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

Transcription

1 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

2 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).

3 credits and agenda agenda geometric definition of flat surfaces analytic definition of flat surfaces relation to Teichmueller discs Veech groups Veech tesselation

4 first definition and examples informal definition/construction of flat surfaces Definition Let P 1,..., P n 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.

5 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)

6 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)

7 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

8 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.

9 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 α 1,..., α s are the angles at a cone point v j F, then the (total) cone angle at v j is θ j = s α i. i=1 A cone point is regular if and only the cone angle equals 2π (lies flat).

10 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.

11 first definition and examples Euler s formula Proposition Suppose a flat surface has genus g and v cone points with cone angles θ j. Then v θ j = 2π(2g 2 + v). j=1

12 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 {u α } on S\F such that the transition maps are affine linear u α (P) = au β (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 {u α } is called a translation structure. If a = ±1 then {u α } is called a demi-translation structure.

13 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 {u α } may be defined as follows u α (P) = P P 0 f 1/q (z)dz If ω is a 1-form then {u α } is a translation structure. If ω is a quadratic differential then {u α } is a demi-translation structure.

14 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 {u α } as follows u α (P) = P P 0 f 1/q (z)dz The flat structure defined above will have G as a group of automorphisms.

15 automorphisms and deformation of structures deformation of structures Let {u α } be defined by a quadratic differential ω, and let g PSL 2 (R). Then {gu α } is a demi-translation structure as gu α (P) = g(au β (P) + b) = agu β (P) + gb) Let S g be the corresponding surface. If g SO(2) then S and S g are conformally equivalent. g S g is a map of H = PSL 2 (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.

16 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.

17 definition of Veech group Veech group and Teichmueller disc The Veech group is essentially the set of all g PSL 2 (R) such that S and S g are conformally equivalent. Specifically interested in those deformations in which the Veech group is a lattice in PSL 2 (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 PSL 2 (Z) acting on the upper half plane.

18 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

19 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.

20 Voronoi and Delaunay Tilings Veech Tesselation -1 The Delaunay decomposition of S g 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.

21 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.

22 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.

23 done references and links William Veech A tessellation associated to a quadratic differential. Preliminary report. Abstracts of the AMS M. Troyanov, On the Moduli Space of Singular Euclidean Surfaces, arxiv:math/ v2 Josh Bowman s website bowman/

24 done Any Questions?

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