Translation surfaces: saddle connections,
triangles and covering constructions
Chenxi Wu
Cornell University
July 29, 2016
Translation surfaces
A (compact) translation surface is:
I
a compact surface with a atlas to R2 (or C), covering all but
finitely many points, whose transition functions are
translations, or
I
a collection of polygons in the plane with sides glued through
translation, or
I
a compact Riemann surface with a holomorphic 1-form.
Examples: flat torus, regular octagon with opposite sides glued.
Applications: Rational polygonal billiards, IET.
Strata of translation surfaces
I
The sets of translation surfaces with the same topological
type are called strata. They are complex affine manifolds
where the charts are period coordinates. They are never
compact and in general not connected.
I
The group SL(2, R) acts on a stratum by post-composing
with the coordinate charts. This action renormalize the
geodesic flow on surface. All SL(2, R) orbits are known to be
complex affine submanifolds (Eskin-Mirzakhani-Mohammadi).
I
The horocycle orbit closures are also important. However,
these are not always affine submanifolds.
Lattice Surfaces
I
Surfaces with closed SL(2, R) orbit are called lattice surfaces.
The simplest of these is the flat torus. Branched covers of the
flat torus that are lattice surfaces are called square-tiled.
I
Non-square tiled lattice surfaces: regular polygon (the first,
discovered by Veech), genus 2 lattice surfaces, prym
eigenform in ℋ(4) or ℋ(6), Bouw-Möller family etc.
Veech Dichotomy
I
Given a direction on a translation surface, one can define the
geodesic flow on the surface in that direction for almost all
points.
I
Veech Dichotomy: The geodesic flow on any lattice surface
must be either uniquely ergodic or completely periodic (i.e.
the surface is decomposed into cylinders in that direction).
I
When it is completely periodic, it is easy to show that
furthermore the moduli of the cylinders must be
commensurable.
Holonomies of saddle connections
I
Example: Flat torus with a marked point.
I
Example: a lattice surface in ℋ(2) corresponding to the
quadratic order of discriminant 13. (p.6)
I
Quadratic growth rate.
I
The holonomies of saddle connections on a non-square-tiled
lattice surfaces can not be uniformly discrete, i.e. the distance
between two distinct points are not bounded below.
I
For square-tiled case, it is a classical result that the set is not
relatively dense.
Holonomies of saddle connections
h + 2l
h′ + 2l ′
h+l
h′ + l ′
h
h′
Minimal area of triangles and virtual triangles
I
Smillie-Weiss, Vorobets showed that the area of triangles and
“virtual triangles” on a lattice surface must be bounded away
from 0 and the converse is also true.
I
Smillie and Weiss described an algorithm that can enumerate
all lattice surfaces according to the area of smallest virtual
triangle, represented through Thurston-Veech construction.
Minimal area of triangles and virtual triangles
We did computations on this area for some known lattice surfaces,
using known enumeration of cylinder decompositions on these
surfaces up to affine action.
·10−2
8
7
6
5
4 AreaVT
3
2
1
AreaT
0.02
0.04
0.06
0.08
0.1
0.12
Thurston-Veech construction
I
If a surface is completely periodic in two different directions,
and all cylinder moduli in each direction are commensurable,
then it can be uniquely determined up to affine transformation
by the intersection pattern of the cylinders and the ratio of
the moduli. This is called a Thurston-Veech construction.
I
The intersection pattern can be represented by a ribbon
graph, called Thurston-Veech diagram.
I
Thurston used it to construct pseudo-Anosov diffeomorphisms.
Algorithm for enumerating lattice surfaces according to
smallest virtual triangle
I
Step I: find all possible cylinder intersection pattern with less
than a certain number of cylinder intersections (i.e.
rectangles).
I
Step II: find all possible ratios of moduli.
I
Step III: check for the area of the smallest virtual triangle.
We improved the bound in Step II and did part of Step III by
computer while the remaining by hand, and found all
non-arithmetic lattice surfaces with area of smallest virtual triangle
at least .05 of the total area. There are 5 of them up to affine
action.
Branched Abelian cover of the flat pillowcase
An important class of example of square-tiled lattice surfaces are
the Abelian branched covers of the pillowcase (which is a half
translation surface i.e. the transition function is a translation or
rotation of 𝜋).
I
They are used in Bouw-Möller’s construction of a
two-parameter infinite family of lattice surfaces, which
includes the Veech & Ward families.
I
Their Lyaponov exponents have been calculated.
I
They can be used to construct interesting examples of
horocycle orbit closures. (See Lucien’s thesis)
I
They are related to the classical problem of hypergeometric
differential equations (See Deligne-Mostow).
Branched Abelian cover of the flat pillowcase
z3
e2
z2
e2
z3
e3
B2
e1
B1
e3
z4
e4
z1
e4
z4
Wollmilchsau:
z3
e12
e23
B02
z2
e02
z3
B01
e03
z4
z3
e34
e22
z1
z2
e04
e12
z4
z3
e33
B12
B11
e13
z4
z3
e04
e32
z1
z2
e14
e22
z4
z3
e03
B22
B21
e23
z4
z3
e14
e02
z1
z2
e24
e32
z4
z3
e13
B32
z4
e24
z1
B31
e33
e34
z4
Branched Abelian cover of the flat pillowcase
Results:
1.
H 1 (M, Σ; C) =
⨁︁
H 1 (𝜌)
𝜌∈Δ
this decomposition is preserved by the action of the affine group
Aff, while the factors may be permuted.
2. The dimension of H 1 (𝜌) is 3 if 𝜌 is the trivial representation and
2 otherwise.
3. The projection to absolution cohomology restricting to each
H 1 (𝜌) is either trivial or does not split.
4. For 𝜌 such that the map in 3. is not 0, a finite index subgroup of
the affine group acts on H 1 (𝜌) as a (Euclidean (when the map
in 3. does not split), hyperbolic or spherical) triangle group.
The case for H 1 (M) was done by Wright.
In the case where the action is a spherical triangle group, we can
decide if it is discrete by ADE classification.
Bouw-Möller surfaces
Bouw-Möller construction. Hooper’s grid graph.
If a lattice surface arises from deforming an abelian branched cover
of flat pillowcase M using a relative cohomology class 𝛼, its affine
diffeomorphism group is with the affine diffeomorphism group of
M, and the deformation also preserves the Thurston-Veech
structure in the two diagonal directions, then this surface be one of
the followings cases:
1. a branched cover of the regular 2n-gon branched at the cone
point.
2. a branched cover of a Bouw-Möller surface.
3. a branched cover of the regular n-gon branched at the
mid-point of edges.
Bouw-Möller surfaces
e
11
12
f
g
13
c
14
10
15
h
h
9
d
b
8
g
f
7
a1
2
6
e
b
3
d
4
c5
Figure: The (5, 3) Bouw-Möller surface.
a
“Flipping”
ℱM
M
IV
III
ℱ
II
IV
III
Q
P
I
II
I
a
dd
c
c
re-glue
bb
a
change the gluing direction
a
dd
d
d
a
c
c
c
c
a
bb
bb
a
Figure: Constructing Thurston-Veech diagram by flipping.
The case 3.
Thank you very much!