Flat Surfaces, Teichmueller Discs, Veech Groups, and the ... · Overview Flat Surfaces Teichmueller...

Overview Flat Surfaces Teichmueller Discs Veech Group and Veech Tesselation

Flat Surfaces, Teichmueller Discs, VeechGroups, and the Veech Tessellation

S. Allen Broughton - Rose-Hulman Institute of TechnologyChris Judge - Indiana University

AMS Regional Meeting at Pennsylvania StateOctober 2009


credits and agenda

some credits

current work is joint with Chris Judge.original and many subsequent investigations by W. Veechpapers 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" inTeichmueller space.Interesting pictures of the Veech tesselation have beendrawn by Josh Bowman (see web link reference at the endof the talk).


credits and agenda

agenda

geometric definition of flat surfacesanalytic definition of flat surfacesrelation to Teichmueller discsVeech groupsVeech tesselation


first definition and examples

informal definition/construction of flat surfaces

DefinitionLet P1, . . . ,Pn be a sequence of polygons such that every sideof every polygon is matched with exactly one side (same edgelength) of another polygon. The match may be to another sideof the same polygon. The compact space S, obtained by gluingthe polygons together via the matching, is called a flat surface.We assume the surface is connected.



flat surfaces - examples - 1

Here are some examples of flat surfaces.

any of the platonic surfacesflat torus

double pentagon (show on the board)




Veech used the ideas of flat surfaces to discuss billiardtrajectoriesconsider the surface formed from the development of aconvex table whose corner angles are rational multiples ofπ

show and tell with PentagonPeriodic.pdf andPentagonDense.pdf (end of .pdf)




the billiard trajectories are geodesics on the developed flatsurfacethe billiard trajectory can be periodic or dense (uniquelyergodic)Veech dichotomy: in certain circumstances all trajectoriesare periodic or uniquely ergodic



flat surface geometry - 1

A flat surface has three types of points:interior points of polygonshinge points where two polygons meet along the interior ofan edgecone points at the vertices of polygonsThe first two types of points are regular points on thesurface. A neigbourhood of a hinge point can be made tolook 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 .




The local geometry is determined as follows:Each regular point has a neighbourhood with the regularflat plane geometry, flattening a hinge as needed.Cone points need a measure of non-regularity, called thecone angle.If α1, . . . , αs are the angles at a cone point vjF , then the(total) cone angle at vj is

θj =s∑

i=1

αi .

A cone point is regular if and only the cone angle equals2π (lies flat).




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.



Euler’s formula

PropositionSuppose a flat surface has genus g and v cone points withcone angles θj . Then

v∑j=1

θj = 2π(2g − 2 + v).


second definition and examples

complex analytic definition

DefinitionA closed Riemann surface S with finite singular point set F hasa 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} iscalled a translation structure.If a = ±1 then {uα} is called a demi-translation structure.


second definition and examples

structures from forms

Let ω = fdzq be a q-differential with divisor in F and atworst simple poles. Then an atlas {uα} may be defined asfollows

uα(P) =

∫ P

P0

f 1/q(z)dz

If ω is a 1-form then {uα} is a translation structure.If ω is a quadratic differential then {uα} is ademi-translation structure.


automorphisms and deformation of structures

automorphism of structures

For any surface with automorphism group G, let ω be aninvariant q-differential.Define as before an atlas {uα} as follows

uα(P) =

∫ P

P0

f 1/q(z)dz

The flat structure defined above will have G as a group ofautomorphisms.



deformation of structures

Let {uα} be defined by a quadratic differential ω, and letg ∈ 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 theappropriate Teichmueller space T.Think of the image as a complex geodesic in T, through Sin the direction ω.



Teichmueller disc

The corresponding disc in T is typically called aTeichmueller disc.The image of a Teichmueller disc in the moduli space is acurve.Otherwise you get something analogous to an irrationalline on a torus.


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 theVeech group is a lattice in PSL2(R).The corresponding disc in T is typically called aTeichmueller disc.The image of a Teichmueller disc in the moduli space is acurve.Otherwise you get something analogous to an irrationalline on a torus.the canonical example is PSL2(Z) acting on the upper halfplane.


Voronoi and Delaunay Tilings

Voronoi Decomposition of a Flat Surface

Given a flat surface S construct the Voronoi tiling withrespect to the singular points F .Show and tell quadrilateral example on board.cells are points closest to a unique singular pointedges are points closest to two singular pointsvertices are points closest to three or more singular points



Delaunay Decomposition of a Flat Surface

The Delaunay decomposition is dual to Voronoidecomposition.The vertices are the singular points.Every polygon in a Delaunay tiling is cyclic (byconstruction).The Delaunay decomposition is a canonical polygonconstruction for a flat surface.Generically the decomposition is by triangles.Show and tell quadrilateral example on board.



Veech Tesselation -1

The Delaunay decomposition of Sg will vary in aTeichmueller disc.Locally constant, generically a triangle decomposition.Transitions are generically quadrilateral flipsShow and tell quadrilateral flip on the board forquadrilateral flip.Mark all the transitions on a hyperbolic disc.Show picture of tesselation by Bowman from the web.



Veech Tesselation - 2

Here is a paraphrasal of a theorem due to Veech.

TheoremThe transition points in the hyperbolic disc are portions ofgeodesics.

back to picture of tesselation by Bowman from the web.



current work

Tangent Star Lemma.Analysis of stratification of Teichmueller space of flatsurfaces by the Delaunay decomposition.Relation of the Veech group and the automorphism groupof the Veech Tessellation.


done

references and links

William Veech A tessellation associated to a quadraticdifferential. Preliminary report. Abstracts of the AMS946-37-61.M. Troyanov, On the Moduli Space of Singular EuclideanSurfaces, arXiv:math/0702666v2Josh Bowman’s websitehttp://www.math.cornell.edu/ bowman/


done

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