Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
Flat Surfaces, Teichmueller Discs, Veech
Groups, and the Veech Tessellation
S. Allen Broughton - Rose-Hulman Institute of Technology
Chris Judge - Indiana University
AMS Regional Meeting at Pennsylvania State
October 2009
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
credits and agenda
some credits
current work is joint with Chris Judge.
original and many subsequent investigations by W. Veech
papers of M. Troyanov are a good background source.
lots of interest in applications of flat surfaces to the study of
"zero divisor strata of quadratic differentials" in
Teichmueller space.
Interesting pictures of the Veech tesselation have been
drawn by Josh Bowman (see web link reference at the end
of the talk).
Overview
Flat Surfaces
Teichmueller Discs
credits and agenda
agenda
geometric definition of flat surfaces
analytic definition of flat surfaces
relation to Teichmueller discs
Veech groups
Veech tesselation
Veech Group and Veech Tesselation
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
first definition and examples
informal definition/construction of flat surfaces
Definition
Let P1 , . . . , Pn be a sequence of polygons such that every side
of every polygon is matched with exactly one side (same edge
length) of another polygon. The match may be to another side
of the same polygon. The compact space S, obtained by gluing
the polygons together via the matching, is called a flat surface.
We assume the surface is connected.
Overview
Flat Surfaces
Teichmueller Discs
first definition and examples
flat surfaces - examples - 1
Here are some examples of flat surfaces.
any of the platonic surfaces
flat torus
double pentagon (show on the board)
Veech Group and Veech Tesselation
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
first definition and examples
flat surfaces - examples - 2
Veech used the ideas of flat surfaces to discuss billiard
trajectories
consider the surface formed from the development of a
convex table whose corner angles are rational multiples of
π
show and tell with PentagonPeriodic.pdf and
PentagonDense.pdf (end of .pdf)
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
first definition and examples
flat surfaces - examples - 3
the billiard trajectories are geodesics on the developed flat
surface
the billiard trajectory can be periodic or dense (uniquely
ergodic)
Veech dichotomy: in certain circumstances all trajectories
are periodic or uniquely ergodic
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
first definition and examples
flat surface geometry - 1
A flat surface has three types of points:
interior points of polygons
hinge points where two polygons meet along the interior of
an edge
cone points at the vertices of polygons
The first two types of points are regular points on the
surface. A neigbourhood of a hinge point can be made to
look like a flat piece of plane by flattening.
The cone points are usually considered to be singular.
They cannot be flattened unless the total angle is 2π.
denote the collection of cone points by F .
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
first definition and examples
flat surface geometry - 2
The local geometry is determined as follows:
Each regular point has a neighbourhood with the regular
flat plane geometry, flattening a hinge as needed.
Cone points need a measure of non-regularity, called the
cone angle.
If α1 , . . . , αs are the angles at a cone point vj F , then the
(total) cone angle at vj is
θj =
s
X
αi .
i=1
A cone point is regular if and only the cone angle equals
2π (lies flat).
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
first definition and examples
flat surface geometry - 3
Here are some examples of cone angles.
Example
A cube has 8 cone points with cone angle 3π/2.
An icosahedron has 12 cone points with cone angle 5π/3.
The torus has no singular cone points.
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
first definition and examples
Euler’s formula
Proposition
Suppose a flat surface has genus g and v cone points with
cone angles θj . Then
v
X
j=1
θj = 2π(2g − 2 + v ).
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
second definition and examples
complex analytic definition
Definition
A closed Riemann surface S with finite singular point set F has
a flat analytic structure if there is an complex analytic atlas {uα }
on S\F such that
the transition maps are affine linear
uα (P) = auβ (P) + b, a, b ∈ C
the transition maps are rigid: |a| = 1.
S is the completion of S\F in the pulled back metric from C
If a = 1 then S is called a translation surface and {uα } is
called a translation structure.
If a = ±1 then {uα } is called a demi-translation structure.
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
second definition and examples
structures from forms
Let ω = fdz q be a q-differential with divisor in F and at
worst simple poles. Then an atlas {uα } may be defined as
follows
Z
P
uα (P) =
f 1/q (z)dz
P0
If ω is a 1-form then {uα } is a translation structure.
If ω is a quadratic differential then {uα } is a
demi-translation structure.
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
automorphisms and deformation of structures
automorphism of structures
For any surface with automorphism group G, let ω be an
invariant q-differential.
Define as before an atlas {uα } as follows
Z
P
uα (P) =
f 1/q (z)dz
P0
The flat structure defined above will have G as a group of
automorphisms.
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
automorphisms and deformation of structures
deformation of structures
Let {uα } be defined by a quadratic differential ω, and let
g ∈ PSL2 (R).
Then {guα } is a demi-translation structure as
guα (P) = g(auβ (P) + b) = aguβ (P) + gb)
Let Sg be the corresponding surface.
If g ∈ SO(2) then S and Sg are conformally equivalent.
g → Sg is a map of H = PSL2 (R)/SO(2) into the
appropriate Teichmueller space T.
Think of the image as a complex geodesic in T, through S
in the direction ω.
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
automorphisms and deformation of structures
Teichmueller disc
The corresponding disc in T is typically called a
Teichmueller disc.
The image of a Teichmueller disc in the moduli space is a
curve.
Otherwise you get something analogous to an irrational
line on a torus.
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
definition of Veech group
Veech group and Teichmueller disc
The Veech group is essentially the set of all g ∈ PSL2 (R)
such that S and Sg are conformally equivalent.
Specifically interested in those deformations in which the
Veech group is a lattice in PSL2 (R).
The corresponding disc in T is typically called a
Teichmueller disc.
The image of a Teichmueller disc in the moduli space is a
curve.
Otherwise you get something analogous to an irrational
line on a torus.
the canonical example is PSL2 (Z) acting on the upper half
plane.
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
Voronoi and Delaunay Tilings
Voronoi Decomposition of a Flat Surface
Given a flat surface S construct the Voronoi tiling with
respect to the singular points F .
Show and tell quadrilateral example on board.
cells are points closest to a unique singular point
edges are points closest to two singular points
vertices are points closest to three or more singular points
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
Voronoi and Delaunay Tilings
Delaunay Decomposition of a Flat Surface
The Delaunay decomposition is dual to Voronoi
decomposition.
The vertices are the singular points.
Every polygon in a Delaunay tiling is cyclic (by
construction).
The Delaunay decomposition is a canonical polygon
construction for a flat surface.
Generically the decomposition is by triangles.
Show and tell quadrilateral example on board.
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
Voronoi and Delaunay Tilings
Veech Tesselation -1
The Delaunay decomposition of Sg will vary in a
Teichmueller disc.
Locally constant, generically a triangle decomposition.
Transitions are generically quadrilateral flips
Show and tell quadrilateral flip on the board for
quadrilateral flip.
Mark all the transitions on a hyperbolic disc.
Show picture of tesselation by Bowman from the web.
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
Voronoi and Delaunay Tilings
Veech Tesselation - 2
Here is a paraphrasal of a theorem due to Veech.
Theorem
The transition points in the hyperbolic disc are portions of
geodesics.
back to picture of tesselation by Bowman from the web.
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
Voronoi and Delaunay Tilings
current work
Tangent Star Lemma.
Analysis of stratification of Teichmueller space of flat
surfaces by the Delaunay decomposition.
Relation of the Veech group and the automorphism group
of the Veech Tessellation.
Overview
Flat Surfaces
Teichmueller Discs
Veech Group and Veech Tesselation
done
references and links
William Veech A tessellation associated to a quadratic
differential. Preliminary report. Abstracts of the AMS
946-37-61.
M. Troyanov, On the Moduli Space of Singular Euclidean
Surfaces, arXiv:math/0702666v2
Josh Bowman’s website
http://www.math.cornell.edu/ bowman/
Overview
Flat Surfaces
Teichmueller Discs
done
Any Questions?
Veech Group and Veech Tesselation
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