Fundamental domains and generators for lattice Veech groups

Transcription

1 Fundamental domains and generators for lattice Veech groups Ronen E. Mukamel August 0 0 Abstract The moduli space QM g of non-zero genus g quadratic differentials has a natural action of G = GL + R/ ± 0 0. The Veech group PSLX q is the stabilizer of X q QM g in G. We describe a new algorithm for finding elements of PSLX q which for lattice Veech groups can be used to compute a fundamental domain and generators. Using our algorithm we give the first explicit examples of generators and fundamental domains for non-arithmetic Veech groups where the genus of H/ PSLX q is greater than zero. Contents Introduction Quadratic differentials Girth differentials and saddle periods Examples Introduction Fix an integer g and let M g be the moduli space of genus g Riemann surfaces. The space M g is a complex orbifold and carries a complete Finsler Teichmüller metric whose geodesics are explicitly described by Teichmüller s theorem. The bundle QM g M g of holomorphic quadratic differentials consists of pairs X q where q is a non-zero holomorphic quadratic differential on X M g. The geodesic flow on M g gives an R- action on QM g which together with the C -action fixing X and rescaling q generates an action of G = GL + R/ ± 0 0 on QM g. The Veech group PSLX q is the stabilizer of X q in G. There are many examples of surfaces whose Veech groups are complicated [HS HL Mc Th]. There are few The research for this paper was supported in part by grant DMS-06 from the National Science Foundation.

2 examples of Veech groups that have been described in their entirety. For instance each integer D congruent to 0 or mod determines a quadratic differential X D q D in QM as in Figure whose Veech group is a lattice in PSL R [Ca Mc]. The homeomorphism type of H/ PSLX D q D is determined in [Mc Ba Mu] but little else is known about PSLX D q D outside of some small values for D. The purpose of this paper is to introduce girth differentials and use them to study the group PSLX q and the quotient H/ PSLX q. We will describe an algorithm for finding elements of PSLX q which for lattice Veech groups gives generators for PSLX q and a fundamental domain for the action of PSLX q on H. Using an implementation of our algorithm in Sage [Mu] we compute the first explicit examples of fundamental domains and generators for non-arithmetic Veech groups where the genus of H/ PSLX q is greater than zero. Theorem.. Fundamental domains and generators for PSLX D q D are encoded in Table for D = and 60. A fundamental domain for PSLX q is depicted in Figure. It is elementary and straightforward to check whether any given element g G is in PSLX D q D by comparing the Delaunay triangulations of X D q D and g X D q D and the tools to do so are available in [Mu]. To prove Theorem. without refering to the results in this paper one simply checks that the claimed generators are in PSLX D q D and that the group they generate has covolume equal to the volume of H/ PSLX D q D which is computed in [Ba]. We will give another proof of Theorem. at the end of this introduction that is independent of [Ba]. The rest of this paper is devoted to explaining the algorithm used to generate Table. Girth differentials and the well-girthed spine. Throughout this paper we fix a quadratic differential X 0 q 0 in QM g and set QL = G X 0 q 0 and L = C \QL. The quotient L is a hyperbolic Riemann surface isomorphic to H/ PSLX 0 q 0 and points in L are quadratic differentials in QL up to scale. We will call a differential X q QL a girth differential if one of its shortest saddle connections is horizontal of length one and for each X q QL we define: NX q = {λ C : X λq is a girth differential}. The set NX q is a non-empty cyclically ordered and finite set. The number of elements in NX q depends only on the differential up to scale [X q] L and equals the product of the number of girth differentials in C q with the orbifold order of [X q] in L. We will call a differential up to scale [X q] L well-girthed if #NX q. Our algorithm for finding elements of PSLX 0 q 0 is based on the study of the level sets for #NX q: L[n] = {[X q] L : #NX q = n} and the well-girthed spine: Spine W G L = n L[n]. Our main theorem shows that the sets L[n] stratify L by easy to understand pieces and L deformation retracts onto Spine W G L.

3 Theorem.. The sets L[n] give a stratification L = n L[n] by disjoint sets with n k L[n] closed for each k and L deformation retracts onto its well-girthed spine: L Spine W G L. Each connected component of L[] is diffeomorphic to the disk or the punctured disk each connected component of L[] is a finite length geodesic arc and the set n L[n] is discrete in L. Examples well-girthed spines are depicted in Figures and. Finite volume orbits. When the volume of L is finite we use the Veech dichotomy to show that L[] contains a neighborhood of each cusp allowing us to prove: Theorem.. If the volume of L is finite then Spine W G L is compact the set n L[n] is finite the set L[] has finitely many connected components and the set L[] consists of # cuspsl components of each of which is diffeomorphic to the punctured disk. Decorated Riemann surfaces. Theorem. should be compared with the decompositions of decorated Riemann surfaces studied in [BE]. A decoration of a finite volume cusped hyperbolic surface X is a subset B X consisting of disjoint and simple horocycles about each cusp of X. Associated to each decoration of X is a spine Spine D X B X similar in spirit to the spine Spine W G L of L consisting of all points in X which have multiple shortest paths to B. For finite volume L one can show that the well-girthed spine Spine W G L is associated to a particular decoration of L. Algorithm. Theorem. is meant to suggest an algorithm for exploring L and finding elements of PSLX 0 q 0. Each differential up to scale [X q] L[k] for k and of orbifold order e in L is connected to k/e neighbors in n L[n] by components of L[]. It is staightforward to compute the neighbors of [X q] L[k] from the saddle periods on X q and we describe an algorithm to do so at the end of Section see also Figure. The graph whose vertices correspond to points in n L[n] and whose edges correspond to components of L[] can be explored and the points in n L[n] and components of L[] can be enumerated by performing a breadth first search using this algorithm. Elements of PSLX 0 q 0 can be computed by finding cycles in this graph. By Theorem. when PSLX 0 q 0 is a lattice the graph above has finitely many vertices and edges and our breadth first search will terminate allowing us to compute a fundamental domain and set of generators for PSLX 0 q 0. Note that this process does not require knowing the volume of H/ PSLX 0 q 0 in advance. In Section we apply our algorithm to several examples of Veech groups of genus two quadratic differentials see also Figures and. If L has orbifold points we mean that there is a homotopy h t : L L defined on the coarse space associated to L and a homotopy h t : L L defined on a good cover π : L L so that h 0 is the identity h t restricts to the identity on π Spine W G L and h is a retraction onto π Spine W G L and π h t = h t π.

4 V= i orbifold point - element of L[] - component of L[] Figure : The Riemann surface L = H/ PSLX q of genus one has nine cusps and three orbifold points of order two. The convex hull of V H top is an ideal -gon middle and a fundamental domain for PSLX q. Our algorithm computes a fundamental domain and generators for PSLX q by performing a breadth first search on the well-girthed spine Spine W G L bottom.

5 Proof of Theorem.. As described in Section the pair S D σ D listed in Table encodes an ideal hyperbolic polygon DS D and generators Γ SD σ D G for a Fuchsian group Γ D with fundamental domain DS D. By comparing Delaunay triangulations as described above we check that each element of Γ SD σ D is in PSLX D q D and conclude that Γ D is finite index in PSLX D q D. To check that PSLX D q D is no larger we enumerate the vertices and edges of the preimage P of the well-girthed spine under the natural map H/Γ D H/ PSLX D q D. By comparing Delaunay triangulations we see that the vertices of P correspond to pairwise distinct points in QM and conclude that Γ D = PSLX D q D. Other algorithms. There are several other methods for computing elements in PSLX 0 q 0. The algorithm described in [Mc] searches for parabolic elements in PSLX 0 q 0 and will find generators for lattice Veech groups only if the quotient H/ PSLX 0 q 0 has genus zero. A stratification of L by hyperbolic polygons based on Delaunay triangulations is studied in [Ve] and an algorithm based on this stratification is suggested in [Bo]. While this algorithm will theoretically give generators and a fundamental domain for arbitrary Veech groups the stratification of L in Theorem. will typically be much simpler than the Delaunary stratification and Theorem. is the first example of an explicit computation of fundamental domains and generators for non-arithmetic Veech groups with higher genus quotients. An algorithm for finding generators of PSLX 0 q 0 when PSLX 0 q 0 is arithmetic i.e. commensurable to PSL Z is given in [Sc] and in [ER] it is shown that every finite index subgroup of Γ = kerpsl Z PSL Z/Z is a Veech group. Notes and references. The spine Spine W G L is also studied in [SW] to give a characterization of surfaces with lattice Veech groups. In particular the retraction L Spine W G L we define in Proposition. also appears there cf. proof of Proposition. in [SW]. Theorem. can be read as a converse to Corollary. of [SW] which shows that if L[] has finitely many components then L has finite volume. Our terminology is meant to suggest a comparison with well-rounded lattices which form a spine for the homogeneous space SL n R/ SL n Z [Ash]. For examples of quadratic differentials with complicated and not necessarily lattice Veech groups see e.g. [Mc Th]. It would be interesting to explore those Veech groups using our algorithm. For further background on the moduli space of quadratic differentials QM g the G-action on QM g and Veech groups see e.g. [MT] or [Zo]. Acknowledgements. I would like to thank C. McMullen and S. Kerckhoff for useful conversations. I would also like to thank C. McMullen for sharing computer code that enumerates the saddle periods on X 0 q 0 on which our algorithm and the computations appearing in this paper rely. The author was supported in part by National Science Foundation grant DMS-06.

6 Quadratic differentials In this section we collect background and fix notation about quadratic differentials and the G-action on QM g. For further background see [MT Zo]. Quadratic differentials. A holomorphic quadratic differential on a Riemann surface X M g is a holomorphic section of the square of the cotangent bundle on X. The collection QX of all holomorphic quadratic differentials on X forms a g - dimensional complex vector space and as X ranges in M g the non-zero elements in QX form a bundle QM g M g. A non-zero differential q QX gives a metric q on X which is flat except for cone singularities at the zeros Zq of q. The differential q also determines a foliation Fq of X \ Zq by horizontal geodesics. A typical way to specify a pair X q is to glue parallel sides of equal length on a collection of polygons {P... P k } in C by transition functions of the form z ±z + c see Section for examples: X q = i P i dz /. Lines in C give geodesics on X and horizontal lines give leaves of Fq. The quadratic differential up to scale represented by X q we will denote by [X q]. Saddle connections and saddle periods. A saddle connection on X q is a q -geodesic γ : [a b] X beginning and ending at Zq and avoiding Zq otherwise. Along γ there are two choices of square roots for q and the complex numbers γ ± q are called periods of γ. The image in C /z z of the set of all saddle periods on X q we will denote by: { } PerX q = ± γ is a saddle q : C /z z. γ connection on X q Note that PerX λ q = {±λv : ±v PerX q} = λ PerX q. The set PerX q is discrete and its preimage in C is bounded away from zero. The set of all shortest saddle periods in PerX q we will denote by: Per m X q = {±v PerX q : ± v ± w for any ± w PerX q}. As in Section we will call X q a girth differential if one of the shortest saddle periods on X q is horizontal of length one i.e. ± Per m X q. Since Per m X q is a discrete subset of a circle it is a non-empty cyclically ordered and finite set. The sets NX q and Per m X q are in natural bijection: Proposition.. The map ±λ λ gives a bijection between Per m X q and NX q. Proof. For any λ C the shortest saddle periods on X q and X λ q are related by λ Per m X q = Per m X λ q. One of the shortest saddle connections on X λ q is horizontal and of length one if and only if ±λ Per m X q. 6

7 GL + R- and G-actions. Let GL + R denote the group of two-by-two matrices with positive determinant and let G be the quotient GL + R/ ± 0 0. The group R acts on C by real-linear maps: GL + a b x + iy = ax + by + cx + dyi. c d This action covers a G-action on the quotient C /z z. For a quadratic differential X q obtained by gluing together the polygons P i C and a matrix A GL + R the quadratic differential A X q is obtained by gluing together the polygons A P i : A X q = i A P i dz /. The matrix 0 0 stabilizes every point in QMg and this GL + R-action also covers a G-action. The Veech group of X q is its stabilizer in G: PSLX q = {A G : A X q = X q}. An important property of saddle periods is that they are equivariant with respect to G i.e.: PerA X q = A PerX q. In particular PSLX q preserves the set PerX q. It is not true that the shortest saddle periods are equivariant with respect to G. Instead we have: Proposition.. Fix any X q QM g the set: UX q = {B G : Per m B X q B Per m X q} contains an open neighborhood of the identity. Proof. The claim follows easily from the fact that PerX q is discrete. Let l 0 be the length of one of the shortest saddle periods on X q and let / > ɛ > 0 be a number so that there are no saddle periods whose lengths belong to the interval l 0 + ɛl 0. We will show that UX q contains the image in G of the set Uɛ/ of matrices that distort lengths by a factor of at most + ɛ/: Uɛ/ = { g GL + R : + ɛ/ v < g v < + ɛ/ v for any v C }. If g Uɛ/ then the length of the shortest saddle period on g X q is at most +ɛ/l 0 and the longest saddle period in g Per m g X q PerX q has length at most +ɛ/ l 0 < +ɛl 0. Based on our choice of ɛ Per m g X q g Per m X q and the image of g in G is in UX q. Proposition. yields the following as a corollary. Proposition.. For any k the set n k L[n] is closed in L. 7

8 Proof. We will show that the complement V = n<k L[n] of n k L[n] is open. If [X q] V then Per m X q contains fewer than k saddle periods. By Proposition. there is a neighborhood U of the identity in G consisting of matrices g with Per m g X q g Per m X q. The image of U in L under g [g X q] is a neighborhood of [X q] and contained in V so V is open. Euclidean similarities and the hyperbolic plane. We will denote by C GL + R the subgroup of Euclidean similarities i.e. the subgroup commuting with SO. We will denote by [A] the coset representative of A in C \ GL + R. We will identify the quotient C \ GL + R with the hyperbolic plane via the bijection H C \ GL + R defined by τ [A τ ] where A τ = Re τ 0 Im τ. In particular there is a unique holomorphic structure and metric on C \ GL + R for which this identification is an isometry. The map f : H L defined by f[a] = [A X 0 q 0 ] is the universal covering map for L. Möbius transformations. The quotient C \ GL + R has a right GL+ R-action and G-action with B acting by [A] [AB]. There is a unique Möbius transformation m B : H H with the property that [A τ B] = [A mb τ]. Namely if B = a b c d then m B τ = aτ b cτ+d. The universal cover f : H L satisfies f m B = f for each B PSLX 0 q 0 and f covers an isometry H/ PSLX 0 q 0 L where B PSLX 0 q 0 acts on the right on H by m B. Geodesics half planes and horoballs. We now state some facts about geodesics half planes and horoballs in H = C \ GL + R that are elementary to verify. For any pair of distinct points ±v and ±w in C /z z with v and w linearly independent over R the set: γ±v ±w = {[A] H : A ±v = A ±w } is the geodesic with endpoints Rev Rew Rev+Rew Imw Imv and Imv+Imw. For such ±v and ±w the set: T ±v ±w = {[A] H : A ±v < A ±w } is an open half plane with boundary T ±v ±w = γ±v ±w. Note if v = λw with λ > real the set γ±v ±w is empty T ±v ±w is empty and T ±w ±v = H. For any ±v C /z z and real number C > 0 the set: B±v C = {[A] H : A ±v deta } C. is the closed horoball tangent to H at Rev/ Imv and of radius C Im v. 8

9 Girth differentials and saddle periods In this section we will prove Theorems. and. by studying the set PerX 0 q 0 of saddle periods on X 0 q 0 and the action of PSLX 0 q 0 on PerX 0 q 0. We will associate to each finite subset S PerX 0 q 0 a set: H m S = {[A] H : Per m A X 0 q 0 = A S}. We will characterize when H m S is non-empty and then prove the following proposition allowing us to establish the claims about the components of L[n] made in Theorem.: Proposition.. Fix S PerX 0 q 0 with H m S non-empty. One of the following holds:. H m S is an open convex subset of H and contained in a horoball #S = ;. H m S is an open geodesic arc of finite length #S = ; or. H m S is a single point #S. The image of H m S under [A] [A X 0 q 0 ] is a connected component of L[#S]. We will then construct an explicit homotopy inverse for the inclusion Spine W G L L using the G-action on QL to complete the proof of Theorem.. The key observation is that the set of girth differentials is closed under multiplication by matrices of the form 0 0 e t for t > 0: Proposition.. If X q QL is a girth differential and t 0 then 0 0 e t X q is also a girth differential. This gives a foliation of the set of girth differentials by intervals which project to geodesic rays in L. The map collapsing these rays to their endpoints gives the desired homotopy inverse. Finally we will conclude this section by turning to the case where L has finite volume. Using the Veech dichotomy we will show that L[] contains a horoball neighborhood of each cusp of L to conclude that Spine W G L is contained in compact set and prove Theorem.. Ellipses. To start we will characterize when H m S is empty. By an ellipse respectively circle in C /z z we will mean the image of an ellipse respectively circle in C invariant under z z. We will say a finite subset S PerX 0 q 0 is supported by the ellipse E C /z z if S = E PerX 0 q 0 and the region bounded by E contains no saddle periods. Equivalently an ellipse E containing S supports S if and only if Per m A X 0 q 0 = A S whenever A E is a circle. The subsets of PerX 0 q 0 supported by ellipses are those for which H m S is nonempty: Proposition.. Fix a finite set S PerX 0 q 0. The set H m S is non-empty if and only if S is supported by an ellipse. 9

10 Proof. First suppose [A] H m S and let E C /z z be the circle passing through Per m A X 0 q 0 = A S. The ellipse A E supports S. Conversely suppose E supports S and let A G be any matrix for which A E is a circle. For such an A Per m A X 0 q 0 = A S and [A] H m S. The collection of subsets of PerX 0 q 0 supported by ellipses: EllX 0 q 0 = {S PerX 0 q 0 : S is supported by an ellipse}. and the collections EllX 0 q 0 [n] = {S EllX 0 q 0 : #S = n} will play an important role in what follows. Convex sets. in H. We now associate to each S EllX 0 q 0 a complete convex set HS Proposition.. For any S EllX 0 q 0 the set: HS = {[A] H : The set A S lies on a circle} is a convex and complete. More specifically HS = H if #S = HS is a geodesic if #S = and HS is a point if #S. Proof. First suppose #S =. For any [A] H A S consists of a single point ±v C /z z which lies on the circle of radius ±v. If #S = i.e. S = {±v ±w} then HS is the geodesic γ±v ± w defined in Section note that v is not a real multiple of w since S is supported by an ellipse. Finally suppose #S. The preimage of S in C consists of at least six points and there is exactly one ellipse E in C /z z containing S. The set HS consists of the single point [A] H for which A E is a circle. Next we study the set H m S defined at the beginning of this section: Proposition.. For any S EllX 0 q 0 the set: H m S = {[A] H : Per m A X q = A S} is a non-empty convex and open subset of HS and is contained in the intersection of horoballs ±v S B ±v AreaX 0q 0. π Proof. The set H m S is non-empty by Proposition.. We will now show that H m S is open in HS. Fix a point [A] H m S. By Proposition. there is a neighborhood U of the identity in G with the property that: Per m BA X 0 q 0 B Per m A X 0 q 0 = BA S whenever B U. The image of U under g : B [BA] is an open neighborhood of [A]. For any B U with gb = [BA] HS Per m BA X 0 q 0 is a subset of BA S a set which lies on a circle. It follows that Per m BA X 0 q 0 = BA S [BA] H m S and gu HS is contained in gu H m S. 0

11 Next we will show that H m S is convex. Recall that for any ±v and ±w in C /z z the set T ±v ±w = {[A] H : A ±v < A ±w } is convex and usually an open half plane. The region H m S satisfies: H m S = HS ±w PerX 0 q 0 ±w S ±v S T ±v ±w and is also convex. We will now show H m S is contained in the closed horoball B±v AreaX 0q 0 π for each ±v S. If A S = Per m A X 0 q 0 then the injectivity radius of the metric flat metric on A X 0 q 0 is at least A ±v for each ±v S. Whenever [A] B±v AreaX 0 q 0 /π the area of A X 0 q 0 is small enough the injectivity radius of the flat metric is smaller than A ±v. A stratification of H. We now show that the sets H m S stratify H and that the image of H m S under the universal covering map f : H L is an open subset of L[#S]. Proposition.6. The sets H m S as S ranges in EllX 0 q 0 are pairwise disjoint and cover H. Proof. First suppose H m S and H m S intersect. If [A] H m S H m S then A S = Per m A X 0 q 0 = A S and since A is invertible S = S. To see that the sets H m S cover H note that [A] is in H m S where S = A Per m A X 0 q 0. Proposition.7. For any S EllX 0 q 0 [n] the image fh m S is an open subset of L[n]. Proof. The set fh m S is contained in L[n] since for any [A] H m S f[a] = [A X 0 q 0 ] has: #NA X 0 q 0 = # Per m A X 0 q 0 = #A S = n. Now fix any [A] H m S. We will show that fh m S contains a neighborhood of f[a] in L[n]. By Proposition. there is a neighborhood U of the identity in G with the property that Per m BA X 0 q 0 B Per m A X 0 q 0 whenever B U. The image of U under g : B [BA X 0 q 0 ] is an open neighborhood of f[a] in L and gb is in L[n] if and only if: Per m BA X 0 q 0 = B Per m A X 0 q 0 = BA S i.e. [BA] H m S. The neighborhood gu L[n] of f[a] is contained in fh m S.

12 Veech group. We now study the action of PSLX 0 q 0 on H. We start by showing that PSLX 0 q 0 permutes the pieces of the stratify of H = S EllX 0 q 0 H ms: Proposition.8. Fix B PSLX 0 q 0 and S EllX 0 q 0. The set B S is in EllX 0 q 0 and the Möbius transformation m B : [A] [AB] restricts to an isometry between H m S and H m B S. Proof. Fix [A] H m S i.e. Per m A X 0 q 0 = A S. If E is any ellipse supporting S then B E supports B S. The coset [AB] = m B [A] is in H m B S since: Per m AB X 0 q 0 = Per m A X 0 q 0 = A S = AB B S. This shows that m B sends H m S into H m B S. Since m B = m B m B restricts to an isometry between H m S and H m B S. Partitions of L and L[n]. By Proposition.8 the Veech group permutes the components of the stratification H = S EllX 0 q 0 H ms giving a stratification of the quotient H/ PSLX 0 q 0 into disjoint sets: H/ PSLX 0 q 0 = H m S/ StabS. [S] EllX 0 q 0 / PSLX 0 q 0 Here [S] is the coset in EllX 0 q 0 / PSLX 0 q 0 containing S and StabS is the stabilizer of S in PSLX 0 q 0. Since the universal covering map f : H L factors through an isometry H/ PSLX 0 q 0 L and fh m S is contained in L[#S] this stratification of H/ PSLX 0 q 0 gives a stratification of L[n] into disjoint sets: L[n] = fh m S. [S] EllX 0 q 0 [n]/ PSLX 0 q 0 By Proposition.6 fh m S is open in L[n] giving: Proposition.9. For each S EllX 0 q 0 [n] the set fh m S is a connected component of L[n]. The map [S] fh m S gives a bijection between the connected components of L[n] and EllX 0 q 0 [n]/ PSLX 0 q 0. Proof. The sets fh m S as S ranges over coset representatives in EllX 0 q 0 [n]/ PSLX 0 q 0 are disjoint and open subsets of L[n]. They are also connected since H m S is convex and therefore path connected. We are now ready to prove Proposition. stated at the beginning of this section: Proof of Proposition.. Fix S PerX 0 q 0 with H m S not empty i.e. S EllX 0 q 0. In Proposition.9 we showed that the image of H m S under [A] [A X 0 q 0 ] is a connected component of L[n]. The remaining claims about H m S follow from Propositions. and.. If #S = H m S is an convex open subset of HS = H and contained in the horoball B ±v AreaX 0 q 0 /π where ±v S. If #S = H m S is an open and convex subset of the geodesic HS and is contained in the intersection ±v S B ±v AreaX 0q 0 π which is compact. Such a set must be an open geodesic arc of finite length. Finally if #S then H m S is a point.

13 Connected components of L[n]. We are now ready to prove the claims in Theorem. about the connected components of L[n]. We start with L[]: Proposition.0. Each connected component of L[] is diffeomorphic to the disk or the annulus. Proof. By Proposition.9 each connected component of L[] is equal to fh m S for some S EllX 0 q 0 []. By Proposition.8 the quotient H m S/ StabS injects into the quotient H/ PSLX 0 q 0 and the restriction of f to H m S covers a bijection from the quotient H m S/ StabS onto fh m S. This bijection is a diffeomorphism since it is a local isometry. We need to show that H m S/ StabS is diffeomorphic to the disk or the annulus. Since H m S is a convex open set in HS = H H m S is diffeomorphic to a disk. The stabilizer StabS is a unipotent subgroup of the discrete group PSLX 0 q 0. Either StabS is trivial and fh m S is diffeomorphic to the disk or StabS is isomorphic to Z acts freely and properly discontinuously on H m S and H m S is diffeomorphic to the annulus. Proposition.. Each connected component of L[] is a finite length geodesic arc. Proof. By Proposition.9 each connected component of L[] is equal to fh m S for some S EllX 0 q 0 []. By Proposition. H m S is an open convex subset of the geodesic HS and is contained in the intersection of a pair of horoballs tangent to distinct points in H. The set H m S is a finite length geodesic arc and so is its image fh m S in L. Proposition.. The set n L[n] is discrete in L. Proof. Let [A] H be any point with f[a] = [A X 0 q 0 ] L[n] for n. Set S = A Per m A X 0 q 0 so that [A] H m S. By Proposition. there is a neighborhood U of the identity in G with the property that Per m BA X 0 q 0 B Per m A X 0 q 0 = BA S whenever B U. The image of U under g : B f[ba] is an open set in L. Suppose gb L[k] with k and choose S S so Per m BA X 0 q 0 = BA S. Since S contains at least three points there is a unique ellipse E supporting S. The matrix B takes the circle A E into the circle BA E and is therefore a Euclidean similarity. In other words the only point in gu n L[n] is f[a]. Retraction and homotopy equivalence. We now show that L deformation retracts onto n L[n] by constructing an explicit homotopy inverse. We start by showing that L[] is foliated by geodesic rays. To do so we will show that the set of girth differentials in QL: GL = {X q QL : X q is a girth differential } is invariant under multiplication by g t = 0 0 e t for t 0. The following Proposition implies Proposition.:

14 Proposition.. There is a continuous function T : GL R 0 so that for any X q GL we have:. g t X q is a girth differential and #Ng t X q = for t > T X q;. g t X q is not a girth differential for t < T X q; and. g t X q is a girth differential and #NX t q t for t = T X q. Proof. For any ±x+iy PerX q with x < the number t 0 = log x y is nonpositive since x +y >. The number t 0 also has the property that 0 0 e ±x+iy t is greater than one for t > t 0 equal to one when t = t 0 and is less than one for t < t 0. It is straightforward to verify that the function T X q given explicitly by: x T X q = sup ±x+iy PerXq log x < satisfies the desired properties. Note that the intersection of PerX q with the strip {x + iy : x < } is non-empty since the area of q is finite. Proposition. shows that L[] is foliated by open half-infinite geodesic rays whose endpoints lie in n L[n]. The map sending [X q] L[] to the endpoint of the leaf of this foliation gives a homotopy inverse for the inclusion n L[n]: Proposition.. The inclusion Spine W G L L is a homotopy equivalence with homotopy inverse: h : L Spine W G L defined by h[x q] = [g T Xq X q] whenever X q GL. Proof. For X q GL and t 0 define: y h t X q = g max tt Xq X q. The functions h t : GL GL are continuous and depend continuously on t. Also h t covers a well a well-defined map h t : L L since whenever C q contains more than one girth differential T X q i = 0 for each i. The function h 0 is the identity map on L and the function h t restricts to the identity on Spine W G L for every t 0. As t tends to infinity h t tends uniformly on compact sets to the retraction h = h. The functions h t gives the desired homotopy equivalence at least between the coarse space associated to the orbifold L and the subset n L[n]. If L has orbifold points we can replace L by a good cover π : L L say by marking the Z/nZ-homology of [X q] L for n large enough. It is straightforward to define a homotopy h t between the identity on L and a retraction onto π n L[] which satisfies π h t = h t π. Note that the orbifold points of L all lie in n L[n] since the orbifold order of [X q] L[n] divides n. Proof of Theorem.. The sets L[n] are clearly disjoint and give a stratification L = n L[n]. The set n k L[n] for each k is closed by Proposition.. The other claims about the components of L[n] are established in Propositions.0. and.. That L[n] L is a homotopy equivalence is established in Proposition.. n.

15 Lattice Veech groups. and L has finite volume. We now turn to the case when PSLX 0 q 0 is a lattice Cusps. A cusp of a Fuchsian group Γ is the fixed point x H of any unipotent element g Γ. The set of all cusps of Γ we will denote by CΓ. A cusp of the quotient V = H/Γ is a Γ-orbit in CΓ and we will denote the set of all cusps on V by CV. When the volume of V is finite the set CV is finite and can be added to V to give a closed surface V = V CV. Cylinder decompositions. It is well known that if PSLX 0 q 0 contains a unipotent element stabilizing the line L = R ±x + iy of slope y/x then there is a collection {γ... γ n } of saddle connections whose periods lie in L and whose complement in X 0 q 0 is a disjoint union of metric cylinders {C... C k }. We will need the following partial converse for lattice Veech groups which is a consequence of the famous Veech dichotomy [Ve]: Theorem. Veech. Suppose PSLX 0 q 0 is a lattice. Each saddle period ±v PerX 0 q 0 is parallel to a cylinder decomposition of X 0 q 0 and is stabilized by a unipotent element in PSLX 0 q 0. The Veech dichotomy allows us to prove: Proposition.. The following three subsets of H are equal: T = CPSLX 0 q 0 T = { x/y H : ±x + iy PerX 0 q 0 } and T = { x/y H : {±x + iy} EllX 0 q 0 }. Proof. The sets T and T are equal by Theorem.. The sets T and T are equal since a collection S = {±v} PerX 0 q 0 of saddle periods consisting of a single saddle period is supported by an ellipse if and only if ±v is the shortest saddle period in the line R ±v. Proposition.6. Suppose L has finite volume and S = {±v} EllX 0 q 0. The set H m S contains B±v K for some K > 0. Proof. When L has finite volume the complement in X 0 q 0 of the set of all saddle connections with periods parallel to ±v is a disjoint union of metric cylinders {C... C k }. Let K > 0 be a number smaller than AreaC i for each i. For any point [A] B±v K the height of the cylinder A C i is larger than A ±v. For such an [A] any saddle period in A X 0 q 0 not parallel to A ±v is longer than A ±v. The saddle periods parallel to A ±v are also longer than A ±v since S is supported by an ellipse so A ±v is the unique shortest saddle period on A X 0 q 0. The point [A] is in H m S. Proposition.7. Suppose L has finite volume. The set L[] has #CL connected components each of which is diffeomorphic to the annulus and L[] contains a horoball around each cusp of L.

16 Proof. By Proposition.9 each connected component of L[] is equal to fh m S for some S = {±v} EllX 0 q 0 []. By the Theorem. ±v is stabilized by a unipotent element of PSLX 0 q 0 and StabS is isomorphic to Z. The component fh m S is diffeomorphic to the annulus H m S/ StabS. To see that L[] has #CL connected components note that the equality of sets T = T from Proposition. gives a bijection between the cusps of PSLX 0 q 0 and elements of EllX 0 q 0 [] which is equivariant with respect to PSLX 0 q 0. So we have: #CL = #CPSLX 0 q 0 / PSLX 0 q 0 = # EllX 0 q 0 []/ PSLX 0 q 0 and the right hand side is in bijection with the number of components of L[] by Proposition.9. Now fix a cusp c of L and let t H be any cusp of PSLX 0 q 0 mapping to c. By Proposition. t = x/y for some S = {±x + iy} EllX 0 q 0. The set H m S whose image under f lies in L[] contains a horoball about t by Proposition.6. Proposition.8. If the volume of L is finite then Spine W G L is compact the set L[n] is finite the set L[] has finitely many components. n Proof. By the previous proposition the set L[] contains a neighborhood of each cusp in L. The spine Spine W G L is closed by Proposition. and is contained in a compact subset of L since it is contained in the complement of L[]. Since n L[n] is a discrete subset of this compact set n L[n] is finite. Since L has finite volume the Euler characteristic of L is finite. The homotopy equivalent set n L[n] also has finite Euler characteristic. But the Euler characteristic of n L[n] is equal the difference between the number of points in n L[n] and the number of components of L[] and possibly a finite contribution from the orbifold points on L so L[] has finitely many components. Proof of Theorem.. The remaining claims about L[n] when L has finite volume are established in Propositions.7 and.8. Algorithm. Fix a point [X q] L[k] with k. We conclude this section by describing an algorithm for finding the neighbors of [X q] in the graph whose vertices are points in n L[n] and whose edges are components of L[]. Choose a matrix A G so [X q] = [A X 0 q 0 ] and set S = A Per m A X 0 q 0 so H m S = {[A]} and fh m S = [X q]. Let ±v... ±v k be the elements in S ordered by the counterclockwise cyclic ordering induced by the unique ellipse supporting S. The components of L[] emanating from [X q] are the geodesic arcs fh m {±v i ±v i+ } where subscripts are read modulo k. One endpoint of H m {±v j ±v j+ } is H m S. To compute the other endpoint of H m {±v j ±v j+ }:. Compute the matrix B G with B ±v j = ± and B ±v j+ = ±i. The cyclic ordering on the v i s ensures that B S lies in the quadrant {±x + iy : xy 0}.. Find the minimum Ecc 0 of the quantity: Ecc±w = x y /xy where ± x + iy = B ±w 6

17 and ±w ranges in the saddle periods for which B ±w = ±x+iy with x y <. The condition on ±w ensures there is an ellipse E passing through ± ±i and B ±w and the quantity Ecc±w measures the eccentricity of E.. The collection of saddle periods: S = { ±w PerX 0 q 0 : B ±w = ±x + iy with x + Ecc 0 xy + y = } contains at least three saddle periods and is supported by an ellipse. The endpoints of H m {±v j ±v j+ } are H m S and H m S. Figure gives an example of a quadratic differential up to scale with #NX q = as well as its three neighbors. Examples In this section we study several examples of Veech groups in genus two using our algorithm. L-shaped polygons. Each integer D satisfying D 0 or mod determines a Euclidean octagon P D as in Figure. Gluing parallel sides of equal length together by translations gives a genus two Riemann surface with a holomorphic quadratic differential: X D q D = P D dz /. An important result of Calta and McMullen is that the Veech group of X D q D is a lattice: Theorem. Calta McMullen. The Veech group PSLX D q D is a lattice. Set QL D = G X D q D and L D = C \QL D. Fundamental domains and generators. We now show how to encode an ideal n-gon DS and generators Γ Sσ for a Fuchsian group with fundamental domain DS by a pair S σ consisting of a finite set S R and a gluing involution σ in the permutation group of S. Let v = x y... v n = xn y n be the elements of S ordered so that x /y < x /y < < x n /y n and set z j = x j + iy j. Define DS H to be the convex hull of the set { x /y... x n /y n } H. Given a pair v j and v k in S there is a unique element Av j v k G satisfying: Av j v k ±z j = ±z k+ and Av j v k ±z j+ = ±z k. The Möbius transformation m Avi v j takes the edge of DS connecting x j /y j to x j+ /y j+ to the edge connecting x k+ /y k+ to x k /y k. If σ is an order two element in the permutation group of S the set: Γ Sσ = {Av i σv i : v i S} generates a Fuchsian group with DS as fundamental domain. 7

18 P P P P v v v w Figure : Gluing parallel sides of equal length on the polygons top gives four genus two differentials X i q i = P i dz / with #NX i q i =. The thick lines in P i give the shortest saddle connections on X i q i. The three neighbors of the differential up to scale [X q ] are [X i q i ] for i as is easily computed by enumerating a small number of the saddle periods on X q bottom generated by code written by C. McMullen. 8

19 +λi i 0 λ b Figure : Each integer D with D 0 or mod determines a Euclidean octagon P D C built out of a square of side length λ = e+ D and a b -rectangle where b = D e / and e = or 0 so that e D mod. Gluing parallel sides of equal length together by translations gives a genus two quadratic differential X D q D = P D dz / with lattice Veech group. Example D=. The Riemann surface L has genus zero three cusps and one orbifold point whose orbifold order is two. Using our algorithm we enumerated the components of L [n] for n. The set L [] consists of three differentials up to scale L [n] is empty for n and L [] has five components. We also computed a pair S σ encoding a fundamental domain and generators for PSLX q : { S = 0 and σ =. } 0 The domain DS is the convex hull of { + 0 } and PSLX q is generated by: { + Γ S σ = ± + 0 ± ± } 0. The domain DS is depicted in Figure. Example D=. The Riemann surface L has genus zero three cusps one orbifold point whose orbifold order is two. Using our algorithm we enumerated the components of L [n] for n. The set L [] see Figure consists of three differentials up to scale L [n] is empty for n and L [] has five components. We also computed a pair S σ encoding a fundamental domain and generators for PSLX q : { S = + 0 } and σ =. 0 9

20 { } The domain DS is the convex hull of 7+ 0 and PSLX q is generated by: 0 ± 6+ 0 ± 0 6+ Γ S σ = ± 6+ 9 The domain DS and the stratification L = n L [n] are drawn in Figure. Example D=. The Riemann surface L has Euler characteristic / genus one nine cusps and three orbifold points each of which has orbifold order two. The set L [n] is empty for n the set L [] consists of points and the set L[] has connected components. The pair S σ listed in Table encodes a fuandmental domain and generators for PSLX q. The domain DS is the ideal -gon equal to the convex hull of: The domain DS and the preimages of L [n] for n are drawn in Figure. A few examples of elements the Veech group of X q include: ± 0 ± ± PSLX q A full list of generators for PSLX q can be easily computed from the pair S σ listed in Table. 0

21 L i L i - / -7/ - 0 Figure : The Riemann surfaces L D for D = top and D = bottom both have genus zero three cusps and one orbifold point of order two. The stratification L D = n L D[n] decomposes L D into easy to understand pieces. Enumerating points in L D [n] for n and components of L D [] allows us to compute a fundamental domain and generators for PSLX D q D.

22 D Fundamental domain and generators for PSLX D q D 8 S = { σ = S = { } } σ = S 8 = { S = { } 0 σ 8 = σ = } Table : Fundamental domains and generators for the Veech group PSLX D q D are encoded by a pair S D σ D consisting of a finite set S D R and an involution σ D in the permutation group of S D as described in Section. The quotient H/ PSLX D q D has genus one for D = 8 and 7 genus two for D = and genus three for D = 6 and 60.

23 D Fundamental domain and generators for PSLX D q D S = { } 0 0 σ = S 6 = { } σ 6 = S 7 = { } σ 7 = S 60 = { } σ 60 =

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