ray from the origin to $z\in\mathcal{O}_{\sigma}$, then
$d(z,\hat{s})=d(z,0)+d(0,\hat{s})$ and certainly $z\notin\partial
H_{\iota}(s)$. All other $z$ lie within an angle of $\pi$ on either side of
$\hat{\iota}(s)$. The ramified covering $\mathcal{O}_{\sigma}\to\mathbb{C}$
sends this sector of angle $2\pi$ isometrically to $\mathbb{C}$, and the
result then follows from the classical case. ∎

######  Definition 12.

For a singularity $\sigma$ we define the convex body of $\sigma$ to be

$\Omega_{\sigma}=\cap_{s}\,H_{\iota}(s)\,,$

where $s$ runs through the set of saddle connections emanating from $\sigma$.

Naturally enough, we say that a subset of any $\mathcal{O}_{i}$ is convex if
it contains the straight line segment joining any two of its points.

######  Proposition 13.

We have equality of sets

$\Omega_{\sigma}=\overline{\iota(\mathcal{C}_{\sigma})}.$

Furthermore, to each edge $e$ of $\mathcal{C}_{\sigma}$ is associated a saddle
connection $s$ emanating from $\sigma$ such that the image of $e$ in
$\overline{\iota(\mathcal{C}_{\sigma})}$ lies on $s^{\perp}$. Letting
$\mathcal{E}_{\sigma}^{\perp}$ denote the collection of all such saddle
connections,

$\Omega_{\sigma}=\cap_{s\in\mathcal{E}_{\sigma}^{\perp}}\,H_{\iota}(s).$

###### Proof.

We first show that
$\overline{\iota(\mathcal{C}_{\sigma})}=\cap_{s\in\mathcal{E}_{\sigma}^{\perp}}\,H_{\iota}(s)$.
Choose an edge $e$ of $C_{\sigma}$, and let $\sigma^{\prime}$ be the
singularity whose Voronoi 2-cell shares this edge. Let $\hat{e}$ in
$\overline{\iota(\mathcal{C}_{\sigma})}$ correspond to $e$ as above. For
$\hat{q}$ either endpoint of $\hat{e}$ by Proposition 8, the polar coordinates
inverse map $j$ extends to have domain including the open disk $D(\hat{q})$,
which it takes to a set of regular points of $(X,\omega)$. With $w$ be as
above, the straight line from $0$ to $w$ has its interior contained in
$D(\hat{q})$ (for at least one of the choices of $\hat{q}$) and since $j$
extends continuously to take $w$ to $\sigma^{\prime}$, there is a
corresponding saddle connection $s$ on $(X,\omega)$ connecting $\sigma$ to
$\sigma^{\prime}$ and such that $\hat{s}=w$. Therefore, $\hat{e}$ is in
$s^{\perp}$. From this it follows that
$\overline{\iota(\mathcal{C}_{\sigma})}=\cap_{s\in\mathcal{E}_{\sigma}^{\perp}}\,H_{\iota}(s)$.

Certainly
$\cap_{s\in\mathcal{E}_{\sigma}^{\perp}}\,H_{\iota}(s)\supset\Omega_{\sigma}$.
If the two convex sets were not equal then there would be some $s$ such that
$s^{\perp}$ meets $\iota(\mathcal{C}_{\sigma})$ and in particular there must
be some endpoint $\hat{q}$ of an edge $\hat{e}$ that is closer to $\hat{s}$
than to $0$. But then $\hat{s}\in D(\hat{q})$. From the proof of Proposition
8, $D(\hat{q})\subset S\cup S^{+}$ and since $0$ is on the boundary of this
union, the line segment from $0$ to $\hat{s}$ is contained in the closure of
$S\cup S^{+}$. Similarly to in the previous paragraph, $s$ is the image under
the extension of $j$ of the line segment. However, this shows that the point
$q$ corresponding to $\hat{q}$ is closer to $\Sigma$ than is $\hat{q}$ to $0$,
a contradiction. Thus, no such $s$ can exist. ∎

###  3.3. Voronoi staples determine $(X,\omega)$

We introduce several key sets for our considerations.

######  Definition 14.

Reversing orientation on any saddle connection $s$ of $(X,\omega)$ results in
a saddle connection $s^{\prime}$. We say that $s,s^{\prime}$ are orientation-
paired, and denote the set of these pairs by $\mathcal{M}(X,\omega)$. We then
call $\hat{\iota}(s),\hat{\iota}(s^{\prime})$ orientation-paired marked
segments, and denote the set of these pairs by
$\widehat{\mathcal{M}}(X,\omega)$. When $s\in\mathcal{E}_{\sigma}^{\perp}$,
let $e$ denote the edge of $\mathcal{C}_{\sigma}$ lying on its perpendicular
bisector, and $e^{\prime}$ be $e$ with the opposite orientation. Letting
$\sigma^{\prime}$ denote the singularity from which $s^{\prime}$ emanates,
$e^{\prime}$ is an edge of $\mathcal{C}_{\sigma^{\prime}}$, from which it
follows that $s^{\prime}\in\mathcal{E}_{\sigma^{\prime}}^{\perp}\,$. We then
call $\\{s,s^{\prime}\\}$ a Voronoi staple, and denote the set of all Voronoi
staples by $\mathcal{S}(X,\omega)$. The corresponding subset of
$\widehat{\mathcal{M}}(X,\omega)$ is called the marked Voronoi staples,
denoted by $\widehat{\mathcal{S}}(X,\omega)$, see Figure 4.

DEFCBA$a$$b$$c$$a$$b$$c$ |  | CBA$b$$c$$a$image in $\mathcal{O}_{2}$DFimage in $\mathcal{O}_{1}$E$c^{\prime}$$a^{\prime}$$b^{\prime}$
---|---|---

Figure 4. Left: Two distinct 2-cells comprising the Voronoi decomposition of
the hexagonal torus subordinate to two removable singularities. Right: The
resulting image of the cells under $\iota$ as well as of the saddle
connections under $\hat{\iota}$. The marked Voronoi staples —
$\\{a,a^{\prime}\\},\\{b,b^{\prime}\\},\\{c,c^{\prime}\\}$ — show how to glue
together the closure of the images of the 2-cells to obtain the surface.

Our choice of terminology is to highlight the implication of Propositions 6
and 13: $(X,\omega)$ can be viewed as the disjoint union of the closures of
the various $\Omega_{\sigma}$ “stapled together” by identifying sides lying on
perpendicular bisectors of members of the same marked Voronoi staple.

######  Theorem 15.

Suppose that both $(X_{1},\omega_{1}),(X_{2},\omega_{2})$ belong to
$\mathcal{H}(d_{1},\dots,d_{s})$ and choose maps $\iota_{1},\iota_{2}$ to
$\mathcal{O}$. Then $(X_{1},\omega_{1}),(X_{2},\omega_{2})$ are equivalent
translation surfaces if and only if $\widehat{\mathcal{S}}(X_{1},\omega_{1})$
and $\widehat{\mathcal{S}}(X_{2},\omega_{2})$ are in the same
$\mathrm{Trans}(\mathcal{O})$-orbit.

###### Proof.

First suppose that there is an element of $\mathrm{Trans}(\mathcal{O})$ taking
$\hat{\iota}_{2}(\,\mathcal{S}(X_{2},\omega_{2})\,)$ to
$\hat{\iota}_{1}(\,\mathcal{S}(X_{1},\omega_{1})\,)$. We can replace
$\iota_{2}$ by its composition with this map and assume equality of these
images. Proposition 13 then shows that for each singularity $\sigma_{1}$ of
$(X_{1},\omega_{1})$ there is a singularity $\sigma_{2}$ of
$(X_{2},\omega_{2})$ so that $\Omega_{\sigma_{1}}=\Omega_{\sigma_{2}}$.
Furthermore, since each identification of corresponding pairs of sides of the
various $\Omega_{\sigma_{1}},\Omega_{\sigma^{\prime}_{1}}$ is indexed by some
Voronoi pair, we find also that the
$\Omega_{\sigma_{2}},\Omega_{\sigma^{\prime}_{2}}$ are identically paired.
Therefore, the resulting surfaces are isometric and in particular translation
equivalent. Proposition 6 then gives that
$(X_{1},\omega_{1}),(X_{2},\omega_{2})$ are translation equivalent.

If $(X_{1},\omega_{1}),(X_{2},\omega_{2})$ are equivalent translation
surfaces, then there is a homeomorphism
$f:(X_{2},\omega_{2})\to(X_{1},\omega_{1})$ that is, off of singularities, a
translation in local coordinates. In particular, $f$ maps Voronoi 2-cells to
Voronoi 2-cells and sends $\mathcal{S}(X_{2},\omega_{2})$ to
$\mathcal{S}(X_{1},\omega_{1})$. The composition $\iota_{1}\circ f$ then
restricts to an isometry of the Voronoi 2-cells of $(X_{2},\omega_{2})$ into
$\mathcal{O}$; by Lemma 4 there is an element of $\mathrm{Trans}(\mathcal{O})$
which composed with $\iota_{2}$ equals $\iota_{1}\circ f$ on the Voronoi
2-cells of $(X_{2},\omega_{2})$. Equality holds then also for the extended
maps, and thus $\widehat{\mathcal{S}}(X_{1},\omega_{1})$ (the image of the
first) is in the $\mathrm{Trans}(\mathcal{O})$ -orbit of
$\widehat{\mathcal{S}}(X_{2},\omega_{2})$. ∎

###  3.4. Membership criterion

The next result applies the above with $(X_{1},\omega_{1})=(X,\omega)$ and
$(X_{2},\omega_{2})=A\cdot(X,\omega)$ to give a criterion for $A$’s membership
in $\mathrm{SL}(X,\omega)$. For this, we use the map
$f_{A}:\mathcal{O}\to\mathcal{O}$ of Lemma 2.

######  Proposition 16 (Membership Criterion, initial form).

Fix $(X,\omega)$ and suppose $A\in\mathrm{SL}_{2}\mathbb{R}$. Then
$A\in\mathrm{SL}(X,\omega)$ if and only if $f_{A}$ sends
$\widehat{\mathcal{S}}(X,\omega)$ to a subset of
$\widehat{\mathcal{M}}(X,\omega)$, up to some element of
$\mathrm{Trans}(\mathcal{O})$.

###### Proof.

We first establish an intermediate result for all
$A\in\mathrm{SL}_{2}\mathbb{R}$, confer Figure 5. For any
$A\in\mathrm{SL}_{2}\mathbb{R}$, let $\hat{\iota}_{A}$ be the (unique up to
translations) analog of $\iota$ for $A\cdot(X,\omega)$. The map
$\mathrm{Id}_{X}:A\cdot(X,\omega)\to(X,\omega)$ has linear part $A^{-1}$, and
sends the orientation-paired saddle connections of $A\cdot(X,\omega)$ set-wise
to those of $(X,\omega)$. Hence $\hat{\iota}\circ\mathrm{Id}_{X}$ sends these
to $\widehat{\mathcal{M}}(X,\omega)$. Composing further with
$\mathrm{Id}_{\mathcal{O}}:{\mathcal{O}}\to A\cdot{\mathcal{O}}$ (whose linear
part is $A$) gives that the orientation-paired saddle connections of
$A\cdot(X,\omega)$ are sent isometrically in pairs into $A\cdot\mathcal{O}$ by
$\mathrm{Id}_{\mathcal{O}}\circ\hat{\iota}\circ\mathrm{Id}_{X}$. Thereafter
composing with $\widehat{F}_{A}$ sends these isometrically into $\mathcal{O}$.
Since $f_{A}=\widehat{F}_{A}\circ\mathrm{Id}_{\mathcal{O}}$, Lemma 4 (arguing
by the density of directions of saddle connections) shows that
$f_{A}\circ\hat{\iota}\circ\mathrm{Id}_{X}$ is $\hat{\iota}_{A}$, up to
$\mathrm{Trans}(\mathcal{O})$-equivalence. Thus, we may assume equality and
hence that $\hat{\iota}_{A}$ sends the orientation-paired saddle connections
of $A\cdot(X,\omega)$ isometrically to
$f_{A}(\,\widehat{\mathcal{M}}(X,\omega)\,)$.

${\mathcal{M}(X,\omega)}$${\mathcal{M}(\,A\cdot(X,\omega)\,)}$${\widehat{\mathcal{M}}(X,\omega)}$${\mathcal{O}}$${A\cdot\mathcal{O}}$${\mathcal{O}}$${\widehat{\mathcal{M}}(\,A\cdot(X,\omega)\,)}$$\scriptstyle{\hat{\iota}}$$\scriptstyle{\hat{\iota}_{A}}$$\scriptstyle{\mathrm{Id}_{X}}$$\scriptstyle{\mathrm{Id}_{\mathcal{O}}}$$\scriptstyle{f_{A}}$$\scriptstyle{\widehat{F}_{A}}$
Figure 5. Up to an element of $\mathrm{Trans}(\mathcal{O})$, $\hat{\iota}_{A}$
sends $\mathcal{M}(\,A\cdot(X,\omega)\,)$ to
$f_{A}(\,\widehat{\mathcal{M}}(X,\omega)\,)$.

($\Rightarrow$) Given $A\in\mathrm{SL}(X,\omega)$, then also
$A^{-1}\in\mathrm{SL}(X,\omega)$. By Theorem 15, we may assume that
$\hat{\iota}(\,\mathcal{S}(X,\omega)\,)$ and
$\hat{\iota}_{A^{-1}}(\,\mathcal{S}(\,A^{-1}\cdot(X,\omega)\,)\,)$ are equal.
From the previous paragraph, this is a subset of
$f_{A^{-1}}(\,\widehat{\mathcal{M}}(X,\omega)\,)$. It follows that
$f_{A}\circ\hat{\iota}(\,\mathcal{S}(X,\omega)\,)$ equals some subset of
$\widehat{\mathcal{M}}(X,\omega)$. This first direction hence holds.

($\Leftarrow$) Again since $\mathrm{SL}(X,\omega)$ is a group, it suffices to
show that $A\in\mathrm{SL}(X,\omega)$ if
$f_{A^{-1}}(\,\widehat{\mathcal{S}}(X,\omega)\,)\subset\widehat{\mathcal{M}}(X,\omega)$.
To show this, we apply $f_{A}$ and find
$\widehat{\mathcal{S}}(X,\omega)\subset
f_{A}(\,\widehat{\mathcal{M}}(X,\omega)\,)=\widehat{\mathcal{M}}(\,A\cdot(X,\omega)\,)$.
Any singularity $\sigma$ of $A\cdot(X,\omega)$ is identified with a fixed
singularity of $(X,\omega)$, naturally also denoted by $\sigma$. Since
$\Omega_{\sigma}(\,A\cdot(X,\omega)\,)$ is the intersection of the half-spaces
for all marked segments associated to $\sigma$, the convex body given by the
intersection over the $s\in\mathcal{E}_{\sigma}^{\perp}(X,\omega)$ contains
$\Omega_{\sigma}(\,A\cdot(X,\omega)\,)$. By Proposition 13, this larger convex
body is $\Omega_{\sigma}(X,\omega)$, and is a copy of the closure of the
Voronoi 2-cell for $\sigma$ on $(X,\omega)$.

Summing over all $\sigma$ shows the natural Lebesgue measure of
$A\cdot(X,\omega)$ is at most the measure of $(X,\omega)$. However, since
$A\in\mathrm{SL}_{2}\mathbb{R}$, the measure of $A\cdot(X,\omega)$ is the same
as that of $(X,\omega)$. Therefore, equality of convex bodies holds for each
$\sigma$. We now apply the construction of Proposition 6 to each of the
collections of $\Omega_{\sigma}(\,A\cdot(X,\omega)\,)$ and
$\Omega_{\sigma}(X,\omega)$. The fact that $f_{A}$ respects orientation-
pairings assures that the resulting two translation surfaces are equivalent.
Therefore, $A\cdot(X,\omega)$ and $(X,\omega)$ are translation equivalent and
hence $A\in\mathrm{SL}(X,\omega)$. ∎

######  Remark 17.

In his Ph.D. dissertation [E], Edwards shows that the
$\mathrm{SL}_{2}\mathbb{R}$ action on $\mathcal{O}$ induces an action on
$\mathrm{Trans}(\mathcal{O})$-equivalent classes of $\mathbb{Z}_{2}$-paired
subsets of $\mathcal{O}$, and that the stabilizer of the class of
$\widehat{\mathcal{M}}(X,\omega)$ is exactly $\mathrm{SL}(X,\omega)$.

For ease, let us say that the length of a Voronoi staple is the length of the
underlying saddle connection (with one of the possible orientations). In the
following, we assume that some $\iota:(X,\omega)\to\mathcal{O}$ has been
chosen. Recall that the maximum singular value of a real matrix $A$ is the
maximum of length of the image of unit vectors under $A$ and thus gives a
bound on how much $A$ can expand lengths.

######  Definition 18.

For each positive real number $r$, let
$\mathcal{P}_{r}=\mathcal{P}_{r}(X,\omega)$ be the set of orientation pairs of
marked segments of length at most $r$.

######  Corollary 19 (Membership Criterion, with bounds).

Let $\ell$ be the maximal length of any Voronoi staple of $(X,\omega)$.
Suppose that $\nu\in[1,\infty)$ and $A\in\mathrm{SL}_{2}\mathbb{R}$ has
maximum singular value at most $\nu$.

Then $A\in\mathrm{SL}(X,\omega)$ if and only if $f_{A}$ maps the set of marked
Voronoi staples of $(X,\omega)$ into some
$\mathrm{Trans}(\mathcal{O})$-translate of $\mathcal{P}_{\nu\ell}$.

###### Sketch.

We use Proposition 16. Since $f_{A}$ has linear part $A$, it expands lengths
by at most the maximum singular value of $A$ and hence the image of any marked
Voronoi staple can have length at most $\nu\ell$. ∎

######  Example 20 (Respecting orientation-pairing is crucial).

It is important to note that an element $A\in\mathrm{SL}_{2}(\mathbb{R})$ may
take marked segments to marked segments, but still fail the membership
criterion, due to it not respecting orientation-pairings. Figure 6 illustrates
this in the case of an $(X,\omega)\in\mathcal{H}(2)$, formed by glueing three
unit squares together in the configuration shown and identifying opposite
sides, where $A=\left(\begin{smallmatrix}1&0\\\ 1&1\end{smallmatrix}\right)$
sends the horizontal marked segment $s$ to the marked segment $\gamma$, but
does not send the paired $s^{\prime}$ to $\gamma^{\prime}$. Nor is there any
element of $\mathrm{Trans}(\mathcal{O})$ that can resolve this. However, the
reader may wish to verify that $B=\left(\begin{smallmatrix}1&0\\\
2&1\end{smallmatrix}\right)$ does fulfill the membership criterion.

|  $s$$\gamma$ |  $\gamma^{\prime}$ |  $s^{\prime}$$f_{A}(s^{\prime})$
---|---|---|---

Figure 6. Illustrating Example 20. Left: Surface formed by identifying
opposite sides, each segment is of length one, is in $\mathcal{H}(2)$. Voronoi
partition indicated in black, dashed. Angle measurement beginning at
horizontal of left middle vertex (first $2\pi$ in single blue, second in
double brown, third in triple black rings). Right: The single connected
component of $\mathcal{O}$, also of singularity of angle $6\pi$, represented
as three copies of $\mathbb{C}$ glued along the positive $x$-axis in cyclic
order left to right. Red arrows show one marked Voronoi staple
$\\{s,s^{\prime}\\}$. Black arrows show one choice, up to action of
$\mathrm{Trans}(\mathcal{O})$, for $\gamma=f_{A}(s)$ and its orientation
paired marked segment, where $A=\left(\begin{smallmatrix}1&0\\\
1&1\end{smallmatrix}\right)$. Blue arrow is $f_{A}(s^{\prime})$. Since
$\gamma^{\prime}\neq f_{A}(s^{\prime})$ (and similarly for all
$\mathrm{Trans}(\mathcal{O})$-images of $\gamma$), $A$ fails the membership
criterion. Also indicated: dots give endpoints of other marked segments,
shaded disks are of radius $\nu\ell$ with $\ell,\nu$ defined as in Corollary
19, of values here $\ell=1$ and $\nu=\nu(\,||A||)=\sqrt{(3+\sqrt{5})/2}$, see
Definition 21.

##  4\. Nesting hyperbolic polygons agreeing on ever larger balls

The previous section presents theoretical results for determining elements of
a specific discrete subgroup of $\mathrm{SL}_{2}\mathbb{R}$. Here we give
results on using finite lists of elements of any given Fuchsian group
$\Gamma\subset\mathrm{PSL}_{2}(\mathbb{R})$ to determine aspects of a
fundamental domain for the action of $\Gamma$ on the Poincaré upper half-plane
model of hyperbolic space, $\mathbb{H}$. Recycling notation, in this section
we use $H,\Omega$ to again indicate half-spaces and their intersections, but
in the context of the $\mathbb{H}$.

Convention. Throughout this section, we assume that $\Gamma$ has no non-
trivial elements fixing $i\in\mathbb{H}$.

(Should in fact such elements exist, then since $\Gamma$ acts properly
discontinuously, conjugating this group by a sufficiently small translation
must result in a group whose stabilizer of $i$ is trivial. This was used to
determine the translation surface underlying the computations for Figure 7.)

The standard construction of the Dirichlet domain (centered at $z=i$) for a
group $\Gamma$ respecting our convention is algorithmic: one takes the
intersection of half-planes one at a time, see [K] for a textbook discussion.
We thus contemplate taking the intersection of such half-planes for all
elements of Frobenius norm of at most a given bound. As the bound goes to
infinity, these convex bodies nest down to the Dirichlet domain for $\Gamma$.

However, more is true. We show that as the Frobenius norm increases there are
ever larger hyperbolic balls within which these convex bodies already match
the Dirichlet domain of the full group. This allows us to use an area
computation to detect if $\Gamma$ is generated by the elements whose norm is
bounded by any explicit constant.

###  4.1. Bounded Frobenius norm elements give Dirichlet domain within bounded
ball

Let $d_{\mathbb{H}}(z,w)$ denote the standard hyperbolic distance between two
points in the upper half-plane. For $A\in\mathrm{SL}_{2}\mathbb{R}$ let
$||A||$ denote the Frobenius norm of $A$, that is
$||A||=\sqrt{\mathrm{tr}(A^{t}A)}$. It is well-known that the Frobenius norm
of $A$ equals the square-root of the sum of the squares of the singular values
of $A$. Note also that the Frobenius norm descends to give a well-defined
function on $\mathrm{PSL}_{2}\mathbb{R}$, which we also call the Frobenius
norm.

######  Definition 21.

Define the function

$\displaystyle\nu:[\sqrt{2},\infty)$ $\displaystyle\to[1,\infty)$
$\displaystyle a$
$\displaystyle\mapsto\sqrt{\dfrac{1}{2}(a^{2}+\sqrt{a^{4}-4}\,)}\,.$

Note that $\nu(||A||)$ gives the maximum singular value of $A$.

The next result is presumably well-known but we do not know of a source where
it is explicitly stated.

######  Lemma 22.

Suppose that
$A\in\mathrm{SL}_{2}\mathbb{R}\setminus\mathrm{SO}_{2}\mathbb{R}$. Then the
hyperbolic distance from $i$ to $A\cdot i$ is given by

$d_{\mathbb{H}}(i,A\cdot i)=2\log\nu(\,||A||\,).$

In particular, the hyperbolic ball centered at $i$ and of radius
$r=\log\nu(\,||A||\,)$ is contained in the half-plane of elements that are
nearer to $i$ than to $A\cdot i$,

$H_{i}(A)\supset B(i,\log\nu(\,||A||\,)).$

######  Definition 23.

For any positive real number $a$, let $\Gamma^{a}$ denote the set of all
elements of $\Gamma$ whose Frobenius norm is at most $a$. Furthermore, for any
subset $S\subset\Gamma$, let $\Omega(\,S\,)=\cap_{A\in S}\,H_{i}(A)$. Thus
when $S$ is a subgroup of $\Gamma$, we have that $\Omega(\,S\,)$ is its
Dirichlet domain centered at $z=i$.

The very construction of Dirichlet domains shows that as $a$ increases, the
$\Omega(\,\Gamma^{a}\,)$ nest down to $\Omega(\,\Gamma\,)$. Since
$a\mapsto\log\nu(a)$ defines an increasing function, the following shows that
within ever growing balls the $\Omega(\,\Gamma^{a}\,)$ fully describe
$\Omega(\,\Gamma\,)$. (Of course, neither must be completely contained in any
such fixed ball.)

######  Proposition 24.

For each $a\geq\sqrt{2}$,

$\Omega(\,\Gamma\,)\cap B(i,\log\nu(a))=\Omega(\,\Gamma^{a}\,)\cap
B(i,\log\nu(a)\,).$

###### Proof.

Since $a\mapsto\log\nu(a)$ is a strictly increasing function, Lemma 22 implies
that also if $a<||A||$ then $H_{i}(A)$ contains the ball
$B(i,\log\nu(\,a\,))$. Since $\Omega(\,\Gamma\,)$ is formed from
$\Omega(\,\Gamma^{a}\,)$ by intersection with all such $H_{i}(A)$, with
$A\in\Gamma$, the result holds. ∎

Figure 7. The shaded region shows where the computed intersection of half-
planes from elements of small Frobenius norm must agree with the Dirichlet
region of the full group, as guaranteed by Proposition 24. Here,
$\Gamma=\mathrm{SL}(M\cdot(X,\omega))$, where $(X,\omega)$ is given in Example
20, see also Figure 6, and $M=\left(\begin{smallmatrix}1&0\\\
1/2&1\end{smallmatrix}\right)$ is used to obtain a surface whose Veech group
has trivial stabilizer of $z=i$. Here $a=\sqrt{257}/4$.

###  4.2. Recognizing a generating set

This subsection gives a theoretical result that leads to a stopping condition
for our algorithm to compute $\mathrm{SL}(X,\omega)$, when this group is a
lattice.

######  Theorem 25.

Suppose that $a\geq\sqrt{2}$ is such that

$\mu_{{}_{\mathbb{H}}}(\,\Omega(\Gamma^{a}\,)\,)<2\,\mu_{{}_{\mathbb{H}}}(\,\Omega(\,\Gamma^{a}\,)\cap
B(i,\log\nu(a)\,)\,).$

Then the subset $\Gamma^{a}$ generates $\Gamma$. In particular, $\Gamma$ is a
lattice.

###### Proof.

Since $\Gamma$ is a discrete group, the subset $\Gamma^{a}$ is finite and
hence $\Omega(\,\Gamma^{a}\,)$ has finitely many sides. By construction there
is a finite subset of $\Gamma^{a}$ pairing these sides. Let $G$ be the
subgroup of $\Gamma$ generated by these side parings. Then

$\mu_{{}_{\mathbb{H}}}(\Omega(G))\leq\mu_{{}_{\mathbb{H}}}(\Omega(\Gamma^{a})).$

Proposition 24 combines with this to give

$\displaystyle\mu_{{}_{\mathbb{H}}}(\Omega(\Gamma))$
$\displaystyle\geq\mu_{{}_{\mathbb{H}}}(\,\Omega(\,\Gamma\,)\cap
B(i,\log\nu(a))\,)$
$\displaystyle=\mu_{{}_{\mathbb{H}}}(\,\Omega(\,\Gamma^{a}\,)\cap
B(i,\log\nu(a))\,)$
$\displaystyle>\frac{1}{2}\mu_{{}_{\mathbb{H}}}(\Omega(\,\Gamma^{a})\,)$
$\displaystyle\geq\frac{1}{2}\mu_{{}_{\mathbb{H}}}(\,\Omega(G)\,).$

Hence, the index of $G$ in the group $\Gamma$ is

$|\Gamma:G|=\frac{\mu_{{}_{\mathbb{H}}}(\Omega(G))}{\mu_{{}_{\mathbb{H}}}(\Omega(\Gamma))}<2.$

Therefore, $G=\Gamma$. It follows that $\Gamma^{a}$ generates $\Gamma$.

The initial hypothesis implies that
$\mu_{{}_{\mathbb{H}}}(\,\Omega(\,\Gamma^{a}\,)\,)$ is finite. We deduce that
$\Gamma$ is finitely generated and has finite co-volume. It is a lattice. ∎

##  5\. From theory to algorithm

We sketch how the results above can be combined to give an algorithm to
compute Veech groups.

###  5.1. Basic steps of algorithm

Given a polygonal presentation of $(X,\omega)$, one can calculate to find a
Voronoi decomposition. Further calculation leads to the Voronoi staples, as
one can explicitly calculate all saddle connections up to any given length.
Thus, the set $\widehat{\mathcal{S}}(X,\omega)$ and the value $\ell$ of
Corollary 19 are knowable with finite calculation.

We test membership using Corollary 19. We can form candidate matrices by
choosing of the saddle connections of two distinct saddle connections and a
choice of image vectors. Beginning with $a=\sqrt{2}$, we choose image vectors
from the holonomy vectors corresponding to elements of
$\mathcal{P}_{\nu(a)\,\ell}$. We then check whether the candidate matrix $A$
is such that $f_{A}$ does send the marked Voronoi staples pairwise into
orientation-paired marked segments, up to an element of
$\mathrm{Trans}(\mathcal{O})$. Exactly when this is the case is
$A\in\mathrm{SL}(X,\omega)$. We complete this round for all possible candidate
matrices for this value of $a$. We then form $\Omega(\Gamma^{a})$ and apply
the obvious test implied by Theorem 25. If this test is positive, then we have
determined all of what is rightly denoted as $\mathrm{PSL}(X,\omega)$. (The
only other way that our algorithm halts is by a user-entered upper bound on
the Frobenius norm of elements to test.) Otherwise, we increase $a$ (say by
doubling its value).

###  5.2. Some technical details

We roughly indicate how some of the technical matters of implementing the
algorithm are addressed.

  * •

From $\mathrm{PSL}(X,\omega)$ to $\mathrm{SL}(X,\omega)$. Passing from
$\mathrm{PSL}(X,\omega)$ to $\mathrm{SL}(X,\omega)$ is merely a matter of
determining if $-I_{2}\in\mathrm{SL}(X,\omega)$. Since $||-I_{2}||=\sqrt{2}$,
this is directly tested using Corollary 19.

  * •

Finding a Veech group that trivially stabilizes $z=i$. The stabilizer inside
of all of $\mathrm{SL}_{2}(\mathbb{R})$ of $z=i$ is
$\mathrm{SO}_{2}(\mathbb{R})$, from which a Fuchsian group $\Gamma$ follows
our convention exactly if
$\Gamma^{\sqrt{2}}\cap\mathrm{SO}_{2}(\mathbb{R})\subset\\{\pm I_{2}\\}$.
Given a translation surface $(X,\omega)$, we test for this by calculating
$\mathrm{PSL}(X,\omega)^{\sqrt{2}}$. Upon failure, we rather find an $M$ such
that $M\cdot(X,\omega)$ trivially stabilizes $z=i$ and compute
$\mathrm{SL}(M\cdot(X,\omega)\,)$ with the algorithm as already sketched;
thereafter, conjugating by $M^{-1}$ gives $\mathrm{SL}(X,\omega)$. (Figure 7
arises from a computation in such a setting.) It is the fact that
$\mathrm{SL}(X,\omega)$ is discrete and hence that $\mathrm{PSL}(X,\omega)$
acts properly discontinuously on $\mathbb{H}$ (in particular, fixed points are
isolated), that implies the existence of some such $M$. In fact, there is some
$n\in\mathbb{N}$ such that $M=\left(\begin{smallmatrix}1&0\\\
1/n&1\end{smallmatrix}\right)$ suffices. (An iterative loop can determine the
smallest value of $n$ for which this holds.)

  * •

“Marked” data from $(X,\omega)$. Values of the map $\hat{\iota}$ to
$\mathcal{O}$ are represented in terms of data structures reflecting the
geometry of $(X,\omega)$: We first choose an enumeration of the singularities;
as illustrated in Figure 6, for each singularity we choose one of the
horizontal rays emanating from it, and measure angles beginning at that marked
horizontal. (The difference of choices of the horizontal is accounted for by
$\mathrm{Trans}(\mathcal{O}_{i})$ as in Lemma 1.) Thereafter, each saddle
connection is recorded as a vector in the plane, along with a record of the
singularity from which it emanates, as well as of which multiple of $2\pi$
gives the copy of $\mathbb{C}$ in which its image lies (compare Figure 6), and
finally the record includes a pointer to its orientation-paired saddle
connection.

  * •

$\mathrm{SL}_{2}(\mathbb{R})$ action on marked segments. When applying the
membership criterion, it is crucial that one notes that for
$A\in\mathrm{SL}_{2}(\mathbb{R})$, $f_{A}$ can send a marked segment from one
copy of $\mathbb{C}$ into one of the neighboring $\mathbb{C}$. This is a
matter of the vector being “pushed” across the positive $x$-axis. As an
example, the reader can verify that in the setting of Example 20 and Figure 6
$f_{A}$ sends $1-i/2$ in the first copy of $\mathbb{C}$ into the second copy
of $\mathbb{C}$.

##  6\. Verified calculations

We have verified the results of running code based upon this algorithm for a
collection of surfaces. These include various “L” surfaces, particular
(double) regular polygons, the famed “eierliegende Wollmilchsau” surface, and
others, see [Sa] for most of these.

###  6.1. An “L”-surface

Here we content ourselves with a brief report on the calculation of the main
running example of this paper. For the translation surface $(X,\omega)$ of
Example 20 and Figure 6, applying the algorithm, we found that
$\mathrm{SL}(X,\omega)$ is generated by the matrices

$S=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},\,A=\begin{pmatrix}2&1\\\
-1&0\end{pmatrix}.$

Note that $S$ squares to give $-I_{2}$ and that $SA$ is equal to
$B=\big{(}\begin{smallmatrix}1&0\\\ 2&1\end{smallmatrix}\big{)}$ (confer the
final sentence of Example 20 ). We can easily recognize that the ideal
triangle of vertices at $-1,0,\infty$ is a fundamental domain for this group:
$S$ sends the $y$-axis to itself (fixing $z=i$) and $A$ sends the geodesic of
“feet” $-1$ and $0$ to the vertical line through $x=-1$. Thus
$\mathrm{PSL}(X,\omega)$ is a Fuchsian triangle group, of signature
$(0;2,\infty,\infty)$ — it is of genus zero, has two cusps and one orbifold
point of index $2$. The fundamental domain has area $\pi$, and
$\mathrm{SL}(X,\omega)$ is certainly a subgroup of
$\mathrm{SL}_{2}(\mathbb{R})$; our calculation thus accords with the first
entry in the table of results in Section 4.2 of [Sch] (the “L”-surface there
is translation equivalent to our $(X,\omega)$ ).

Note that with $M$ as in the caption of Figure 7, $M^{-1}$ sends the Dirichlet
domain centered at $z=i$ for $\mathrm{PSL}(M\cdot(X,\omega)\,)$ suggested in
that figure to the Dirichlet domain at $z=M^{-1}\cdot i$ for
$\mathrm{PSL}(X,\omega)$. This image is a pentagon with ideal vertices at
$-1,0$; the sides meeting at $-1$ are paired by $A$, those meeting at $0$ by
$B$, the remaining side passes through $z=i$ and is paired with itself by $S$.

###  6.2. Investigating an infinitely generated Veech group

McMullen [Mc] gave an infinite family of translation surfaces in
$\mathcal{H}(1,1)$, each of whose Veech group is infinitely generated (an
“infinitely generated” group is one which cannot be finitely generated). Our
algorithm can of course find elements in order of Frobenius norm for such
groups; we investigated $\mathrm{SL}(X,\omega)$ for his surface corresponding
to $a=1+\sqrt{3}$, see Figure 8 (and the figures in [Mc]). Each of the
vertical and the horizontal directions of this surface determine respective
parabolic elements $P_{1}=\begin{pmatrix}1&0\\\
1&1\end{pmatrix},P_{2}=\begin{pmatrix}1&2+a\\\
0&1\end{pmatrix}\in\mathrm{SL}(X,\omega)$. Intriguingly enough, we found that
all of the $25,010$ elements of $\mathrm{SL}(X,\omega)$ up to Frobenius norm
256 lie in the subgroup generated by $-I_{2}$ and the parabolic elements
$P_{1}$ and $P_{2}$.

This leads us to ask for the study of the general growth behavior in terms of
$x$ of the minimal number of elements within an infinitely generated group to
generate the subgroup containing all elements of Frobenius norm up to $x$.

In the meantime, we give an explicit element of
$\mathrm{SL}(X,\omega)\setminus\langle-I_{2},P_{1},P_{2}\rangle$. The
Dirichlet domain of the Fuchsian $\langle P_{1},P_{2}\rangle$ based at $z=i$
is easily seen to have vertical sides at $x=\pm(2+a)/2$ as well as sides whose
feet at infinity are $-2,0$ and $0,2$ and free sides $(-(2+a)/2,-2)$ and
$(2,(2+a)/2)$. By Theorem 9.2 of [Mc], any direction of a separatrix
containing a Weierstrass point corresponds to a parabolic element of
$\mathrm{SL}(X,\omega)$. Now, any such direction of slope whose inverse lies
in a free side of a fundamental domain cannot be the fixed point of any
element of $\langle P_{1},P_{2}\rangle$. In particular, using the polygonal
presentation, again see Figure 8, the visible geodesic path from $(0,0)$ to
the Weierstrass point at $(2+3a/2,1+a/2)$ that corresponds to a parabolic
fixed point of $\mathrm{PSL}(X,\omega)$ but not of $\langle
P_{1},P_{2}\rangle$. Let $Q\in\mathrm{PSL}(X,\omega)$ be the corresponding
element. Then the cylinder decomposition in the direction leads to
$Q=\begin{pmatrix}-254-147\sqrt{3}&657+379\sqrt{3}\\\
-99-57\sqrt{3}&256+147\sqrt{3}\end{pmatrix}$. Conjugation by $P_{2}^{-1}$ then
gives the element in $\mathrm{SL}(X,\omega)\setminus\langle-
I_{2},P_{1},P_{2}\rangle$ of smallest norm that we have found to date.

Figure 8. Cylinder composition in a parabolic direction not in $\langle
P_{1},P_{2}\rangle$ for McMullen’s $a=1+\sqrt{3}$ example of infinitely
generated Veech group. Here $P_{1},P_{2}$ are parabolic elements corresponding
to the vertical and horizontal directions. The surface, formed by identifying
opposite facing sides, is in $\mathcal{H}(1,1)$. Marked points are the
vertices, which give rise to the cone singularities, and the Weierstrass
points, see [Mc].

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