Origamis with non congruence Veech groups
Gabriela Schmithüsen
In this article we give an introduction to origamis (often also called square-tiled
surfaces) and their Veech groups. As main theorem we prove that in each genus
there exist origamis, whose Veech groups are non congruence subgroups of SL2(Z).
The basic idea of an origami is to obtain a topological surface from a few combina-
torial data by gluing finitely many Euclidean unit squares according to specified
rules. These surfaces come with a natural translation structure. One assigns in
general to a translation surface a subgroup of GL2(R) called the Veech group. In
the case of surfaces defined by origamis, the Veech groups are finite index sub-
groups of SL2(Z). These groups are the objects we study in this article.
One motivation to be interested in Veech groups is their relation to Teichmüller
disks and Teichmüller curves, see e.g. the article [H 06] of F. Herrlich in the
same volume: A translation surface of genus g defines in a geometric way an
embedding of the upper half plane into the Teichmüller space Tg of closed Rie-
mann surfaces of genus g. The image is called Teichmüller disk. Its projection to
the moduli space Mg is sometimes a complex algebraic curve, called Teichmüller
curve. More precisely this happens, if and only if the Veech group is a lattice in
SL2(R). In this case the algebraic curve can be determined from the Veech group
up to birationality.
It is hard to determine the Veech group for a general translation surface. How-
ever, if the translation surface comes from an origami there is a special approach
to this problem. It is based on the idea of describing origamis by finite index sub-
groups of F2, the free group in two generators. This leads to a characterization of
origami Veech groups as the images in SL2(Z) of certain subgroups of Aut(F2),
the automorphism group of F2.
Using this approach we will calculate Veech groups of two origamis explicitly.
They turn out to be non congruence groups. Starting from these examples we
obtain infinite sequences of origamis all of whose Veech groups are non congruence
groups. This leads to the following theorem.
Theorem 1. Each moduli space Mg (g ≥ 2) contains an origami curve whose
Veech group is a non congruence group.
In Section 1 we introduce origamis and present different equivalent ways to de-
scribe them. In Section 2 we give a glance on the mathematical context. We
describe, how an origami defines a family of translation surfaces and explain
roughly , how one obtains a Teichmüller curve in moduli space starting from
http://arxiv.org/abs/0704.0416v1
an origami. We introduce Veech groups and shortly point out their relation to
Teichmüller curves. In Section 3 we turn to Veech groups of origamis and present
a characterization of them in terms of automorphisms of the free group F2 in two
generators. We use this characterization to calculate two examples explicitly.
Finally, in Section 4 we show that these two examples produce Veech groups that
are non congruence groups and give a method to construct out of them infinite
sequences of Veech groups that are again non congruence groups.
The first part (Section 1 - Section 3) of this article is meant to give a handy in-
troduction to origamis and an overview on some of our results about their Veech
groups. In the second part we state and prove Theorem 1 based on the results in
the PhD thesis [S 05] of the author.
For a broader introduction and overview on origamis and Teichmüller curves
as well as for references to the larger context, we refer the the reader e.g. to
[HeSc 06], [S 04] and [S 05].
Acknowledgments: I would like to thank Frank Herrlich for his support in
respect of the content and for his proof reading, Stefan Kühnlein for helpful dis-
cussions and suggestions especially on non congruence groups and the organizers
of the conference for giving me the opportunity to contribute to these proceedings.
This work was partially supported by a fellowship within the Postdoc-Programme
of the German Academic Exchange Service (DAAD).
1 ORIGAMIS 3
1 Origamis
There are several ways to define origamis. We start with the somehow playful
description that we have learned from [Lo 05], where also the name origami was
introduced: An origami is obtained by gluing the edges of finitely many copies
Q1, . . . , Qd of the Euclidean square Q via translations according to the following
rules:
• Each left edge shall be identified to a right edge and vice versa.
• Similarly, each upper edge shall be identified to a lower one.
• The arising closed surface X shall be connected.
We only study what is called oriented origamis in [Lo 05] and call them just
origamis.
Example 1.1.
a) The simplest example is the origami that is made from only one square.
There is precisely one possibility to glue its edges according to the rules.
One obtains a torus E. We call this origami the trivial origami O0.
Figure 1: The trivial origami. Opposite edges are glued.
Observe that the four vertices of the square are all identified and become
one point on the closed surface E. We call this point ∞.
b) We now consider an origami made from four squares, see Figure 2. Some
identifications of the edges are already done in the picture. For all other
edges those having same labels are glued. The origami is called L(2, 3) for
obvious reasons.
2 3 4
a b c
Figure 2: The origami L(2, 3). Opposite edges are glued.
1 ORIGAMIS 4
Observe that in this case the vertices labeled with • and the vertices labeled
with ◦ are respectively identified and become two points on the closed surface
X. By calculating the Euler characteristic one obtains, that the genus of
the surface X is 2.
c) Finally, we consider an example with five squares, see Figure 3. Here,
edges with same labels are identified. For the unlabeled edges, those which
are opposite to each other are glued. We call the origami D.
1 2 3
• • •
Figure 3: The origami D. Edges with the same label and unlabeled edges
that are opposite are glued.
In this case, we obtain the three identification classes ◦, ⋆ and • for the
vertices. The genus of the closed surface X is again 2.
Origamis as coverings of a torus
Observe, that the trivial origami O0 from Example 1.1 a) is universal in the
following sense: If X is the closed surface that arises from an arbitrary origami
O and E the torus that arises from O0, then we have a natural map X → E by
mapping each of the unit squares of the origami O that form the surface X to
the one unit square of O0 that forms the torus E. This map is a covering that is
unramified except over the one point ∞ ∈ E. Conversely, given a closed surface
X together with such a covering p : X → E, we obtain a decomposition of X
into squares by cutting X along the preimages of the edges of the one square of
O0 that forms E. This motivates the following definition of origamis.
Definition 1.2. An origami O of genus g and degree d is a covering p : X → E
of degree d from a closed, oriented (topological) surface X of genus g to the torus
E that is ramified over at most one marked point ∞ ∈ E.
Remark that we have fixed here one torus E and one point ∞ ∈ E. In particular
we may furthermore fix a point M 6= ∞ on E and a set of standard generators
of the fundamental group π1(E,M) that do not pass through ∞. That way we
obtain a fixed isomorphism
∗) ∼= F2, (1)
1 ORIGAMIS 5
where E∗ = E−∞ and F2 = F2(x, y) is the free group in two generators x and y.
Describing E by gluing the edges of the unit square via translations, we choose
M to be the midpoint of the unit square and the standard generators to be the
horizontal and the vertical simply closed curve through M , see Figure 4.
Figure 4: Generators of π1(E
Example 1.3. In Example 1.1, in a) the covering is the identity id : E → E.
In b) we have a covering p : X → E of degree 4 that is ramified in the two points
labeled by • and ◦. Recall that the genus of X is 2.
In c) we have a covering p : X → E of degree 5 ramified in the two points labeled
by • and ⋆. Observe that though the point on X labeled by ◦ is a preimage of ∞,
the covering is not ramified in this point. The genus of X is again 2.
Definition 1.4. We say that two origamis O1 = (p1 : X1 → E) and O2 =
(p2 : X2 → E) are equivalent, if there is a homeomorphism ϕ : X1 → X2 with
p1 = p2 ◦ ϕ.
Description by a pair of permutations
An origami O = p : (X → E) of degree d defines (up to conjugation in Sd)
• a homomorphism m : F2 = F2(x, y) → Sd or equivalently
• a pair of permutations (σa, σb) in Sd
as follows:
Let M1, . . . , Md be the preimages of the point M (defined as above) under p.
Furthermore, let
m : π1(E
∗,M) → Sym(M1, . . . ,Md)
be the monodromy map defined by p, i.e. for the closed path c ∈ π1(E
∗,M) the
point Mi is mapped to Mj by m(c) if and only if the lift of the curve c to X via
p, that starts in Mi, ends in Mj.
Choosing an isomorphism Sym(M1, . . . ,Md) ∼= Sd and using the isomorphism
∗) ∼= F2 fixed in (1) makes m into a homomorphism from F2 to Sd. We set
σa = m(x) and σb = m(y).
Observe that this homomorphism depends on the chosen isomorphism to Sd and
on the choice of the origami in its equivalence class only up to conjugation in Sd.
Therefore we consider two homomorphisms m1 : F2 → Sd and m2 : F2 → Sd to
be equivalent, if they are conjugated by an element in Sd. Similarly we call two
pairs (σa, σb) and (σ
b) in Sd equivalent, if they are simultaneously conjugated,
i.e. there is some s ∈ Sd such that σa = sσ
−1 and σb = sσ
1 ORIGAMIS 6
Example 1.5. In Example 1.1 we obtain for the origami L(2, 3) in b) the mon-
odromy homomorphism
m : F2 → S4, x 7→ (2 3 4) and y 7→ (2 1),
and thus σa = (2 3 4) and σb = (2 1).
For the origami D in c) we similarly obtain the permutations
σa = (1 2 3) and σb = (1 4 5)(2 3).
Description as finite index subgroups of F2
Origamis can be equivalently described as finite index subgroups of F2, the free
group in two generators, as stated in the following remark. The characterization
of the Veech groups of origamis is mainly based on this observation.
Remark 1.6. We have a one-to-one correspondence:
origamis up to equivalence ↔ finite index subgroups of F2 up to conjugacy.
More precisely, this correspondence is given as follows:
Let O = (p : X → E) be an origami. Define E∗ = E − {∞} and X∗ =
X − p−1(∞). Thus we may restrict p to the unramified covering p : X∗ → E∗.
This defines an embedding of the corresponding fundamental groups:
U = π1(X
∗) ⊆ π1(E
∗) ∼= F2
Again we use the fixed isomorphism in (1), see also Figure 4. Changing the
origami in its equivalence class leads to a conjugation of U with an element in
F2. The index of the subgroup of F2 is the degree d of the covering p.
Conversely, given a finite index subgroup U of F2 we retrieve the origami in
the following way: Let v : Ẽ∗ → E∗ be a universal covering of E∗. By the
theorem of the universal covering, π1(E
∗) is isomorphic to Deck(Ẽ∗/E∗), the
group of deck transformations of Ẽ∗/E∗. Furthermore, the finite index subgroup
U of Deck(Ẽ∗/E∗) corresponds to an unramified covering p : X∗ → E∗ of finite
degree. This can be extended to a covering X → E, where X is a closed surface.
Example 1.7. In Example 1.1, we obtain the following subgroups of F2:
In a), X∗ is the once punctured torus itself and U = F2.
In b), X∗ is a genus 2 surface with 2 punctures. Thus U = π1(X
∗) is a free group
of rank 5. Keeping in mind that we use the identification π1(E
∗) ∼= F2 = F2(x, y)
described in Figure 4, one can read off from the picture in Figure 2 that
U = < x3, xyx−1, x2yx−2, yxy−1, y2 >
In c), X∗ is a genus 2 surface with three punctures. Thus U is a free group of
rank 6. More precisely, we read off the picture in Figure 3, that
U = < x3, xyx−2, x2yx−1, yxy−1, y2xy−2, y3 >
2 TRANSLATION STRUCTURES AND VEECH GROUPS 7
Description as a finite graph
Finitely, sometimes it is convenient to describe an origami O = (p : X → E)
as a finite, oriented labeled graph: Namely, let U be the finite index subgroup
of F2 (unique up to conjugation) that corresponds to O as described in the last
paragraph. Then we represent the origami by the Cayley-Graph of U ⊆ F2:
The vertices of the graph are the coset representatives. They are labeled with a
representative of the coset. The edges are labeled with x and y. For each vertex
(with label w ∈ F2) there is an x-edge from it to the vertex that belongs to the
coset of wx. And similarly there is a y-edge to the vertex that belongs to the
coset wy.
Example 1.8. The following figure shows the Cayley-graph for the origami L(2, 3)
from Example 1.1:
?>=<89:;ȳ
GFED@ABCīd
x // ?>=<89:;x̄ x //
�� GFED@ABC
Figure 5: Graph for O = L(2, 3).
2 Translation structures and Veech groups
Translation structures
Recall that an atlas on a surface is called translation atlas, if all transition maps
are translations. An origamiO = (p : X → E) naturally defines an SL2(R)-family
of translation structures µA (A ∈ SL2(R)) on X
∗ = X − p−1(∞) as follows:
• As first step, observe that each A ∈ SL2(R) naturally defines a translation
structure ηA on the torus E itself by identifying it with C/ΛA, where
and ΛA is the lattice <
> in C (2)
• Then define the translation structure µA on X
∗ by lifting ηA via p, i.e.
µA = p
Using the first description of an origami that we gave by gluing squares, we obtain
the translation structure µI (where I is the identity matrix), if we identify the
squares with the Euclidean unit square in C. We obtain µA for a general matrix
2 TRANSLATION STRUCTURES AND VEECH GROUPS 8
A ∈ SL2(R) from this by identifying the squares with the parallelogram spanned
by the two vectors
Thus the SL2(R)-variations of the translation structure µI can be thought of as
affine shearing of the unit squares, see Figure 6.
Figure 6: Sheared translation structure for the origami L(2, 3).
From an origami to a Teichmüller curve in the moduli space
By the SL2(R)-family of translation structures, the origami O = (p : X → E)
defines a specific complex algebraic curve called Teichmüller curve in the moduli
space Mg of closed Riemann surfaces of genus g. We state this construction here
only briefly as motivation and refer e.g. to the overview article [HeSc 06] for a
detailed description and links to references. A particular nice configuration of
such Teichmüller curves is described in [H 06] in this volume.
The Teichmüller curve in Mg is obtained from the origami in the following way:
• The translation structure µA described in the previous paragraph is in par-
ticular a complex structure on the surface X∗ which can be extended to the
closed surface X . The Riemann surface (X, µA) together with the identity
map id : X → X as marking then defines a point in the Teichmüller space
Tg. Thus we obtain the map: ι̂ : SL2(R) → Tg, A 7→ [(X, µA), id].
• If A ∈ SO2(R), then the affine map z 7→ A ·z is holomorphic. Thus the map
ι̂ factors through SO2(R). Furthermore using that SL2(R) modulo SO2(R)
is isomorphic to the upper half plane H, one obtains a map
ι : H ∼= SO2(R)\SL2(R) → Tg
In fact, this map is an embedding that is in the same time holomorphic and
isometric. A map with this property is called Teichmüller embedding and
its image ∆ in Teichmüller space is called a Teichmüller disk or a geodesic
disk.
2 TRANSLATION STRUCTURES AND VEECH GROUPS 9
• Finally, one may compose the map ι with the projection to the moduli space
Mg. The image of ∆ in Mg is a complex algebraic curve. A curve in Mg
that arises like this as the image of a Teichmüller disk is called Teichmüller
curve.
Note: More generally, one obtains a Teichmüller disk ∆ in a similar way starting
from an arbitrary translation surface (or a bit more general: from a flat surface).
However, the image of such a disk ∆ in moduli space is not always a complex
algebraic curve; in fact its Zariski closure tends to be of higher dimension. It
is an interesting question how to decide whether a translation surface leads to a
Teichmüller curve. One possible answer to this is given by the Veech group which
we introduce in the following paragraph.
Veech groups
Let X∗ be a connected surface and µ a translation structure on it. One assigns to
it a subgroup of GL2(R) called Veech group as described in the following: We con-
sider the group Aff+(X∗, µ) of all orientation preserving affine diffeomorphisms,
i.e. orientation preserving diffeomorphisms that are locally affine maps of the
plane C, see Figure 7. Here – and throughout the whole article – we identify C
with R2 by the map z 7→ (Re(z) , Im(z))t. Thus an affine diffeomorphism f can
be written in terms of local charts as
f : z = (Re(z), Im(z))t 7→ A · (Re(z), Im(z))t+ z0 with A ∈ GL2(R) and z0 ∈ C.
Observe that A does not depend on the chart, since µ is a translation structure.
Thus one obtains a well defined map
D : Aff+(X∗, µ) → GL2(R), f 7→ A
called Derivative map.
Definition 2.1. The Veech group Γ(X∗, µ) of the translation surface (X∗, µ) is
the image of the derivative map D:
Γ(X∗, µ) = D(Aff+(X∗, µ))
z 7→ Az + z0
Figure 7: An affine diffeomorphism of a translation surface
3 VEECH GROUPS OF ORIGAMIS 10
Example 2.2. Let (X∗, µ) be C/ΛI with the natural translation structure. Here
I is the identity matrix and ΛI is the corresponding lattice as defined in (2).
An affine diffeomorphisms of C/ΛI lifts to an affine diffeomorphism of C respect-
ing the lattice. Conversely, each such diffeomorphism descends to C/ΛI. Thus,
we have in this case
Γ(X∗, µ) = SL2(Z).
Veech groups and Teichmüller curves
As indicated in the paragraph about Teichmüller curves, the Veech group “knows”
whether a translation surface defines a Teichmüller curve in moduli space or not.
More precisely, one has the following statement:
Fact: Let X be a surface of genus g and X∗ = X−{P1, . . . , Pn} for finitely many
points P1, . . . , Pn on X . Furthermore let µ be a translation structure on X
Then (X∗, µ) defines a Teichmüller curve C if and only if the Veech group Γ(X∗, µ)
is a lattice in SL2(R). In this case, the curve C is (antiholomorphic) birational
to H/Γ(X∗, µ).
We describe the relation to Teichmüller curves here just as motivation and in
order to give a glance at the general frame. We have therefore resumed theorems
contributed by several authors condensed in what is here called “fact”. A good
access to it can be found e.g. in [EG 97] or [Z 06]. A broader overview on
Veech groups of translation surfaces is given e.g. in [HuSc 01] and in [Le 02].
Teichmüller disks, Teichmüller curves and Veech groups have intensively been
studied by numerous authors, starting from Thurston [T 88] and Veech himself
[V 89]. We refer to [S 04] and [HeSc 06] for more comprehensive overviews on
references.
3 Veech groups of origamis
Let O = p : (X → E) be an origami. We have seen in Section 2 that O defines an
SL2(R)-family of translation structures µA (A ∈ SL2(R)) on X
∗ = X − p−1(∞).
The corresponding Veech groups are not very different. In fact, they are all
conjugated to each other. More precisely, we have:
Γ(X∗, µA) = AΓ(X
∗, µI)A
Thus, we may restrict to the case where A = I which justifies the following
definition.
Definition 3.1. The Veech group Γ(O) of the origamiO is defined to be Γ(X∗, µI).
From Example 2.2 it follows that the Veech group of the trivial origami O0 (de-
fined in Example 1.1) is SL2(Z). For a general origami one can show that Γ(O)
3 VEECH GROUPS OF ORIGAMIS 11
is a finite index subgroup of SL2(Z). In fact, also the converse is true as it was
shown by Gutkin and Judge in [GJ 00]: A Veech group is a finite index subgroup
of SL2(Z) if and only if it comes from an origami.
From this it follows in particular by the Fact presented in Section 2 on page 10
that an origami always defines a Teichmüller curve in the moduli space.
Characterization of origami Veech groups
Recall from Section 1 that an origami O corresponds (up to equivalence) to a
finite index subgroup U of F2 = F2(x, y), the free group in two generators (up to
conjugation). This description enables us to give a characterization of its Veech
group entirely in terms of F2 and its automorphisms.
For this we need the following two ingredients:
• Let β̂ : Aut(F2) → Out(F2) ∼= GL2(Z) be the natural projection. The
fact that we only consider orientation preserving diffeomorphisms applies
to only taking automorphisms of Aut(F2) that are mapped to elements in
SL2(Z). We denote Aut
+(F2) = β̂
−1(SL2(Z)) and restrict to the map
β̂ : Aut+(F2) → SL2(Z).
• Let Stab(U) = {γ ∈ Aut+(F2)|γ(U) = U}
Using these ingredients, it was shown in [S 04] that Veech groups of origamis can
be described as stated in the following theorem.
Theorem 2 (Proposition 1 in [S 04]). For the Veech group Γ(O) of the origami
O holds:
Γ(O) = β̂(Stab(U))
Let us make two comments on this description:
One consequence is, that one obtains an algorithm that can calculate the Veech
group of an arbitrary origami explicitly. This algorithm is described in detail in
[S 04].
As an other consequence, we have now a characterization of all origami Veech
groups as stated in the following corollary.
Corollary 3.2. A finite index subgroup of SL2(Z) occurs as origami Veech group
if and only if it is the image of the stabilizing group Stab(U) ⊆ Aut+(F2) for
some finite index subgroup U in F2.
Thus the question, which finite index subgroups of SL2(Z) are Veech groups be-
comes roughly speaking the same as the question which subgroups of Aut+(F2)
are such stabilizing groups. So far, there is no general answer known.
3 VEECH GROUPS OF ORIGAMIS 12
In [S 05] it was shown that many congruence subgroups of SL2(Z) are Veech
groups. Recall that a congruence group of level n is a subgroup of SL2(Z) that is
the full preimage of some subgroup of SL2(Z/nZ) under the natural homomor-
phism SL2(Z) → SL2(Z/nZ) and n shall be minimal with this property. For prime
level congruence groups the following statement is shown in [S 05, Theorem 4]
Theorem 3. Let p be prime. All congruence groups Γ of level p are Veech groups
except possibly p ∈ {2, 3, 5, 7, 11} and Γ has index p in SL2(Z).
This result is generalized to a statement for arbitrary n in [S 05, Theorem 5]
Presenting the Veech group Γ and the quotient H/Γ for an origami
As mentioned above, using Theorem 2 the Veech group of an origami can be
calculated explicitly. The Veech groups are described as subgroups of SL2(Z)
by generators and coset representatives. We use for the notation that SL2(Z) is
generated by S and T , with
and T =
Recall furthermore from the discussion on Veech groups and Teichmüller curves
in Section 2 on page 10 that for a Veech group Γ we are in particular interested
in the quotient H/Γ, since this quotient is birational to the corresponding Te-
ichmüller curve. Here Γ acts as Fuchsian group on the upper half plane H, which
is endowed with the Poincaré metric.
Since an origami Veech group Γ is a finite index subgroup of SL2(Z), the quotient
H/Γ comes with a natural triangulation. More precisely, we choose the funda-
mental domain for the action of SL2(Z) on H that is the geodesic pseudo-triangle
∆ with vertices P = −1
i, Q = 1
i and P∞ = ∞.
Figure 8: Fundamental domain of SL2(Z).
The surface H/SL2(Z) is obtained by identifying the vertical edges P∞ and Q∞
via T and the edge PQ with itself (with fixed point i) via S.
For an arbitrary subgroup Γ of SL2(Z) of finite index we obtain a fundamental
domain as a union of translates of the triangle ∆: for each coset A we take the
3 VEECH GROUPS OF ORIGAMIS 13
triangle A(∆), where A is a representative of the coset. The identification of
the edges is given by the elements in Γ. Gluing the edges gives the quotient
surface H/Γ, filling in the cusps leads to a closed Riemann surface endowed with
a triangulation. We draw stylized pictures of the fundamental domains that
indicate the triangles (see Figure 9 and 10). The triangles are labeled with a
coset representative, edges that are identified are labeled with the same letter
and vertices that are identified with the same number. Vertices that come from
cusps (i.e. points at ∞) are marked with •.
In particular, one can read off from these stylized pictures the genus and the
number of cusps of the quotient surface H/Γ.
Two examples: the origami L(2,3) and the origami D
The origami L(2,3):
In [S 04, Example 3.5] the Veech group is calculated as follows:
Γ(L(2, 3)) = <
More precisely, one obtains the generators presented as products of S and T as
well as a list of coset representatives.
• List of generators:
= T 3,
= TST 2ST−1T−1,
= TSTST−1S,
= T 2STST−1S−1T−2,
• List of representatives:
I, T, S, T 2, TS, ST, T 2S, TST, T 2ST
Hence, Γ(L(2, 3)) is a subgroup of index 9 in SL2(Z).
The stylized picture of the quotient H/Γ(L(2, 3)) is determined in [S 04, Example
3.6] and is shown here in Figure 9.
3 VEECH GROUPS OF ORIGAMIS 14
TTSTT
Figure 9: Fundamental domain of Γ(L(2, 3)).
From this one can read off that the genus of the quotient H/Γ(L(2, 3)) is 0 and
that it has 3 cusps, namely the vertices labeled by 1,4 and 5. It follows in par-
ticular that the corresponding Teichmüller curve has genus 0.
The origami D:
The Veech group of the origami D is calculated in [S 05, Section 7.3.2]. It has
index 24 in SL2(Z) and the following generators:
= −I, A1 =
= T 3,
= ST 6S−1, A3 =
−7 16
= (T 2S)T 4(T 2S)−1.
= (TS)T 4(TS)−1, A5 =
−20 11
= (TST 2S)T 5(TST 2S)−1,
−18 −5
= (ST 3S)T 2(ST 3S)−1,
The following is a system of cosets representatives:
I , T , S , T 2 , TS , ST , T 2S , TST , ST 2 , STS , T 2ST , TST 2 ,
ST 5 , ST 3 , T 2S , TST 3 , TST 2S , ST 4 , ST 3S , TST 2ST−1 ,
TST 2ST−2 , TST 2ST−3 ; TST 2ST−4 , ST 3ST
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 15
The corresponding origami curve C(D) has genus 0. It is shown with its natu-
ral triangulation in Figure 10. It has six cusps, namely C1, C2, C3, C4, C5 and C6.
TT TTS
TTSTT
STTSTT
TSTTS
TSTTST−1
3ST ST
Figure 10: The origami curve to D.
4 Veech groups that are non congruence groups
Theorem 3 implies that there are many congruence groups which are Veech
groups. How about non congruence groups? In this section we will see that
the Veech groups for the two examples, the origami L(2, 3) and the origami D,
studied in the last paragraph are both non congruence groups. Furthermore,
we give a construction that produces for both of them an infinite sequence of
origamis whose Veech group is a non congruence group. We use this in order to
prove our main theorem.
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 16
An other generalization of the example L(2, 3) was given by Hubert and Lelièvre
in [HL 05], where they show for certain “L-shaped” origamis or square-tiled sur-
faces, how they are called there, that their Veech groups are non congruence
groups. These surfaces are all of genus 2, hence it follows that there are infinitely
many origamis of genus 2 whose Veech group is a non congruence group.
Recall that a group is a congruence group, whose level is a divisor of n, if and
only if it contains the principal congruence group
Γ(n) = {
mod n} = kernel(proj : SL2(Z) → SL2(Z/nZ))
In [S 04, Proposition 3.8] it was shown using a proof of Stefan Kühnlein that the
Veech group of L(2, 3) is a non congruence group. The basic tool for this is the
general level that is defined for any subgroup Γ of SL2(Z) as follows: For each
cusp we define its amplitude to be the smallest natural number n such that there
is an element of Γ conjugated in SL2(Z) to the matrix
which fixes the cusp. Observe that this is equal to the number of triangles around
the vertex that represents the cusp in our stylized picture of the quotient surface
(see Figures 9 and 10). The general level of Γ is the least common multiple of the
amplitudes of all its cusps. A theorem of Wohlfahrt [W 64, Theorem 2] states
that the level and the general level of a congruence group coincide.
The amplitude of the three cusps of H/Γ(L(2, 3)) labeled with 1, 4 and 5 in Fig-
ure 9 is 3, 2 and 4 respectively. Hence, the general level of Γ(L(2, 3)) is 12. Then
it is shown in the proof that Γ(L(2, 3)) does not contain Γ(12) which gives the
contradiction.
The same method can be used in order to show that Γ(D) is a non congruence
group. We here carry out the proof for it. Observe from Figure 10 that the six
cusps C1, . . . , C6 have the amplitude 3, 6, 4, 4, 5 and 2, respectively. Thus the
general level is 60.
Proposition 4.1. The Veech group Γ(D) is a non congruence group.
Proof. Suppose that Γ = Γ(D) is a congruence group. Since the general level of
Γ is 60, we have by the theorem of Wohlfahrt mentioned above, that Γ(60) is a
subgroup of Γ.
We will use the following facts, which can be checked e.g. in Figure 10:
∈ Γ, A6 =
−18 −5
∈ Γ and T =
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 17
In order to verify this in Figure 10, use that
A1 = T
3 and A6 = S
−1T 2S−1T−1S−1TS−1T−3S−1.
We will find an element in Γ whose projection to SL2(Z/60Z) is equal to that of
T , which gives us the desired contradiction.
Recall that
SL2(Z/60Z) ∼= SL2(Z/4Z)× SL2(Z/3Z)× SL2(Z/5Z).
We identify in the following these two groups. Furthermore we denote by p4, p3,
p5 and p60 the projection from SL2(Z) to SL2(Z/4Z), SL2(Z/3Z), SL2(Z/5Z) and
SL2(Z/60Z), respectively. Then p60 = p4 × p3 × p5.
We have
p60(A1) = (
) and
p60(A6) = (
The order of p4(A1) in SL2(Z/4Z) is 4, the order of p3(A1) in SL2(Z/3Z) is 1
and the order of p5(A1) in SL2(Z/5Z) is 5. We also say: The order of p60(A1) is
(4, 1, 5). Since 7 ≡ 3 mod 4 and 7 ≡ 2 mod 5 we have
p60(A
1) = (
) (4)
Furthermore:
p60(A
6) = (
and with the same notation as above p60(A
6) has the order (1, 3, 5). Thus
p60(A
6 ) = (
). (5)
From (4) and (5) it follows that
p60(A
6 · A
1) = (
) = p60(
) = p60(T )
But A206 · A
1 ∈ Γ and T /∈ Γ, thus Γ(60) = ker(p60) cannot be contained in Γ.
Therefore, Γ cannot be a congruence group of level 60. Contradiction!
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 18
Sequences of origamis with non congruence Veech groups
Starting from the origamis L(2, 3) and D we will define respectively a sequence
On, such that for each n ∈ N the Veech group Γ(On) again is a non congruence
group. The basic idea is to “copy and paste”: we will cut the origami along a
segment, take n copies of it and glue them along the cuts.
In Figure 11 we show the origami On for L(2, 3):
1 3 4
5 7 8
. . . 4n-7 4n-5 4n-4
4n-3 4n-1 4n
Figure 11: n copies of L(2, 3). Opposite edges are glued.
Using the description of an origami by a pair of permutations from Section 1, On
is given as:
σa = (1 3 4 5 7 8 9 11 12 . . . 4n−3 4n−1 4n), σb = (1 2)(5 6) . . . (4n−3 4n−2).
Observe that the genus of On is n + 1 and it has 2n cusps: n of order 3 (all n
marked by • in Figure 11), and n of order 1 (all n marked by ◦ in Figure 11).
Finally, we want to present the origami On by the finite index subgroup Hn =
∗) of F2, that corresponds to On by Remark 1.6.
Recall from Example 1.7 that for O1 = L(2, 3), we obtain the free group of rank 5:
U = H1 =< g1 = x
3, g2 = xyx
−1, g3 = x
2yx−2, g4 = yxy
−1, g5 = y
2 >= F5.
The group Hn is obtained as as follows:
Hn = < g
1 , g
1 gj g
1 ∈ F5 | i ∈ {0, . . . , n− 1} and j ∈ {2, . . . , 5} >
In Figure 12, we show the origami Dn:
1 2 3
6 7 8
. . . . . . . . .
5n-4 5n-3 5n-2
b1 a2
b2 an
• • •
* • • •
• • •
Figure 12: n copies of D. Edges with the same label or
unlabeled opposite edges are glued.
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 19
The pair of permutations describing Dn is:
σa = (1 2 3 6 7 8 . . . 5n− 4 5n− 3 5n− 2),
σb = (1 4 5)(6 9 10) . . . (5n− 4 5n− 1 5n)(2 3)(7 8) . . . (5n− 3 5n− 2)
The genus of Dn is 2n and it has n+ 2 cusps: 2 of order 2n (marked as • and ⋆)
and n of order 1 (all n marked by ◦).
Again, we present On by the corresponding finite index subgroup Hn of F2. We
have from Example 1.7 that U = H1 = F6, the free group of rank 6:
U =< g′1 = x
3, g′2 = xyx
−2, g′3 = x
2yx−1, g′4 = yxy
−1, g′5 = y
2xy−2, g′6 = y
3 > = F6
And similarly as above, we obtain:
Hn = < g
1 , g
1 ∈ F6 | i ∈ {0, . . . , n− 1} and j ∈ {2, . . . , 6} >
We will see in the following that for both sequences all Veech groups Γ(On) are
non congruence groups. More precisely, we will show:
Proposition 4.2. For both sequences On the following holds:
• Γ(On) ⊆ Γ(O1), which is for both sequences a non congruence group.
• More generally one has:
n divides m ⇒ Γ(Om) ⊆ Γ(On).
• Different origamis in one sequence have different Veech groups, i.e.:
Γ(On) 6= Γ(Om) for n 6= m.
To prove this, let us detect that we are in the following more general setting.
Setting A:
• Let U be a finite index subgroup of F2. Then U is a free group of rank k
for some k ≥ 2, i.e.
U = < g1, . . . , gk > = Fk
• Let α : Fk → Z be the projection w 7→ ♯g1w
where ♯g1w is the number of g1 in the word w = w(g1, . . . , gk) with g
counted as −1.
• Let Hn be the kernel of pn ◦ α, where pn : Z → Z/nZ is the natural
projection, i.e.
Hn = < g
1 , g
1 gj g
1 ∈ Fk | i ∈ {0, . . . , n− 1} and j ∈ {2, . . . , k} > .
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 20
• Finally, let H0 be the kernel of α, i.e.:
Hn = << g2, . . . , gk >>U ,
is the normal subgroup in U generated by g2, . . . , gk.
Observe that we are in this setting with
U = π1(X
∗) =< x3, xyx−1, x2yx−2, yxy−1, y2 > for the origami L(2, 3) and
U = π1(X
∗) =< x3, xyx−2, x2yx−1, yxy−1, y2xy−2, y3 > for the origami D.
In order to prove the properties in Proposition 4.2, we will need that U fulfills
the following a bit technical condition:
Property B: Let U =< g1, . . . , gk > (k ≥ 2) be as above a finite index
subgroup of F2 of rank k and {wi}i∈I a system of coset representatives with
w1 = id. Suppose that U has the following property:
∀ j ∈ I − {1} : wj << g2, . . . , gk >>U w
j 6⊆ U.
One can check by hand that for both origamis, L(2, 3) and D, this property is
fulfilled. In this setting we obtain the following conclusions.
Proposition 4.3. Let n ∈ N ∪ {0}. Let U be a finite index subgroup of F2
fulfilling property B. With the notations from Setting A, we have:
a) The normalizer of Hn in F2 is equal to U : NormF2(Hn) = U
b) Stab
(Hn) ⊆ StabAut+(F2)
c) Recall that U = Fk, the free group in k generators.
Let βn : Aut(Fk) → GLk(Z/nZ) be the natural projection.
Then Stab
(Hn) is equal to
β−1n ({A = (ai,j)1≤i,j≤k ∈ GLk(Z/nZ)| a1,2 = . . . = a1,k = 0}) ∩ G.
Here we use the notation Z/(0Z) = Z thus β0 is the natural projection
Aut(Fk) → GLk(Z).
Proof.
By definition Hn is normal in U , i.e. U ⊆ NormF2(Hn).
Let now w be an element of F2\U . Hence, w = wj ·u for some j ∈ I−{1}, u ∈ U .
By Property B, there exists some h0 ∈ << g2, . . . , gk >>U = H0, such that
wjh0w
j 6∈ U . Therefore we have w(u
−1h0u)w
−1 6∈ U . But u−1h0u ∈ H0 ⊆ Hn,
since H0 is normal in U . This shows that w 6∈ NormF2(Hn).
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 21
This follows from a), since for a subgroup H of F2 in general holds:
(H) ⊆ Stab
(NormF2(H)), see e.g. [S 06, Remark 3.1].
Define M = {A = (ai,j)1≤i,j≤k ∈ GLk(Z/nZ)| a1,2 = . . . = a1,k = 0}.
Let γ ∈ G. We have to show that γ(Hn) = Hn if and only if βn(γ) ∈ M .
Let furthermore pkn : Fk → (Z/nZ)
k be the natural projection.
Consider the following commutative diagram:
Hn = p
n(Hn) ⊆ (Z/nZ)
βn(γ) // (Z/nZ)k ⊇ pkn(Hn) = Hn
Since pkn is surjective and Hn is the full preimage of Hn = p
n(Hn), it follows that
γ(Hn) = Hn if and only if βn(γ)(Hn) = Hn.
Observe finally that:
Hn = {(0, x2, . . . , xk) ∈ (Z/nZ)
k} and
StabGLk(Z/nZ)(Hn) = { A = (ai,j)1≤i,j≤k ∈ GLk(Z/nZ)|
(y1, . . . , yk) = A · (0, x2, . . . , xk) ⇒ y1 = 0 }
= {A = (ai,j) ∈ GLk(Z/nZ)| a1,2 = . . . = a1,k = 0}
Theorem 2 suggests the following notation.
Definition 4.4. Let U be a subgroup of F2.
With β̂ : Aut+(F2) → SL2(Z) as in Theorem 2, we define
Γ(U) = β̂(Stab
and call Γ(U) the Veech group of U .
We now obtain from Proposition 4.3 the following conclusions.
Corollary 4.5. Suppose that we are in the same situation as in Proposition 4.3,
in particular that U is a finite index subgroup of F2 fulfilling property B. Then
we have for all n ∈ N:
a) Stab
(H0) ⊆ StabAut+(F2)
(Hn) and Γ(H0) ⊆ Γ(Hn).
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 22
b) If m ∈ N with n|m, then:
(Hm) ⊆ StabAut+(F2)
(Hn) and Γ(Hm) ⊆ Γ(Hn).
(H0) =
(Hn) and Γ(H0) =
Γ(Hn)
Proof.
a) and b):
Let γ ∈ G. By Proposition 4.3 we have that
∀n ∈ N : γ ∈ Stab
(Hn) ⇔ βn(γ) = A = (ai,j)
with a1,2 ≡ . . . ≡ a1,k ≡ 0 mod n
and γ ∈ Stab
(H0) ⇔ β0(γ) = A = (ai,j)
with a1,2 = . . . = a1,k = 0.
Thus we have for all n ∈ N and for all m ∈ N with n|m, that
(H0) ⊆ StabAut+(F2)(Hm) ⊆ StabAut+(F2)(Hn).
We have in particular by the definition of the Veech group of a subgroup of F2:
Γ(H0) ⊆ Γ(Hm) ⊆ Γ(Hn).
⊆ follows from a). ⊇ follows from Remark [S 06, Remark 3.1].
We now return to the language of origamis: Let O be an origami, U the corre-
sponding subgroup of F2. Define for U the subgroups Hn (n ∈ N) as in Setting
A and let On be the origamis corresponding to the groups Hn.
By Corollary 4.5 and Theorem 2 we obtain immediately the following result.
Proposition 4.6. If U has the Property B, then
∀n ∈ N : Γ(On) ⊆ Γ(O) and ∀n,m ∈ N : n|m ⇒ Γ(Om) ⊆ Γ(On).
In particular, if Γ(O) is a non congruence group, each Γ(On) is a non congruence
group. Thus in this case, we obtain infinitely many origamis whose Veech group
is a non congruence group.
In order to conclude Proposition 4.2, it is now just left to prove the last item. But
this follows , since we have (see [S 05]) for both sequences On, the one coming
from the origami L(2, 3) and the one coming from the origami D, that
∈ Γ(On) ⇔ 3n divides s. (6)
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 23
This finishes the proof of Proposition 4.2.
Furthermore, Theorem 1 follows from Proposition 4.2.
Remark: From Corollary 4.5 and (6) it follows that Γ(H0) has infinite index in
SL2(Z). Furthermore it is non trivial, since it contains
for L(2, 3) respectively B3 =
for D.
REFERENCES 24
References
[EG 97] C.J. Earle, F.P. Gardiner: Teichmüller disks and Veech’s F -structures.
American Mathematical Society. Contemporary Mathematics 201, 1997 (p.
165–189).
[GJ 00] E. Gutkin, C. Judge: Affine mappings of translation surfaces. Duke
Mathematical Journal 103 No. 2, 2000 (p. 191–212).
[HeSc 06] F. Herrlich, G. Schmithüsen: On the boundary of Teichmüller disks
in Teichmüller and in Schottky space. To appear in Handbook of Teichmüller
theory. Ed. A. Papadopoulos, European Mathematical Society, 2006.
[H 06] F. Herrlich: A comb of origami curves in M3. Proceedings of Symposium
on Transformation Groups, Yokohama, November 2006.
[HL 05] P. Hubert, S. Lelièvre: Noncongruence subgroups in H(2). International
Mathematics Research Notices 2005, No.1 , 2005 (p. 47–64).
[HuSc 01] P. Hubert, T. Schmidt: Invariants of translation surfaces. Annales de
l’Institut Fourier 51 No. 2, 2001 (p. 461–495).
[Le 02] S. Lelièvre: Veech surfaces associated with rational billiards. Preprint,
2002. arXiv:math.GT/0205249.
[Lo 05] P. Lochak: On arithmetic curves in the moduli space of curves. J. Inst.
Math. Jussieu 4, No. 3, 2005 (p. 443–508).
[S 04] G. Schmithüsen: An algorithm for finding the Veech group of an origami.
Experimental Mathematics 13 No. 4, 2004 (p. 459–472).
[S 05] G. Schmithüsen: Veech Groups of Origamis. Dissertation (PhD thesis),
Karlsruhe 2005. Elektronisches Volltextarchiv EVA Universität Karlsruhe.
http://www.ubka.uni-karlsruhe.de/eva/
[S 06] Schmithüsen,G.: Examples for Veech groups of origamis. In: The Geome-
try of Riemann Surfaces and Abelian Varieties. Contemp. Math. 397, 2006 (p.
193–206).
[T 88] W. Thurston: On the geometry and dynamics of diffeomorphisms of sur-
faces. Bulletin (New Series) of the American Mathematical Society 19 No. 2,
1988 (p. 417–431).
[V 89] W.A. Veech: Teichmüller curves in moduli space, Eisenstein series and
an application to triangular billiards. Inventiones Mathematicae 97 No.3 1989,
(p. 553–583).
REFERENCES 25
[W 64] K. Wohlfahrt: An Extension of F. Klein’s Level Concept. Illinois Journal
of Mathematics 8, 1964 (p. 529–535).
[Z 06] A. Zorich: Flat Surfaces in collection ”Frontiers in Number Theory,
Physics and Geometry. Volume 1: On random matrices, zeta functions and
dynamical systems”, Ed. P. Cartier, B. Julia, P. Moussa, P. Vanhove, Springer-
Verlag, 2006 (p. 439–586).
	Origamis
	Translation structures and Veech groups
	Veech groups of origamis
	Veech groups that are non congruence groups

  As main result we show that for each g > 1 there is some translation surface
of genus g whose Veech group is a non congruence subgroup of SL(2,Z). We use
origamis/square-tiled surfaces to produce our examples. The article is divided
into two parts: In the first part we introduce translation surfaces, origamis,
Veech groups and Teichmueller curves and show for two origamis in genus 2 that
their Veech groups are non congruence groups; in the second part we provide a
technique that produces sequences of origamis whose Veech groups are
decreasing. This is used to prove the main result.

Origamis with non congruence Veech groups
Gabriela Schmithüsen
In this article we give an introduction to origamis (often also called square-tiled
surfaces) and their Veech groups. As main theorem we prove that in each genus
there exist origamis, whose Veech groups are non congruence subgroups of SL2(Z).
The basic idea of an origami is to obtain a topological surface from a few combina-
torial data by gluing finitely many Euclidean unit squares according to specified
rules. These surfaces come with a natural translation structure. One assigns in
general to a translation surface a subgroup of GL2(R) called the Veech group. In
the case of surfaces defined by origamis, the Veech groups are finite index sub-
groups of SL2(Z). These groups are the objects we study in this article.
One motivation to be interested in Veech groups is their relation to Teichmüller
disks and Teichmüller curves, see e.g. the article [H 06] of F. Herrlich in the
same volume: A translation surface of genus g defines in a geometric way an
embedding of the upper half plane into the Teichmüller space Tg of closed Rie-
mann surfaces of genus g. The image is called Teichmüller disk. Its projection to
the moduli space Mg is sometimes a complex algebraic curve, called Teichmüller
curve. More precisely this happens, if and only if the Veech group is a lattice in
SL2(R). In this case the algebraic curve can be determined from the Veech group
up to birationality.
It is hard to determine the Veech group for a general translation surface. How-
ever, if the translation surface comes from an origami there is a special approach
to this problem. It is based on the idea of describing origamis by finite index sub-
groups of F2, the free group in two generators. This leads to a characterization of
origami Veech groups as the images in SL2(Z) of certain subgroups of Aut(F2),
the automorphism group of F2.
Using this approach we will calculate Veech groups of two origamis explicitly.
They turn out to be non congruence groups. Starting from these examples we
obtain infinite sequences of origamis all of whose Veech groups are non congruence
groups. This leads to the following theorem.
Theorem 1. Each moduli space Mg (g ≥ 2) contains an origami curve whose
Veech group is a non congruence group.
In Section 1 we introduce origamis and present different equivalent ways to de-
scribe them. In Section 2 we give a glance on the mathematical context. We
describe, how an origami defines a family of translation surfaces and explain
roughly , how one obtains a Teichmüller curve in moduli space starting from
http://arxiv.org/abs/0704.0416v1
an origami. We introduce Veech groups and shortly point out their relation to
Teichmüller curves. In Section 3 we turn to Veech groups of origamis and present
a characterization of them in terms of automorphisms of the free group F2 in two
generators. We use this characterization to calculate two examples explicitly.
Finally, in Section 4 we show that these two examples produce Veech groups that
are non congruence groups and give a method to construct out of them infinite
sequences of Veech groups that are again non congruence groups.
The first part (Section 1 - Section 3) of this article is meant to give a handy in-
troduction to origamis and an overview on some of our results about their Veech
groups. In the second part we state and prove Theorem 1 based on the results in
the PhD thesis [S 05] of the author.
For a broader introduction and overview on origamis and Teichmüller curves
as well as for references to the larger context, we refer the the reader e.g. to
[HeSc 06], [S 04] and [S 05].
Acknowledgments: I would like to thank Frank Herrlich for his support in
respect of the content and for his proof reading, Stefan Kühnlein for helpful dis-
cussions and suggestions especially on non congruence groups and the organizers
of the conference for giving me the opportunity to contribute to these proceedings.
This work was partially supported by a fellowship within the Postdoc-Programme
of the German Academic Exchange Service (DAAD).
1 ORIGAMIS 3
1 Origamis
There are several ways to define origamis. We start with the somehow playful
description that we have learned from [Lo 05], where also the name origami was
introduced: An origami is obtained by gluing the edges of finitely many copies
Q1, . . . , Qd of the Euclidean square Q via translations according to the following
rules:
• Each left edge shall be identified to a right edge and vice versa.
• Similarly, each upper edge shall be identified to a lower one.
• The arising closed surface X shall be connected.
We only study what is called oriented origamis in [Lo 05] and call them just
origamis.
Example 1.1.
a) The simplest example is the origami that is made from only one square.
There is precisely one possibility to glue its edges according to the rules.
One obtains a torus E. We call this origami the trivial origami O0.
Figure 1: The trivial origami. Opposite edges are glued.
Observe that the four vertices of the square are all identified and become
one point on the closed surface E. We call this point ∞.
b) We now consider an origami made from four squares, see Figure 2. Some
identifications of the edges are already done in the picture. For all other
edges those having same labels are glued. The origami is called L(2, 3) for
obvious reasons.
2 3 4
a b c
Figure 2: The origami L(2, 3). Opposite edges are glued.
1 ORIGAMIS 4
Observe that in this case the vertices labeled with • and the vertices labeled
with ◦ are respectively identified and become two points on the closed surface
X. By calculating the Euler characteristic one obtains, that the genus of
the surface X is 2.
c) Finally, we consider an example with five squares, see Figure 3. Here,
edges with same labels are identified. For the unlabeled edges, those which
are opposite to each other are glued. We call the origami D.
1 2 3
• • •
Figure 3: The origami D. Edges with the same label and unlabeled edges
that are opposite are glued.
In this case, we obtain the three identification classes ◦, ⋆ and • for the
vertices. The genus of the closed surface X is again 2.
Origamis as coverings of a torus
Observe, that the trivial origami O0 from Example 1.1 a) is universal in the
following sense: If X is the closed surface that arises from an arbitrary origami
O and E the torus that arises from O0, then we have a natural map X → E by
mapping each of the unit squares of the origami O that form the surface X to
the one unit square of O0 that forms the torus E. This map is a covering that is
unramified except over the one point ∞ ∈ E. Conversely, given a closed surface
X together with such a covering p : X → E, we obtain a decomposition of X
into squares by cutting X along the preimages of the edges of the one square of
O0 that forms E. This motivates the following definition of origamis.
Definition 1.2. An origami O of genus g and degree d is a covering p : X → E
of degree d from a closed, oriented (topological) surface X of genus g to the torus
E that is ramified over at most one marked point ∞ ∈ E.
Remark that we have fixed here one torus E and one point ∞ ∈ E. In particular
we may furthermore fix a point M 6= ∞ on E and a set of standard generators
of the fundamental group π1(E,M) that do not pass through ∞. That way we
obtain a fixed isomorphism
∗) ∼= F2, (1)
1 ORIGAMIS 5
where E∗ = E−∞ and F2 = F2(x, y) is the free group in two generators x and y.
Describing E by gluing the edges of the unit square via translations, we choose
M to be the midpoint of the unit square and the standard generators to be the
horizontal and the vertical simply closed curve through M , see Figure 4.
Figure 4: Generators of π1(E
Example 1.3. In Example 1.1, in a) the covering is the identity id : E → E.
In b) we have a covering p : X → E of degree 4 that is ramified in the two points
labeled by • and ◦. Recall that the genus of X is 2.
In c) we have a covering p : X → E of degree 5 ramified in the two points labeled
by • and ⋆. Observe that though the point on X labeled by ◦ is a preimage of ∞,
the covering is not ramified in this point. The genus of X is again 2.
Definition 1.4. We say that two origamis O1 = (p1 : X1 → E) and O2 =
(p2 : X2 → E) are equivalent, if there is a homeomorphism ϕ : X1 → X2 with
p1 = p2 ◦ ϕ.
Description by a pair of permutations
An origami O = p : (X → E) of degree d defines (up to conjugation in Sd)
• a homomorphism m : F2 = F2(x, y) → Sd or equivalently
• a pair of permutations (σa, σb) in Sd
as follows:
Let M1, . . . , Md be the preimages of the point M (defined as above) under p.
Furthermore, let
m : π1(E
∗,M) → Sym(M1, . . . ,Md)
be the monodromy map defined by p, i.e. for the closed path c ∈ π1(E
∗,M) the
point Mi is mapped to Mj by m(c) if and only if the lift of the curve c to X via
p, that starts in Mi, ends in Mj.
Choosing an isomorphism Sym(M1, . . . ,Md) ∼= Sd and using the isomorphism
∗) ∼= F2 fixed in (1) makes m into a homomorphism from F2 to Sd. We set
σa = m(x) and σb = m(y).
Observe that this homomorphism depends on the chosen isomorphism to Sd and
on the choice of the origami in its equivalence class only up to conjugation in Sd.
Therefore we consider two homomorphisms m1 : F2 → Sd and m2 : F2 → Sd to
be equivalent, if they are conjugated by an element in Sd. Similarly we call two
pairs (σa, σb) and (σ
b) in Sd equivalent, if they are simultaneously conjugated,
i.e. there is some s ∈ Sd such that σa = sσ
−1 and σb = sσ
1 ORIGAMIS 6
Example 1.5. In Example 1.1 we obtain for the origami L(2, 3) in b) the mon-
odromy homomorphism
m : F2 → S4, x 7→ (2 3 4) and y 7→ (2 1),
and thus σa = (2 3 4) and σb = (2 1).
For the origami D in c) we similarly obtain the permutations
σa = (1 2 3) and σb = (1 4 5)(2 3).
Description as finite index subgroups of F2
Origamis can be equivalently described as finite index subgroups of F2, the free
group in two generators, as stated in the following remark. The characterization
of the Veech groups of origamis is mainly based on this observation.
Remark 1.6. We have a one-to-one correspondence:
origamis up to equivalence ↔ finite index subgroups of F2 up to conjugacy.
More precisely, this correspondence is given as follows:
Let O = (p : X → E) be an origami. Define E∗ = E − {∞} and X∗ =
X − p−1(∞). Thus we may restrict p to the unramified covering p : X∗ → E∗.
This defines an embedding of the corresponding fundamental groups:
U = π1(X
∗) ⊆ π1(E
∗) ∼= F2
Again we use the fixed isomorphism in (1), see also Figure 4. Changing the
origami in its equivalence class leads to a conjugation of U with an element in
F2. The index of the subgroup of F2 is the degree d of the covering p.
Conversely, given a finite index subgroup U of F2 we retrieve the origami in
the following way: Let v : Ẽ∗ → E∗ be a universal covering of E∗. By the
theorem of the universal covering, π1(E
∗) is isomorphic to Deck(Ẽ∗/E∗), the
group of deck transformations of Ẽ∗/E∗. Furthermore, the finite index subgroup
U of Deck(Ẽ∗/E∗) corresponds to an unramified covering p : X∗ → E∗ of finite
degree. This can be extended to a covering X → E, where X is a closed surface.
Example 1.7. In Example 1.1, we obtain the following subgroups of F2:
In a), X∗ is the once punctured torus itself and U = F2.
In b), X∗ is a genus 2 surface with 2 punctures. Thus U = π1(X
∗) is a free group
of rank 5. Keeping in mind that we use the identification π1(E
∗) ∼= F2 = F2(x, y)
described in Figure 4, one can read off from the picture in Figure 2 that
U = < x3, xyx−1, x2yx−2, yxy−1, y2 >
In c), X∗ is a genus 2 surface with three punctures. Thus U is a free group of
rank 6. More precisely, we read off the picture in Figure 3, that
U = < x3, xyx−2, x2yx−1, yxy−1, y2xy−2, y3 >
2 TRANSLATION STRUCTURES AND VEECH GROUPS 7
Description as a finite graph
Finitely, sometimes it is convenient to describe an origami O = (p : X → E)
as a finite, oriented labeled graph: Namely, let U be the finite index subgroup
of F2 (unique up to conjugation) that corresponds to O as described in the last
paragraph. Then we represent the origami by the Cayley-Graph of U ⊆ F2:
The vertices of the graph are the coset representatives. They are labeled with a
representative of the coset. The edges are labeled with x and y. For each vertex
(with label w ∈ F2) there is an x-edge from it to the vertex that belongs to the
coset of wx. And similarly there is a y-edge to the vertex that belongs to the
coset wy.
Example 1.8. The following figure shows the Cayley-graph for the origami L(2, 3)
from Example 1.1:
?>=<89:;ȳ
GFED@ABCīd
x // ?>=<89:;x̄ x //
�� GFED@ABC
Figure 5: Graph for O = L(2, 3).
2 Translation structures and Veech groups
Translation structures
Recall that an atlas on a surface is called translation atlas, if all transition maps
are translations. An origamiO = (p : X → E) naturally defines an SL2(R)-family
of translation structures µA (A ∈ SL2(R)) on X
∗ = X − p−1(∞) as follows:
• As first step, observe that each A ∈ SL2(R) naturally defines a translation
structure ηA on the torus E itself by identifying it with C/ΛA, where
and ΛA is the lattice <
> in C (2)
• Then define the translation structure µA on X
∗ by lifting ηA via p, i.e.
µA = p
Using the first description of an origami that we gave by gluing squares, we obtain
the translation structure µI (where I is the identity matrix), if we identify the
squares with the Euclidean unit square in C. We obtain µA for a general matrix
2 TRANSLATION STRUCTURES AND VEECH GROUPS 8
A ∈ SL2(R) from this by identifying the squares with the parallelogram spanned
by the two vectors
Thus the SL2(R)-variations of the translation structure µI can be thought of as
affine shearing of the unit squares, see Figure 6.
Figure 6: Sheared translation structure for the origami L(2, 3).
From an origami to a Teichmüller curve in the moduli space
By the SL2(R)-family of translation structures, the origami O = (p : X → E)
defines a specific complex algebraic curve called Teichmüller curve in the moduli
space Mg of closed Riemann surfaces of genus g. We state this construction here
only briefly as motivation and refer e.g. to the overview article [HeSc 06] for a
detailed description and links to references. A particular nice configuration of
such Teichmüller curves is described in [H 06] in this volume.
The Teichmüller curve in Mg is obtained from the origami in the following way:
• The translation structure µA described in the previous paragraph is in par-
ticular a complex structure on the surface X∗ which can be extended to the
closed surface X . The Riemann surface (X, µA) together with the identity
map id : X → X as marking then defines a point in the Teichmüller space
Tg. Thus we obtain the map: ι̂ : SL2(R) → Tg, A 7→ [(X, µA), id].
• If A ∈ SO2(R), then the affine map z 7→ A ·z is holomorphic. Thus the map
ι̂ factors through SO2(R). Furthermore using that SL2(R) modulo SO2(R)
is isomorphic to the upper half plane H, one obtains a map
ι : H ∼= SO2(R)\SL2(R) → Tg
In fact, this map is an embedding that is in the same time holomorphic and
isometric. A map with this property is called Teichmüller embedding and
its image ∆ in Teichmüller space is called a Teichmüller disk or a geodesic
disk.
2 TRANSLATION STRUCTURES AND VEECH GROUPS 9
• Finally, one may compose the map ι with the projection to the moduli space
Mg. The image of ∆ in Mg is a complex algebraic curve. A curve in Mg
that arises like this as the image of a Teichmüller disk is called Teichmüller
curve.
Note: More generally, one obtains a Teichmüller disk ∆ in a similar way starting
from an arbitrary translation surface (or a bit more general: from a flat surface).
However, the image of such a disk ∆ in moduli space is not always a complex
algebraic curve; in fact its Zariski closure tends to be of higher dimension. It
is an interesting question how to decide whether a translation surface leads to a
Teichmüller curve. One possible answer to this is given by the Veech group which
we introduce in the following paragraph.
Veech groups
Let X∗ be a connected surface and µ a translation structure on it. One assigns to
it a subgroup of GL2(R) called Veech group as described in the following: We con-
sider the group Aff+(X∗, µ) of all orientation preserving affine diffeomorphisms,
i.e. orientation preserving diffeomorphisms that are locally affine maps of the
plane C, see Figure 7. Here – and throughout the whole article – we identify C
with R2 by the map z 7→ (Re(z) , Im(z))t. Thus an affine diffeomorphism f can
be written in terms of local charts as
f : z = (Re(z), Im(z))t 7→ A · (Re(z), Im(z))t+ z0 with A ∈ GL2(R) and z0 ∈ C.
Observe that A does not depend on the chart, since µ is a translation structure.
Thus one obtains a well defined map
D : Aff+(X∗, µ) → GL2(R), f 7→ A
called Derivative map.
Definition 2.1. The Veech group Γ(X∗, µ) of the translation surface (X∗, µ) is
the image of the derivative map D:
Γ(X∗, µ) = D(Aff+(X∗, µ))
z 7→ Az + z0
Figure 7: An affine diffeomorphism of a translation surface
3 VEECH GROUPS OF ORIGAMIS 10
Example 2.2. Let (X∗, µ) be C/ΛI with the natural translation structure. Here
I is the identity matrix and ΛI is the corresponding lattice as defined in (2).
An affine diffeomorphisms of C/ΛI lifts to an affine diffeomorphism of C respect-
ing the lattice. Conversely, each such diffeomorphism descends to C/ΛI. Thus,
we have in this case
Γ(X∗, µ) = SL2(Z).
Veech groups and Teichmüller curves
As indicated in the paragraph about Teichmüller curves, the Veech group “knows”
whether a translation surface defines a Teichmüller curve in moduli space or not.
More precisely, one has the following statement:
Fact: Let X be a surface of genus g and X∗ = X−{P1, . . . , Pn} for finitely many
points P1, . . . , Pn on X . Furthermore let µ be a translation structure on X
Then (X∗, µ) defines a Teichmüller curve C if and only if the Veech group Γ(X∗, µ)
is a lattice in SL2(R). In this case, the curve C is (antiholomorphic) birational
to H/Γ(X∗, µ).
We describe the relation to Teichmüller curves here just as motivation and in
order to give a glance at the general frame. We have therefore resumed theorems
contributed by several authors condensed in what is here called “fact”. A good
access to it can be found e.g. in [EG 97] or [Z 06]. A broader overview on
Veech groups of translation surfaces is given e.g. in [HuSc 01] and in [Le 02].
Teichmüller disks, Teichmüller curves and Veech groups have intensively been
studied by numerous authors, starting from Thurston [T 88] and Veech himself
[V 89]. We refer to [S 04] and [HeSc 06] for more comprehensive overviews on
references.
3 Veech groups of origamis
Let O = p : (X → E) be an origami. We have seen in Section 2 that O defines an
SL2(R)-family of translation structures µA (A ∈ SL2(R)) on X
∗ = X − p−1(∞).
The corresponding Veech groups are not very different. In fact, they are all
conjugated to each other. More precisely, we have:
Γ(X∗, µA) = AΓ(X
∗, µI)A
Thus, we may restrict to the case where A = I which justifies the following
definition.
Definition 3.1. The Veech group Γ(O) of the origamiO is defined to be Γ(X∗, µI).
From Example 2.2 it follows that the Veech group of the trivial origami O0 (de-
fined in Example 1.1) is SL2(Z). For a general origami one can show that Γ(O)
3 VEECH GROUPS OF ORIGAMIS 11
is a finite index subgroup of SL2(Z). In fact, also the converse is true as it was
shown by Gutkin and Judge in [GJ 00]: A Veech group is a finite index subgroup
of SL2(Z) if and only if it comes from an origami.
From this it follows in particular by the Fact presented in Section 2 on page 10
that an origami always defines a Teichmüller curve in the moduli space.
Characterization of origami Veech groups
Recall from Section 1 that an origami O corresponds (up to equivalence) to a
finite index subgroup U of F2 = F2(x, y), the free group in two generators (up to
conjugation). This description enables us to give a characterization of its Veech
group entirely in terms of F2 and its automorphisms.
For this we need the following two ingredients:
• Let β̂ : Aut(F2) → Out(F2) ∼= GL2(Z) be the natural projection. The
fact that we only consider orientation preserving diffeomorphisms applies
to only taking automorphisms of Aut(F2) that are mapped to elements in
SL2(Z). We denote Aut
+(F2) = β̂
−1(SL2(Z)) and restrict to the map
β̂ : Aut+(F2) → SL2(Z).
• Let Stab(U) = {γ ∈ Aut+(F2)|γ(U) = U}
Using these ingredients, it was shown in [S 04] that Veech groups of origamis can
be described as stated in the following theorem.
Theorem 2 (Proposition 1 in [S 04]). For the Veech group Γ(O) of the origami
O holds:
Γ(O) = β̂(Stab(U))
Let us make two comments on this description:
One consequence is, that one obtains an algorithm that can calculate the Veech
group of an arbitrary origami explicitly. This algorithm is described in detail in
[S 04].
As an other consequence, we have now a characterization of all origami Veech
groups as stated in the following corollary.
Corollary 3.2. A finite index subgroup of SL2(Z) occurs as origami Veech group
if and only if it is the image of the stabilizing group Stab(U) ⊆ Aut+(F2) for
some finite index subgroup U in F2.
Thus the question, which finite index subgroups of SL2(Z) are Veech groups be-
comes roughly speaking the same as the question which subgroups of Aut+(F2)
are such stabilizing groups. So far, there is no general answer known.
3 VEECH GROUPS OF ORIGAMIS 12
In [S 05] it was shown that many congruence subgroups of SL2(Z) are Veech
groups. Recall that a congruence group of level n is a subgroup of SL2(Z) that is
the full preimage of some subgroup of SL2(Z/nZ) under the natural homomor-
phism SL2(Z) → SL2(Z/nZ) and n shall be minimal with this property. For prime
level congruence groups the following statement is shown in [S 05, Theorem 4]
Theorem 3. Let p be prime. All congruence groups Γ of level p are Veech groups
except possibly p ∈ {2, 3, 5, 7, 11} and Γ has index p in SL2(Z).
This result is generalized to a statement for arbitrary n in [S 05, Theorem 5]
Presenting the Veech group Γ and the quotient H/Γ for an origami
As mentioned above, using Theorem 2 the Veech group of an origami can be
calculated explicitly. The Veech groups are described as subgroups of SL2(Z)
by generators and coset representatives. We use for the notation that SL2(Z) is
generated by S and T , with
and T =
Recall furthermore from the discussion on Veech groups and Teichmüller curves
in Section 2 on page 10 that for a Veech group Γ we are in particular interested
in the quotient H/Γ, since this quotient is birational to the corresponding Te-
ichmüller curve. Here Γ acts as Fuchsian group on the upper half plane H, which
is endowed with the Poincaré metric.
Since an origami Veech group Γ is a finite index subgroup of SL2(Z), the quotient
H/Γ comes with a natural triangulation. More precisely, we choose the funda-
mental domain for the action of SL2(Z) on H that is the geodesic pseudo-triangle
∆ with vertices P = −1
i, Q = 1
i and P∞ = ∞.
Figure 8: Fundamental domain of SL2(Z).
The surface H/SL2(Z) is obtained by identifying the vertical edges P∞ and Q∞
via T and the edge PQ with itself (with fixed point i) via S.
For an arbitrary subgroup Γ of SL2(Z) of finite index we obtain a fundamental
domain as a union of translates of the triangle ∆: for each coset A we take the
3 VEECH GROUPS OF ORIGAMIS 13
triangle A(∆), where A is a representative of the coset. The identification of
the edges is given by the elements in Γ. Gluing the edges gives the quotient
surface H/Γ, filling in the cusps leads to a closed Riemann surface endowed with
a triangulation. We draw stylized pictures of the fundamental domains that
indicate the triangles (see Figure 9 and 10). The triangles are labeled with a
coset representative, edges that are identified are labeled with the same letter
and vertices that are identified with the same number. Vertices that come from
cusps (i.e. points at ∞) are marked with •.
In particular, one can read off from these stylized pictures the genus and the
number of cusps of the quotient surface H/Γ.
Two examples: the origami L(2,3) and the origami D
The origami L(2,3):
In [S 04, Example 3.5] the Veech group is calculated as follows:
Γ(L(2, 3)) = <
More precisely, one obtains the generators presented as products of S and T as
well as a list of coset representatives.
• List of generators:
= T 3,
= TST 2ST−1T−1,
= TSTST−1S,
= T 2STST−1S−1T−2,
• List of representatives:
I, T, S, T 2, TS, ST, T 2S, TST, T 2ST
Hence, Γ(L(2, 3)) is a subgroup of index 9 in SL2(Z).
The stylized picture of the quotient H/Γ(L(2, 3)) is determined in [S 04, Example
3.6] and is shown here in Figure 9.
3 VEECH GROUPS OF ORIGAMIS 14
TTSTT
Figure 9: Fundamental domain of Γ(L(2, 3)).
From this one can read off that the genus of the quotient H/Γ(L(2, 3)) is 0 and
that it has 3 cusps, namely the vertices labeled by 1,4 and 5. It follows in par-
ticular that the corresponding Teichmüller curve has genus 0.
The origami D:
The Veech group of the origami D is calculated in [S 05, Section 7.3.2]. It has
index 24 in SL2(Z) and the following generators:
= −I, A1 =
= T 3,
= ST 6S−1, A3 =
−7 16
= (T 2S)T 4(T 2S)−1.
= (TS)T 4(TS)−1, A5 =
−20 11
= (TST 2S)T 5(TST 2S)−1,
−18 −5
= (ST 3S)T 2(ST 3S)−1,
The following is a system of cosets representatives:
I , T , S , T 2 , TS , ST , T 2S , TST , ST 2 , STS , T 2ST , TST 2 ,
ST 5 , ST 3 , T 2S , TST 3 , TST 2S , ST 4 , ST 3S , TST 2ST−1 ,
TST 2ST−2 , TST 2ST−3 ; TST 2ST−4 , ST 3ST
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 15
The corresponding origami curve C(D) has genus 0. It is shown with its natu-
ral triangulation in Figure 10. It has six cusps, namely C1, C2, C3, C4, C5 and C6.
TT TTS
TTSTT
STTSTT
TSTTS
TSTTST−1
3ST ST
Figure 10: The origami curve to D.
4 Veech groups that are non congruence groups
Theorem 3 implies that there are many congruence groups which are Veech
groups. How about non congruence groups? In this section we will see that
the Veech groups for the two examples, the origami L(2, 3) and the origami D,
studied in the last paragraph are both non congruence groups. Furthermore,
we give a construction that produces for both of them an infinite sequence of
origamis whose Veech group is a non congruence group. We use this in order to
prove our main theorem.
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 16
An other generalization of the example L(2, 3) was given by Hubert and Lelièvre
in [HL 05], where they show for certain “L-shaped” origamis or square-tiled sur-
faces, how they are called there, that their Veech groups are non congruence
groups. These surfaces are all of genus 2, hence it follows that there are infinitely
many origamis of genus 2 whose Veech group is a non congruence group.
Recall that a group is a congruence group, whose level is a divisor of n, if and
only if it contains the principal congruence group
Γ(n) = {
mod n} = kernel(proj : SL2(Z) → SL2(Z/nZ))
In [S 04, Proposition 3.8] it was shown using a proof of Stefan Kühnlein that the
Veech group of L(2, 3) is a non congruence group. The basic tool for this is the
general level that is defined for any subgroup Γ of SL2(Z) as follows: For each
cusp we define its amplitude to be the smallest natural number n such that there
is an element of Γ conjugated in SL2(Z) to the matrix
which fixes the cusp. Observe that this is equal to the number of triangles around
the vertex that represents the cusp in our stylized picture of the quotient surface
(see Figures 9 and 10). The general level of Γ is the least common multiple of the
amplitudes of all its cusps. A theorem of Wohlfahrt [W 64, Theorem 2] states
that the level and the general level of a congruence group coincide.
The amplitude of the three cusps of H/Γ(L(2, 3)) labeled with 1, 4 and 5 in Fig-
ure 9 is 3, 2 and 4 respectively. Hence, the general level of Γ(L(2, 3)) is 12. Then
it is shown in the proof that Γ(L(2, 3)) does not contain Γ(12) which gives the
contradiction.
The same method can be used in order to show that Γ(D) is a non congruence
group. We here carry out the proof for it. Observe from Figure 10 that the six
cusps C1, . . . , C6 have the amplitude 3, 6, 4, 4, 5 and 2, respectively. Thus the
general level is 60.
Proposition 4.1. The Veech group Γ(D) is a non congruence group.
Proof. Suppose that Γ = Γ(D) is a congruence group. Since the general level of
Γ is 60, we have by the theorem of Wohlfahrt mentioned above, that Γ(60) is a
subgroup of Γ.
We will use the following facts, which can be checked e.g. in Figure 10:
∈ Γ, A6 =
−18 −5
∈ Γ and T =
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 17
In order to verify this in Figure 10, use that
A1 = T
3 and A6 = S
−1T 2S−1T−1S−1TS−1T−3S−1.
We will find an element in Γ whose projection to SL2(Z/60Z) is equal to that of
T , which gives us the desired contradiction.
Recall that
SL2(Z/60Z) ∼= SL2(Z/4Z)× SL2(Z/3Z)× SL2(Z/5Z).
We identify in the following these two groups. Furthermore we denote by p4, p3,
p5 and p60 the projection from SL2(Z) to SL2(Z/4Z), SL2(Z/3Z), SL2(Z/5Z) and
SL2(Z/60Z), respectively. Then p60 = p4 × p3 × p5.
We have
p60(A1) = (
) and
p60(A6) = (
The order of p4(A1) in SL2(Z/4Z) is 4, the order of p3(A1) in SL2(Z/3Z) is 1
and the order of p5(A1) in SL2(Z/5Z) is 5. We also say: The order of p60(A1) is
(4, 1, 5). Since 7 ≡ 3 mod 4 and 7 ≡ 2 mod 5 we have
p60(A
1) = (
) (4)
Furthermore:
p60(A
6) = (
and with the same notation as above p60(A
6) has the order (1, 3, 5). Thus
p60(A
6 ) = (
). (5)
From (4) and (5) it follows that
p60(A
6 · A
1) = (
) = p60(
) = p60(T )
But A206 · A
1 ∈ Γ and T /∈ Γ, thus Γ(60) = ker(p60) cannot be contained in Γ.
Therefore, Γ cannot be a congruence group of level 60. Contradiction!
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 18
Sequences of origamis with non congruence Veech groups
Starting from the origamis L(2, 3) and D we will define respectively a sequence
On, such that for each n ∈ N the Veech group Γ(On) again is a non congruence
group. The basic idea is to “copy and paste”: we will cut the origami along a
segment, take n copies of it and glue them along the cuts.
In Figure 11 we show the origami On for L(2, 3):
1 3 4
5 7 8
. . . 4n-7 4n-5 4n-4
4n-3 4n-1 4n
Figure 11: n copies of L(2, 3). Opposite edges are glued.
Using the description of an origami by a pair of permutations from Section 1, On
is given as:
σa = (1 3 4 5 7 8 9 11 12 . . . 4n−3 4n−1 4n), σb = (1 2)(5 6) . . . (4n−3 4n−2).
Observe that the genus of On is n + 1 and it has 2n cusps: n of order 3 (all n
marked by • in Figure 11), and n of order 1 (all n marked by ◦ in Figure 11).
Finally, we want to present the origami On by the finite index subgroup Hn =
∗) of F2, that corresponds to On by Remark 1.6.
Recall from Example 1.7 that for O1 = L(2, 3), we obtain the free group of rank 5:
U = H1 =< g1 = x
3, g2 = xyx
−1, g3 = x
2yx−2, g4 = yxy
−1, g5 = y
2 >= F5.
The group Hn is obtained as as follows:
Hn = < g
1 , g
1 gj g
1 ∈ F5 | i ∈ {0, . . . , n− 1} and j ∈ {2, . . . , 5} >
In Figure 12, we show the origami Dn:
1 2 3
6 7 8
. . . . . . . . .
5n-4 5n-3 5n-2
b1 a2
b2 an
• • •
* • • •
• • •
Figure 12: n copies of D. Edges with the same label or
unlabeled opposite edges are glued.
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 19
The pair of permutations describing Dn is:
σa = (1 2 3 6 7 8 . . . 5n− 4 5n− 3 5n− 2),
σb = (1 4 5)(6 9 10) . . . (5n− 4 5n− 1 5n)(2 3)(7 8) . . . (5n− 3 5n− 2)
The genus of Dn is 2n and it has n+ 2 cusps: 2 of order 2n (marked as • and ⋆)
and n of order 1 (all n marked by ◦).
Again, we present On by the corresponding finite index subgroup Hn of F2. We
have from Example 1.7 that U = H1 = F6, the free group of rank 6:
U =< g′1 = x
3, g′2 = xyx
−2, g′3 = x
2yx−1, g′4 = yxy
−1, g′5 = y
2xy−2, g′6 = y
3 > = F6
And similarly as above, we obtain:
Hn = < g
1 , g
1 ∈ F6 | i ∈ {0, . . . , n− 1} and j ∈ {2, . . . , 6} >
We will see in the following that for both sequences all Veech groups Γ(On) are
non congruence groups. More precisely, we will show:
Proposition 4.2. For both sequences On the following holds:
• Γ(On) ⊆ Γ(O1), which is for both sequences a non congruence group.
• More generally one has:
n divides m ⇒ Γ(Om) ⊆ Γ(On).
• Different origamis in one sequence have different Veech groups, i.e.:
Γ(On) 6= Γ(Om) for n 6= m.
To prove this, let us detect that we are in the following more general setting.
Setting A:
• Let U be a finite index subgroup of F2. Then U is a free group of rank k
for some k ≥ 2, i.e.
U = < g1, . . . , gk > = Fk
• Let α : Fk → Z be the projection w 7→ ♯g1w
where ♯g1w is the number of g1 in the word w = w(g1, . . . , gk) with g
counted as −1.
• Let Hn be the kernel of pn ◦ α, where pn : Z → Z/nZ is the natural
projection, i.e.
Hn = < g
1 , g
1 gj g
1 ∈ Fk | i ∈ {0, . . . , n− 1} and j ∈ {2, . . . , k} > .
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 20
• Finally, let H0 be the kernel of α, i.e.:
Hn = << g2, . . . , gk >>U ,
is the normal subgroup in U generated by g2, . . . , gk.
Observe that we are in this setting with
U = π1(X
∗) =< x3, xyx−1, x2yx−2, yxy−1, y2 > for the origami L(2, 3) and
U = π1(X
∗) =< x3, xyx−2, x2yx−1, yxy−1, y2xy−2, y3 > for the origami D.
In order to prove the properties in Proposition 4.2, we will need that U fulfills
the following a bit technical condition:
Property B: Let U =< g1, . . . , gk > (k ≥ 2) be as above a finite index
subgroup of F2 of rank k and {wi}i∈I a system of coset representatives with
w1 = id. Suppose that U has the following property:
∀ j ∈ I − {1} : wj << g2, . . . , gk >>U w
j 6⊆ U.
One can check by hand that for both origamis, L(2, 3) and D, this property is
fulfilled. In this setting we obtain the following conclusions.
Proposition 4.3. Let n ∈ N ∪ {0}. Let U be a finite index subgroup of F2
fulfilling property B. With the notations from Setting A, we have:
a) The normalizer of Hn in F2 is equal to U : NormF2(Hn) = U
b) Stab
(Hn) ⊆ StabAut+(F2)
c) Recall that U = Fk, the free group in k generators.
Let βn : Aut(Fk) → GLk(Z/nZ) be the natural projection.
Then Stab
(Hn) is equal to
β−1n ({A = (ai,j)1≤i,j≤k ∈ GLk(Z/nZ)| a1,2 = . . . = a1,k = 0}) ∩ G.
Here we use the notation Z/(0Z) = Z thus β0 is the natural projection
Aut(Fk) → GLk(Z).
Proof.
By definition Hn is normal in U , i.e. U ⊆ NormF2(Hn).
Let now w be an element of F2\U . Hence, w = wj ·u for some j ∈ I−{1}, u ∈ U .
By Property B, there exists some h0 ∈ << g2, . . . , gk >>U = H0, such that
wjh0w
j 6∈ U . Therefore we have w(u
−1h0u)w
−1 6∈ U . But u−1h0u ∈ H0 ⊆ Hn,
since H0 is normal in U . This shows that w 6∈ NormF2(Hn).
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 21
This follows from a), since for a subgroup H of F2 in general holds:
(H) ⊆ Stab
(NormF2(H)), see e.g. [S 06, Remark 3.1].
Define M = {A = (ai,j)1≤i,j≤k ∈ GLk(Z/nZ)| a1,2 = . . . = a1,k = 0}.
Let γ ∈ G. We have to show that γ(Hn) = Hn if and only if βn(γ) ∈ M .
Let furthermore pkn : Fk → (Z/nZ)
k be the natural projection.
Consider the following commutative diagram:
Hn = p
n(Hn) ⊆ (Z/nZ)
βn(γ) // (Z/nZ)k ⊇ pkn(Hn) = Hn
Since pkn is surjective and Hn is the full preimage of Hn = p
n(Hn), it follows that
γ(Hn) = Hn if and only if βn(γ)(Hn) = Hn.
Observe finally that:
Hn = {(0, x2, . . . , xk) ∈ (Z/nZ)
k} and
StabGLk(Z/nZ)(Hn) = { A = (ai,j)1≤i,j≤k ∈ GLk(Z/nZ)|
(y1, . . . , yk) = A · (0, x2, . . . , xk) ⇒ y1 = 0 }
= {A = (ai,j) ∈ GLk(Z/nZ)| a1,2 = . . . = a1,k = 0}
Theorem 2 suggests the following notation.
Definition 4.4. Let U be a subgroup of F2.
With β̂ : Aut+(F2) → SL2(Z) as in Theorem 2, we define
Γ(U) = β̂(Stab
and call Γ(U) the Veech group of U .
We now obtain from Proposition 4.3 the following conclusions.
Corollary 4.5. Suppose that we are in the same situation as in Proposition 4.3,
in particular that U is a finite index subgroup of F2 fulfilling property B. Then
we have for all n ∈ N:
a) Stab
(H0) ⊆ StabAut+(F2)
(Hn) and Γ(H0) ⊆ Γ(Hn).
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 22
b) If m ∈ N with n|m, then:
(Hm) ⊆ StabAut+(F2)
(Hn) and Γ(Hm) ⊆ Γ(Hn).
(H0) =
(Hn) and Γ(H0) =
Γ(Hn)
Proof.
a) and b):
Let γ ∈ G. By Proposition 4.3 we have that
∀n ∈ N : γ ∈ Stab
(Hn) ⇔ βn(γ) = A = (ai,j)
with a1,2 ≡ . . . ≡ a1,k ≡ 0 mod n
and γ ∈ Stab
(H0) ⇔ β0(γ) = A = (ai,j)
with a1,2 = . . . = a1,k = 0.
Thus we have for all n ∈ N and for all m ∈ N with n|m, that
(H0) ⊆ StabAut+(F2)(Hm) ⊆ StabAut+(F2)(Hn).
We have in particular by the definition of the Veech group of a subgroup of F2:
Γ(H0) ⊆ Γ(Hm) ⊆ Γ(Hn).
⊆ follows from a). ⊇ follows from Remark [S 06, Remark 3.1].
We now return to the language of origamis: Let O be an origami, U the corre-
sponding subgroup of F2. Define for U the subgroups Hn (n ∈ N) as in Setting
A and let On be the origamis corresponding to the groups Hn.
By Corollary 4.5 and Theorem 2 we obtain immediately the following result.
Proposition 4.6. If U has the Property B, then
∀n ∈ N : Γ(On) ⊆ Γ(O) and ∀n,m ∈ N : n|m ⇒ Γ(Om) ⊆ Γ(On).
In particular, if Γ(O) is a non congruence group, each Γ(On) is a non congruence
group. Thus in this case, we obtain infinitely many origamis whose Veech group
is a non congruence group.
In order to conclude Proposition 4.2, it is now just left to prove the last item. But
this follows , since we have (see [S 05]) for both sequences On, the one coming
from the origami L(2, 3) and the one coming from the origami D, that
∈ Γ(On) ⇔ 3n divides s. (6)
4 VEECH GROUPS THAT ARE NON CONGRUENCE GROUPS 23
This finishes the proof of Proposition 4.2.
Furthermore, Theorem 1 follows from Proposition 4.2.
Remark: From Corollary 4.5 and (6) it follows that Γ(H0) has infinite index in
SL2(Z). Furthermore it is non trivial, since it contains
for L(2, 3) respectively B3 =
for D.
REFERENCES 24
References
[EG 97] C.J. Earle, F.P. Gardiner: Teichmüller disks and Veech’s F -structures.
American Mathematical Society. Contemporary Mathematics 201, 1997 (p.
165–189).
[GJ 00] E. Gutkin, C. Judge: Affine mappings of translation surfaces. Duke
Mathematical Journal 103 No. 2, 2000 (p. 191–212).
[HeSc 06] F. Herrlich, G. Schmithüsen: On the boundary of Teichmüller disks
in Teichmüller and in Schottky space. To appear in Handbook of Teichmüller
theory. Ed. A. Papadopoulos, European Mathematical Society, 2006.
[H 06] F. Herrlich: A comb of origami curves in M3. Proceedings of Symposium
on Transformation Groups, Yokohama, November 2006.
[HL 05] P. Hubert, S. Lelièvre: Noncongruence subgroups in H(2). International
Mathematics Research Notices 2005, No.1 , 2005 (p. 47–64).
[HuSc 01] P. Hubert, T. Schmidt: Invariants of translation surfaces. Annales de
l’Institut Fourier 51 No. 2, 2001 (p. 461–495).
[Le 02] S. Lelièvre: Veech surfaces associated with rational billiards. Preprint,
2002. arXiv:math.GT/0205249.
[Lo 05] P. Lochak: On arithmetic curves in the moduli space of curves. J. Inst.
Math. Jussieu 4, No. 3, 2005 (p. 443–508).
[S 04] G. Schmithüsen: An algorithm for finding the Veech group of an origami.
Experimental Mathematics 13 No. 4, 2004 (p. 459–472).
[S 05] G. Schmithüsen: Veech Groups of Origamis. Dissertation (PhD thesis),
Karlsruhe 2005. Elektronisches Volltextarchiv EVA Universität Karlsruhe.
http://www.ubka.uni-karlsruhe.de/eva/
[S 06] Schmithüsen,G.: Examples for Veech groups of origamis. In: The Geome-
try of Riemann Surfaces and Abelian Varieties. Contemp. Math. 397, 2006 (p.
193–206).
[T 88] W. Thurston: On the geometry and dynamics of diffeomorphisms of sur-
faces. Bulletin (New Series) of the American Mathematical Society 19 No. 2,
1988 (p. 417–431).
[V 89] W.A. Veech: Teichmüller curves in moduli space, Eisenstein series and
an application to triangular billiards. Inventiones Mathematicae 97 No.3 1989,
(p. 553–583).
REFERENCES 25
[W 64] K. Wohlfahrt: An Extension of F. Klein’s Level Concept. Illinois Journal
of Mathematics 8, 1964 (p. 529–535).
[Z 06] A. Zorich: Flat Surfaces in collection ”Frontiers in Number Theory,
Physics and Geometry. Volume 1: On random matrices, zeta functions and
dynamical systems”, Ed. P. Cartier, B. Julia, P. Moussa, P. Vanhove, Springer-
Verlag, 2006 (p. 439–586).
	Origamis
	Translation structures and Veech groups
	Veech groups of origamis
	Veech groups that are non congruence groups
