Veech groups of flat structures on Riemann surfaces
aa r X i v : . [ m a t h . G T ] J u l Veech groups of flat structures on Riemann surfaces
Yoshihiko Shinomiya
Abstract.
In this paper, we construct new examples of Veech groups byextending Schmith¨usen’s method for calculating Veech groups of origamis toVeech groups of unramified finite coverings of regular 2 n -gons. We calculatethe Veech groups of certain Abelian coverings of regular 2 n -gons by using analgebraic method.
1. Introduction
The Teichm¨uller disk is a holomorphic isometric embedding of an upper-halfplane H (or a unit disk) into a Teichm¨uller space. All such embeddings are con-structed by flat structures on Riemann surfaces and SL(2 , R )-orbit on flat struc-tures. To study the image of a Teichm¨uller disk into the moduli space, we considerthe stabilizer of the Teichm¨uller disk in the mapping class group. Veech [ Vee89 ]showed that this stabilizer is regarded as the group of all affine diffeomorphismson a corresponding flat structure and its action can be represented by a Fuchsiangroup which acts on H . The Fuchsian group is called a Veech group.The first non-trivial examples of Veech groups were given by Veech [ Vee89 ] and[
Vee91 ]. His examples are constructed by gluing two congruent regular polygonsalong one side and identifying the parallel sides of the resulting polygons. How-ever, not so many examples are known other than Veech’s. Recently, Schmith¨usen[
Sch04 ] showed an algorithm for finding Veech groups of “origami”. An origamiis an unramified finite covering of a once punctured torus constructed by a unitsquare. We apply her method to unramified finite coverings of regular 2 n -gonsinstead of the unit square to obtain other examples of Veech groups. Veech groupsof universal coverings play an important role in her method. We call these groupsuniversal Veech groups.In this paper, we determine the universal Veech groups of 2 n -gons and givean algorithm to calculate Veech groups of finite Abelian coverings of 2 n -gons. Inthe case of origamis, Schmith¨usen connected the Veech groups of origamis withsubgroups of SL(2 , Z ). She showed that the calculations of Veech groups stop infinitely many steps. In our case, for the Veech groups of Abelian coverings of 2 n -gons whose degree is d , we connect them with subgroups of SL( n, Z d ). We showthat the calculations of Veech groups of certain Abelian coverings can be done byusing the corresponding subgroups of SL( n, Z d ).
2. Definitions
Let X be a Riemann surface of type ( g, n ) with 3 g − n > Definition 2.1 (Holomorphic quadratic differential) . A holomorphic quadratic dif-ferential ϕ on X is a tensor whose restriction to every coordinate neighborhood( U, z ) is the form f dz , here f is a holomorphic function on U .We define | ϕ | to be the differential 2-form on X whose restriction to every coordi-nate neighborhood ( U, z ) has the form | f | dxdy if ϕ equals f dz in U . We say ϕ isintegrable if its norm || ϕ || = Z Z X | ϕ | is finite.We fix an integrable holomorphic quadratic differential ϕ . Denote by X ′ theRiemann surface constructed from X by removing zeros of ϕ . Definition 2.2 (Flat structure) . A flat structure u on X ′ is an atlas of X ′ whichsatisfies the following conditions.(1) Local coordinates of u are compatible with the orientation on X ′ inducedby its Riemann surface structure.(2) For coordinate neighborhoods ( U, z ) and (
V, w ) of u with U ∩ V = φ , thetransition function is the form w = ± z + c in z ( U ∩ V ) for some c ∈ C .(3) u is maximal with respect to (1) and (2).The holomorphic quadratic differential ϕ determines a flat structure u ϕ on X ′ as follows.For each p ∈ X ′ , we can choose an open neighborhood U such that z ( p ) = Z pp √ ϕ is a well-defined and injective function of U . This function is holomorphic in U since ϕ is a holomorphic quadratic differential. If ( U, z ) and (
V, w ) are pairs of suchneighborhoods and functions with U ∩ V = φ , then we have dw = ϕ = dz in U ∩ V . Hence w = ± z + c in z ( U ∩ V ) for some c ∈ C . The flat structure u ϕ is themaximal flat structure which contains such pairs. Definition 2.3 (Affine group of ϕ ) . The affine group
Af f + ( X, ϕ ) of the integrableholomorphic quadratic differential ϕ is the group of all quasiconformal mappings f of X onto itself which satisfy f ( X ′ ) = X ′ and are affine with respect to theflat structure u ϕ . This means that for ( U, z ) , ( V, w ) ∈ u ϕ with f ( U ) ⊆ V , thehomeomorphism w ◦ f ◦ z − is the form z Az + c for some A ∈ GL(2 , R ) and c ∈ C .This A is uniquely determined up to the sign since u ϕ is a flat structure. And A is always in SL(2 , R ) since || ϕ || = R X | ϕ | = R X f ∗ ( | ϕ | ) = det( A ) || ϕ || . Thus wehave a group homomorphism D : Af f + ( X, ϕ ) → PSL(2 , R ). EECH GROUPS OF FLAT STRUCTURES ON RIEMANN SURFACES 3
Definition 2.4 (Veech group of ϕ ) . We call Γ(
X, ϕ ) = D ( Af f + ( X, ϕ )) the Veechgroup of ϕ . Remark.
Veech groups are discrete subgroups of PSL(2 , R ) (see [ EG97 ]).
3. Examples of Veech groups
In this section, we see two examples of Veech groups. The first example is anew example of Veech groups. The second one is the main target of this paper.The purpose of this paper is to determine Veech groups of some coverings of thesecond one. To do this, we need to determine the Veech group of the second one.
Example 3.1.
Let X be a surface constructed as Figure 1. We induce an uniqueconformal structure on X such that the quadratic differential dz on the interiorof the rectangle of Figure 1 extends to a holomorphic quadratic differential ϕ on X . Then X is a Riemann surface of type (2 ,
0) and vertices of four squares be-come two points on X . These points are zeros of ϕ of order 2. We can see that (cid:18) (cid:19) and (cid:18) (cid:19) define elements in Af f + ( X, ϕ ) as Figure 2. HenceΓ = (cid:28)(cid:20)(cid:18) (cid:19)(cid:21) , (cid:20)(cid:18) (cid:19)(cid:21)(cid:29) is a subgroup of the Veech group Γ( X, ϕ ). Sinceevery element in
Af f + ( X, ϕ ) must preserve the set of all lattice points, Γ(
X, ϕ ) isa subgroup of PSL(2 , Z ). It is known that (cid:28)(cid:20)(cid:18) (cid:19)(cid:21) , (cid:20)(cid:18) (cid:19)(cid:21)(cid:29) is the con-gruence subgroup of level 2 and has index 6 in PSL(2 , Z ). Hence Γ( X, ϕ ) is eitherΓ or PSL(2 , Z ). However, (cid:18) (cid:19) cannot be an element in Γ( X, ϕ ). ThereforeΓ(
X, ϕ ) must be Γ.
Figure 1.
The next example is given by Earle and Gardiner ([
EG97 ]).
Example 3.2.
Fix n ≥ n be a regular 2 n -gon. We assume that Π n has two horizontal sides, lengths of the sides are 1 and its vertices are removed. Weidentify each side of Π n with the opposite parallel side by an Euclidean translation(see Figure 3) and denote the resulting surface by P n . We induce an uniqueconformal structure on P n such that the quadratic differential dz on the interiorof Π n extends to a holomorphic quadratic differential ϕ n on P n . If n is even, then P n is a Riemann surface of type ( n ,
1) and if n is odd, then P n is a Riemann surfaceof type ( n − , R n = (cid:18) cos πn − sin πn sin πn cos πn (cid:19) and T n = (cid:18) π n (cid:19) YOSHIHIKO SHINOMIYA
Figure 2. induce elements in
Af f + ( P n , ϕ n ). The action of R n on P n is the rotationabout the center of Π n of angle πn . To see the action of T n on P n , we cut P n along all horizontal segments which connect the vertices of Π n . If n is even, P n is decomposed into n cylinders and the action of T n is the composition of thesquare of the right Dehn twist along a core curve of the cylinder which contains thecenter of Π n and the right Dehn twists along core curves of the other cylinders.If n is odd, P n is decomposed into n − cylinders and the action of T n is thecomposition of the right Dehn twists along core curves of all cylinders. Thus Γ = h [ R n ] , [ T n ] i is a subgroup of the Veech group Γ( P n , ϕ n ). It is easy to see that Γ isa ( n, ∞ , ∞ ) triangle group. Since only discrete group that contains Γ is a (2 , n, ∞ )triangle group (see [ EG97 ] and [
Sin72 ]) and this cannot be Γ( P n , ϕ n ), we haveΓ( P n , ϕ n ) = h [ R n ] , [ T n ] i . Figure 3.
4. Veech groups of coverings of P n and Universal Veech group of P n Fix n ≥
4. Let P n be the same Riemann surface as in Example 3.2 and p : X → P n be an unramified finite covering mapping. Set ϕ X = p ∗ ϕ n , here ϕ n is the holomorphic quadratic differential on P n defined in Example 3.2. Ourpurpose is to calculate the Veech group Γ( X, ϕ X ). We denote Γ( X, ϕ X ) by Γ( X ) EECH GROUPS OF FLAT STRUCTURES ON RIEMANN SURFACES 5 hereafter. Schmith¨usen constructed an algorithm for calculating Veech groups oforigamis ([
Sch04 ]). We apply her method to our case.Let p n : e X n → P n be the universal covering mapping and set e ϕ n = p ∗ n ϕ n .Note that || e ϕ n || = + ∞ . However, we can define the flat structure u e ϕ n on e X n and the affine group Af f + ( e X n , e ϕ n ) in the same manner as the case of inte-grable holomorphic quadratic differentials. Moreover, we have a homomorphism D : Af f + ( e X n , e ϕ n ) → PGL(2 , R ). Set Γ( e X n ) = Im( D ) ∩ PSL(2 , R ). Definition 4.1 (Universal Veech group of P n ) . We call Γ( e X n ) the universal Veechgroup of P n . Remark.
Let X be an unramified finite covering of P n . Then for each f ∈ Af f + ( X, ϕ X ), there exists a lift e f ∈ Af f + ( e X n , e ϕ n ) with D ( e f ) = D ( f ). HenceΓ( X ) is a subgroup of Γ( e X n ).The following idea is due to Schmith¨usen ([ Sch04 ]). For each finite covering X of P n , we take Γ( X ) as follows.Γ( X ) = n [ A ] ∈ Γ( e X n ) | ∃ e f ∈ Af f + ( e X n , e ϕ n ) s . t . D ( e f ) = [ A ] , e f is a lift of ahomeomorphism of X onto itself o = n [ A ] ∈ Γ( e X n ) | ∃ e f ∈ Af f + ( e X n , e ϕ n ) s . t . D ( e f ) = [ A ] , e f ∗ (Gal( e X n /X )) =Gal( e X n /X ) o .To understand Γ( X ), we determine Γ( e X n ). The following theorem is a maintheorem of this paper. Theorem 4.2.
For all n ≥ , Γ( e X n ) = h [ R n ] , [ T n ] i = Γ( P n ) . For the proof of theorem, we represent A ∈ SL(2 , R ) by A = (cid:18) r cos α ( A ) s cos β ( A ) r sin α ( A ) s sin β ( A ) (cid:19) for some r, s > α ( A ) , β ( A ) with 0 ≤ α ( A ) < β ( A ) < π . And set θ ( A ) = β ( A ) − α ( A ). This θ ( A ) means the angle of A (cid:18) (cid:19) and A (cid:18) (cid:19) .The following two lemmas give the proof of the theorem. Lemma 4.3.
For [ A ] ∈ Γ( e X n ) with | cot θ ( A ) | > cot π n , there exists k, l ∈ Z suchthat | cot θ ( A ) | > | cot θ ( T l n R k n A ) | . Lemma 4.4.
For [ A ] ∈ Γ( e X n ) with | cot θ ( A ) | > cot π n , there exists B ∈ h R n , T n i such that cot π n ≥ | cot θ ( BA ) | . Proof of theorem 4.2. Γ( P n ) ⊆ Γ( e X n ) is clear since Γ( e X n ) is the uni-versal Veech group of P n . We show Γ( e X n ) ⊆ Γ( P n ). By Lemma 4.4, for each[ A ] ∈ Γ( e X n ), there exists [ B ] ∈ h [ R n ] , [ T n ] i such thatcot π n ≥ | cot θ ( BA ) | .If we map Q of Figure 4 by an affine transformation BA , the image is parallelogramwhose vertices correspond to vertices of 2 n -gons and which has no such points inits interior. Moreover, it has the same area as Q and each angle θ of its verticessatisfies π/ n ≤ θ ≤ π − π/ n . We can see that such parallelograms are only YOSHIHIKO SHINOMIYA Q , Q , Q and Q of Figure 4 up to the image of them by [ R n ] and [ T n ]. Then BA is either (cid:18) (cid:19) , (cid:18) π n (cid:19) , (cid:18) π n − tan π n (cid:19) or (cid:18) π n − tan π n (cid:19) .However, it does not happen except for the case that AB is the identity I = (cid:18) (cid:19) since every vertex of 2 n -gons must be mapped to a vertex. Hence BA = I and so [ A ] = [ B − ] ∈ h [ R n ] , [ T n ] i = Γ( P n ). (cid:3) Figure 4.Proof of Lemma 4.3.
We consider two cases : (a) cot θ ( A ) > cot π n and (b) − cot θ ( A ) > cot π n Case (a) : There exists k ∈ Z such that B = R k n A satisfies either 0 ≤ α ( B ) < π n or π − π n ≤ α ( B ) < π . We define the function F β ( B ) α ( B ) ( x ) = 4 cot π n · sin β ( B ) sin α ( B )sin( β ( B ) − α ( B )) · x +2 cot π n · sin( β ( B ) + α ( B ))sin( β ( B ) − α ( B )) · x + cot θ ( B )of x ∈ R . Note that F β ( B ) α ( B ) ( l ) = cot (cid:0) β ( T l n B ) − α ( T l n B ) (cid:1) = cot θ ( T l n B ) for each l ∈ Z .(a)-1 : If 0 ≤ α ( B ) < π n , there exists e f ∈ Af f + ( e X n , e ϕ n ) with D ( e f ) =[ B ]. And e f maps the rectangle Q of Figure 4 to a parallelogram whose verticescorrespond to vertices of 2 n -gons and which has no such points in its interior. Hencewe have 0 ≤ α ( B ) < β ( B ) ≤ π n . From this, if α ( B ) = 0, then F β ( B )0 ( x ) = 2 cot π n · x + cot θ ( B ) EECH GROUPS OF FLAT STRUCTURES ON RIEMANN SURFACES 7 and F β ( B )0 ( −
12 ) = − cot π n + cot θ ( B ) > . So there exists a negative integer l such that | F β ( B )0 ( l ) | < | F β ( B )0 ( m ) | for all m ∈ { , − , · · · , l + 1 , l − } .Now we have | cot θ ( A ) | = | F β ( B )0 (0) | > | F β ( B )0 ( l ) | = | cot θ ( T l n R k n A ) | .If 0 < α ( B ) < β ( B ) ≤ π n , then F β ( B ) α ( B ) ( x ) is a quadratic function of x and the axisof F β ( B ) α ( B ) is x = − cot α ( B ) + cot β ( B )4 cot π n < − F β ( B ) α ( B ) ( −
12 ) > . Hence there exists a negative integer l such that | F β ( B ) α ( B ) ( l ) | < | F β ( B ) α ( B ) ( m ) | for all m ∈ { , − , · · · , l + 1 , l − } .And we have | cot θ ( A ) | = | F β ( B ) α ( B ) (0) | > | F β ( B ) α ( B ) ( l ) | = | cot θ ( T l n R k n A ) | .(a)-2 : If π − π n ≤ α ( B ) < π , we have π − π n ≤ α ( B ) < β ( B ) ≤ π and F β ( B ) α ( B ) ( x ) = F ( π − α ( B ))( π − β ( B )) ( − x ) . By using the argument of (a)-1, we have | cot θ ( A ) | > | cot θ ( T l n R k n A ) | for some l ∈ Z . Case (b) : We apply the same argument as in the Case (a) to the angle of twovectors A (cid:18) (cid:19) and A (cid:18) − (cid:19) . Then we have | cot θ ( A ) | > | cot θ ( T l n R k n A ) | forsome k, l ∈ Z . (cid:3) Proof of Lemma 4.4.
Let [ A ] be an element in Γ( e X n ) with | cot θ ( A ) | > cot π n . From the proof of Lemma 4.3, we obtain A = T l n R k n A with | cot θ ( A ) | < | cot θ ( A ) | for some k ∈ Z and l ∈ Z − { } . If | cot θ ( A ) | > cot π n , then we obtain A = T l n R k n A with | cot θ ( A ) | < | cot θ ( A ) | for some k , l ∈ Z − { } from theproof of Lemma 4.3 again. We repeat this operation. If there exists m ∈ N suchthat cot π n ≥ | cot θ ( A m ) | holds, then B = A m A − is what we want. Supposethat | cot θ ( A m ) | > cot π n holds for every m ∈ N . Then we have an infinite sequence { A m } in h R n , T n i · A with | cot θ ( A m − ) | > | cot θ ( A m ) | > cot π n for all m . Werepresent A m by A m = (cid:18) r m cos α m s m cos β m r m sin α m s m sin β m (cid:19) for some r m , s m > ≤ α m < β m < π with r m s m sin( β m − α m ) = 1. Foreach m , there exists e f m ∈ Af f + ( e X n , e ϕ n ) such that D ( e f m ) = [ A m ] and e f m mapsEuclidean segment which connect vertices of 2 n -gons to other segment. Thus wehave YOSHIHIKO SHINOMIYA r m = (cid:12)(cid:12)(cid:12) A m (cid:18) (cid:19) (cid:12)(cid:12)(cid:12) ≥ s m = (cid:12)(cid:12)(cid:12) A m (cid:18) (cid:19) (cid:12)(cid:12)(cid:12) ≥ r m s m = 1sin( β m − α m ) ≤ β − α ) . Hence { α m } , { β m } , { r m } and { s m } are bounded and there exists a subsequence { A m i } of { A m } such that A m i converges to some A ∞ ∈ SL(2 , R ). Since { A m i } isin a discrete set h R n , T n i · A , there exists i ∈ N such that A m i = A ∞ for all i ≥ i . However, this contradicts the construction of the sequence { A m } . Hencethere exists m ∈ N such that cot π n ≥ | cot θ ( A m ) | . (cid:3)
5. Calculation of Veech groups
Let X be an unramified finite covering of P n . By theorem 4.2, we can writeΓ( X ) as follows.Γ( X ) = n [ A ] ∈ h [ R n ] , [ T n ] i | ∃ e f ∈ Af f + ( e X n , e ϕ n ) s.t. D ( e f ) = [ A ] , e f ∗ (Gal( e X n /X )) = Gal( e X n /X ) o .Let z be the point of P n which corresponds to the center of the 2 n -gon Π n as in Example 3.2 and z be one of the preimages of z in X . Let { x , x , · · · , x n } be the system of generators of π ( P n , z ) as Figure 5. Then R n and T n definethe following automorphisms γ R n and γ T n on π ( P n , z ) (see Example 3.2). γ R n : (cid:26) x i x i +1 ( i = 1 , , · · · , n − x n x − . If n is even, γ T n : x x x − n +2 − i x i x − n +2 − i x i ( i = 2 , , · · · , n ) x i ( x − n +2 − i x i ) · · · ( x − n − x )( x − n x ) x x i ( i = 2 , , · · · , n ) x n +1 ( x − n +2 x n ) · · · ( x − n − x )( x − n x ) x x n +1 and if n is odd, γ T n : x − n +1 − i x i x − n +1 − i x i ( i = 1 , , · · · , n − ) .x i ( x − n +1 − i x i ) · · · ( x − n − x )( x − n x ) x i ( i = 1 , , · · · , n − ) x n +12 ( x − n +32 x n − ) · · · ( x − n − x )( x − n x ) x n +12 . Since Gal( e X n /P n ) < Ker( D ), Ker( D ) / Gal( e X n /P n ) = { [ id ] , [ e h n ] } for some e h ∈ Af f + ( e X n , e ϕ n ) with D ( e h ) = [ R n ] and each element in Gal( e X n /P n ) defines aninner automorphism of Gal( e X n /P n ) ∼ = π ( P n , z ), the action of each element of Af f + ( e X n , e ϕ n ) on π ( P n , z ) can be represented by a composition of γ R n , γ T n and inner automorphisms of π ( P n , z ).Hence we have the following. Proposition 5.1.
For e f ∈ Af f + ( e X n , e ϕ n ) , following two are equivalent. Here A is one of elements in D ( e f ) . • The mapping e f satisfies e f ∗ (Gal( e X n /X )) = Gal( e X n /X ) . • There exists one of the preimages z ∈ X of z such that γ A ( π ( X, z )) = π ( X, z ) or γ − A ( π ( X, z )) = π ( X, z ) . EECH GROUPS OF FLAT STRUCTURES ON RIEMANN SURFACES 9 >> Figure 5.
By using this condition, we can determine whether [ A ] is in Γ( X ) or not foreach [ A ] ∈ h [ R n ] , [ T n ] i .Now we can calculate the Veech group Γ( X ) of an unramified finite covering X of P n by using the following method. Schmith¨usen([ Sch04 ]) also use this methodto the calculations of Veech groups of origamis. The calculation is done on thefollowing tree which we explain below. ✬✫ ✩✪
Calculation of Γ( X )(Reidemeister-Schreier method).Given an unramified finite covering X of P n .Let Rep and
Gen be empty sets.Add [ I ] to Rep . Set A = I .Loop:Set B = A · T n , C = A · R n .Check whether B is already represented by Rep :For each [ D ] in Rep , check whether [ B ] · [ D ] − is in Γ( X ).If so, add [ B ] · [ D ] − to Gen .If none is found, add [ B ] to Rep .Do the same for C instead of B .If there exists a successor of A in Rep ,let A be this successor and go to the beginning of the loop.If not, finish the loop.Result: Gen : a list of generators of Γ( X ). Rep : a list of coset representatives in h [ R n ] , [ T n ] i . Proposition 5.2.
Let X be an unramified finite covering of P n . Then we havethe following properties. (1) Any two elements in
Rep belong to different cosets. (2)
The calculation stops in finitely many steps. (3)
In the end, each coset is represented by a member of
Rep . (4) In the end, Γ( X ) is generated by the elements in Gen . Proof. (1) is clear and we can see a proof of (3), (4) in [
Sch04 ]. (2) isequivalent to what Γ( X ) is a finite index subgroup of h [ R n ] , [ T n ] i . By the nextproposition , we conclude that Γ( X ) and Γ( P n ) = h [ R n ] , [ T n ] i are commensu-rable. Hence Γ( X ) is a finite index subgroup of h [ R n ] , [ T n ] i . Since all elementsin Rep belong to different cosets of Γ( X ) in h [ R n ] , [ T n ] i , ♯ Rep cannot be greaterthan this index and hence the calculation of Γ( X ) stops in finitely many steps. (cid:3) For a Riemann surface X and a holomorphic quadratic differential ϕ , denoteby C ( X, ϕ ) the set of all zeros of ϕ and punctures of X . Proposition 5.3. ([ GJ96 ] and [ GJ00 ] . ) Let p : X → Y be a covering mappingbetween Riemann surfeces. Let ϕ X be a holomorphic quadratic differential on X and set ϕ Y = p ∗ ϕ X . Suppose that p ( C ( Y, ϕ Y )) = C ( X, ϕ X ) and p − ( C ( X, ϕ X )) = C ( Y, ϕ Y ) . Then the Veech groups Γ( X, ϕ X ) and Γ( Y, ϕ Y ) are commensurable. Example 5.4.
Let X be the covering of P as Figure 6. We calculate the Veechgroup Γ( X ).The fundamental group of X is π ( X, z ) = (cid:10) x , x , x , x x , x x , x − x x , x − x x (cid:11) . Loop 1 :
Rep = { [ I ] } , Gen = φ , A = I , B = T , C = R .We check [ B ] · [ I ] − = [ T ] ; the homomorphism γ T maps the generators EECH GROUPS OF FLAT STRUCTURES ON RIEMANN SURFACES 11 * Figure 6. of π ( X, z ) as follows γ T : x x x x − x x x x x − x x x x x x x − x x x x x x − x x x x x − x x x − x − x x x x x − x x x − x − x x x x . By taking z as a base point, all images represent closed curves. Hence[ T ] is an element in Γ( X ) and add [ T ] in Gen .Now
Rep = { [ I ] } , Gen = { [ T ] } .We check [ C ] · [ I ] − = [ R ] ; there is no point of X such that γ R n ( x ) = x or γ − R n ( x ) = x − represent closed curves with the point as a base point.Hence [ R ] is not in Γ( X ). We add [ R ] in Rep .Now
Rep = { [ I ] , [ R ] } , Gen = { [ T ] } and R is a successor of A = I and is in Rep . We set A = R .Loop 2 : Rep = { [ I ] , [ R ] } , Gen = { [ T ] } , A = R , B = R T , C = R .We check [ B ] · [ I ] − = [ R T ] ; it is not in Γ( X ).We check [ B ] · [ R ] − = [ R T R − ] ; the homomorphism γ R T R − is theform γ R T R − : x x x − x − x − x x x x x x x x x x x x and maps the generators of π ( X, z ) as follows γ R T R − : x x x − x − x − x − x − x x x x x x x x x x x x x x x x x x x − x − x − x − x x x x x x − x x x − x − x − x − x x x x x x x x x x − x − x − . By taking z as a base point, all images represent closed curves. Hence[ R T R − ] is an element in Γ( X ) and add [ R T R − ] in Gen .Now
Rep = { [ I ] , [ R ] } , Gen = { [ T ] , [ R T R − ] } .We check [ C ] · [ I ] − = [ R ] ; the homomorphism γ R maps the generatorsof π ( X, z ) as follows γ R : x x x x x x − x x x x − x x x − x x − x x x − x x x − x x x − x − x . By taking z as a base point, all images represent closed curves. Hence[ R ] is an element in Γ( X ) and add [ R ] in Gen .We check [ C ] · [ R ] − = [ R ] ; it is not in Γ( X ).Now, Rep = { [ I ] , [ R ] } , Gen = { [ T ] , [ R T R − ] , [ R ] } and there is nosuccessor of A = R in Rep . We finish the loop.Result :
Rep = { [ I ] , [ R ] } , Gen = { [ T ] , [ R T R − ] , [ R ] } .As a result, Γ( X ) = (cid:10) [ T ] , [ R T R − ] , [ R ] (cid:11) and coset representatives in h [ R ] , [ T ] i is { [ I ] , [ R ] } . Remark.
In the case of origamis, Schmith¨usen showed that the calculations alwaysstop by connecting the Veech groups of origami with subgroups of SL(2 , Z )(see[ Sch04 ]). In our case, for certain Abelian coverings of 2 n -gons, we connect theVeech groups with subgroups of SL( n, Z d ) and calculate the Veech groups by usingthe corresponding matrices. It is seen in section 7.
6. Calculation of H / Γ( X )Let X be an unramified finite covering of P n . Assume that the calculationof Γ( X ) by the Reidemeister-Schreier method stopped. Then Gen is a list ofgenerators of Γ( X ) and Rep is a list of coset representatives in h [ R n ] , [ T n ] i .Let D be the fundamental domain of h [ R n ] , [ T n ] i in H as Figure 7. Then F = Int (cid:16) [ [ A ] ∈ Rep [ A ]( D ) (cid:17) is a fundamental domain of Γ( X ). Here [ A ] means a M¨obius transformation.By reading Gen , we can know all pairs of sides of F which are identified bythe action of Γ( X ). We call sides of [ A ]( D ) which correspond to ( − cot π n , i ),(cot π n , i ), ( − cot π n , i ∞ ) and (cot π n , i ∞ ) the R − -side, the R -side, the T − -sideand the T -side of [ A ], respectively. Proposition 6.1.
Assume that
Rep = { [ A ] , [ A ] , · · · , [ A k ] } . Then for each i, j ∈{ , , · · · k } , • The T -side of [ A j ] and the T − -side of [ A i ] are identified if and only if [ A j T n A − i ] ∈ Γ( X ) . EECH GROUPS OF FLAT STRUCTURES ON RIEMANN SURFACES 13 ( Figure 7. • The R -side of [ A j ] and the R − -side of [ A i ] are identified if and only if [ A j R n A − i ] ∈ Γ( X ) . We give a triangulation of H / Γ( X ) by decomposing D as Figure 8. Then thenumber of triangles and sides are 2 · ♯ Rep and 3 · ♯ Rep , respectively. Moreover, wecan calculate the number v of vertices by using Proposition 6.1. When we calculate v , we decompose v as v = v ∞ + v cot + v cone . Here v ∞ is the number of verticescorresponding to ∞ of D , v cot is the number of vertices corresponding to ± cot π n of D and v cone is the number of vertices corresponding to i of D . Then H / Γ( X )has genus (2 + ♯ Rep − v ) / v ∞ + v cot punctures. We can also calculate thenumber of cone points and their orders in the calculation of v cone . ( Figure 8.
Example 6.2.
Let X be the Riemann surface as Figure 6. At the end of thecalculation of Γ( X ), we have Gen = { [ T ] , [ R T R − ] , [ R ] } and Rep = { [ I ] , [ R ] } . Since [ T ] = [ I · T · I − ] is in Γ( X ), the T -side of [ I ] and the T − -side of [ I ] areidentified by Γ( X ). In the same way the T -side of [ R ] and the T − -side of [ R ]are identified and the R -side of [ R ] and the R − -side of [ I ] are identified. Hence H / Γ( X ) has no genus, three punctures and one cone point with order 2 (see Figure9). ( Figure 9.
7. Veech groups of Abelian coverings
In this section, we show that the calculation of Veech group Γ( X ) by theReidemeister-Schreier method always stops if X is a finite Abelian covering of P n .And we show that the calculations of Veech groups of certain Abelian coverings canbe done by using the corresponding subgroups of SL( n, Z d ).Recall that if Γ( X ) is a finite index subgroup of h [ R n ] , [ T n ] i , then the calcula-tion of Γ( X ) stops by the proof of Proposition 5.2. We have a partial answer aboutthe stop of calculations. Theorem 7.1.
Let X be a finite Abelian covering of P n , that is, X is a finiteGalois covering of P n and Gal(
X/P n ) is an Abelian group. Then the calculationof Γ( X ) stops. Proof.
Recall that z is the point of P n which corresponds to the centerof the 2 n -gon Π n as in Example 3.2 and z ∈ X is one of the preimages of z .Since X is a Galois covering, for each w ∈ Gal( e X n /P n ) = h x , x , · · · , x n i , w isin π ( X, z ) if and only if w is in π ( X, z ) for all z ∈ X . Hence [ A ] is in Γ( X ) ifand only if γ A or γ − A fix π ( X, z ) for each[ A ] ∈ h [ R n ] , [ T n ] i .As Gal( X/P n ) ∼ = π ( P n , z ) /π ( X, z ) is an Abelian group, x i x j = x j x i and x i x j = x j x i · w for some w ∈ π ( X, z ). Moreover, set d = lcm { ord( x ) , ord( x ) , · ·· , ord( x n ) } , then x di ∈ π ( X, z ) for all i and all z ∈ X .Set ( e , e , ··· , e n ) = I n . We consider the homomorphism ν : Gal( e X n /P n ) → Z nd ; x i e i . Then there exists a homomorphism Φ d : h γ T , γ R i → SL( n, Z d ) such EECH GROUPS OF FLAT STRUCTURES ON RIEMANN SURFACES 15 that the following diagram is commutative.Gal( e X n /P n ) ν (cid:15) (cid:15) γ A / / Gal( e X n /P n ) ν (cid:15) (cid:15) Z nd Φ d ( A ) / / Z nd Set V = ν ( π ( X, z )). For each [ A ] ∈ h [ R n ] , [ T n ] i , if [ A ] satisfies Φ d ( A )( V ) = V ,then [ A ] ∈ Γ( X ).Now we conclude that ♯ Rep ≤ ♯ SL( n, Z d ) at every step of the calculation.Suppose that ♯ Rep > ♯
SL( n, Z d ) happens at some step of the calculation. Thenthere exists two distinct elements [ A ] and [ B ] in Rep such that Φ d ( A ) = Φ d ( B ).Since Φ d ( AB − ) = I n stabilizes V , [ A ] · [ B ] − is in Γ( X ). However, since [ A ] and [ B ]are distinct elements in Rep , [ A ] · [ B ] − is not in Γ( X ). This is a contradiction. (cid:3) From the proof of theorem 7.1, we have the following.
Corollary 7.2.
Let X be a finite Abelian covering of P n . If there exists d ∈ N such that { ord( x ) , ord( x ) , · · · , ord( x n ) } = { d } or { , d } , then [ A ] ∈ Γ( X ) if andonly if Φ d ( A )( V ) = V for each [ A ] ∈ h [ R n ] , [ T n ] i . Example 7.3.
Let X be the covering of P the same as Figure 6. Then X satisfiesthe assumption of Corollary 7.2. The fundamental group of X is π ( X, z ) = (cid:10) x , x , x , x x , x x , x − x x , x − x x (cid:11) and V = h e , e , e + e i Z . By Corollary 7.2, for [ A ] ∈ h [ R ] , [ T ] i , [ A ] is in Γ( X ) if and only if Φ ( A ) satisfiesthe followings : (cid:26) Φ ( A ) , + Φ ( A ) , + Φ ( A ) , + Φ ( A ) , ≡ ( A ) ,j + Φ ( A ) ,j ≡ j = 2 , .
8. Examples.
Finally we show some examples of Veech groups that are calculated by themethod of this paper.
Example 8.1.
Let X be the double covering of P as Figure 10. Then X is aRiemann surface of type (3 , R = [ R ], T = [ T ]. Then • For [ A ] ∈ h R, T i , [ A ] is in Γ( X ) if and only if Φ ( A ) ,j ≡ j =2 , , • Γ( X )= D T, RT R − , RT RT ( RT R ) − , ( RT ) ( RT RT R ) − , ( RT ) R T ( RT R ) − , ( RT ) R T ( RT RT R ) − ,RT R T ( RT RT R ) − , RT R T ( RT R ) − ,RT R T R, R T R − , R ( RT R T ) − E , • Γ( X ) \ h R, T i = (cid:26) I, R, RT, R , RT R, RT RT, RT R , RT RT R,RT RT R , RT RT R , RT R , RT R T (cid:27) and • H / Γ( X ) is a Riemann surface of type (0 , Figure 10.
Example 8.2.
Let X be the covering of P as Figure 11. Then X is a Riemannsurface of type (5 , R = [ R ], T = [ T ]. Then • For [ A ] ∈ h R, T i , [ A ] is in Γ( X ) if and only if Φ ( A ) satisfies the followings: X i =1 (Φ ( A ) i, − Φ ( A ) i, ) ≡ X i =3 (Φ ( A ) i, − Φ ( A ) i, ) (mod 4) , X i =1 (Φ ( A ) i, + Φ ( A ) i,j ) ≡ X i =3 (Φ ( A ) i, + Φ ( A ) i,j ) (mod 4)( j = 3 , , • Γ( X ) = (cid:10) T, R ( RT ) − , RT R − , RT RT ( RT R ) − , RT R (cid:11) , • Γ( X ) \ h R, T i = { I, R, RT, RT R } and • H / Γ( X ) is a Riemann surface of type (0 , Figure 11.
Example 8.3. n ≥
2. Let X n be the double covering of P n as Figure 12. Thatis, X n is constructed by gluing two regular 4 n -gons. Labels of small and capitalletters appear in turn. The sides whose labels are capital letters are identified withthe opposite sides of another polygon and others are identified with the oppositesides of the same polygon. Then X n is a Riemann surface of type (2 n − , EECH GROUPS OF FLAT STRUCTURES ON RIEMANN SURFACES 17 • For [ A ] ∈ h [ R n ] , [ T n ] i , [ A ] is in Γ( X n ) if and only if Φ ( A ) satisfies thefollowings : n X i =1 Φ ( A ) i − , j ≡ j = 1 , · · · , n ) , n X i =1 (Φ ( A ) i − , + Φ ( A ) i − , j − ) ≡ j = 2 , · · · , n ) , • Γ( X n ) = (cid:10) [ T n ] , [ R n T n R − n ] , [ R n ] (cid:11) , • Γ( X n ) \ h [ R n ] , [ T n ] i = { [ I ] , [ R n ] } and • H / Γ( X n ) is an orbifold which has no genus, 3 punctures and one conepoint whose order is n . Figure 12.
Example 8.4.
For each n ≥
2, let X n be the double covering of P n as Figure13. That is, horizontal and vertical sides of two polygons are identified with theopposite sides of another polygon and others are identified with the opposite sidesof the same polygon. Then X n is a Riemann surface of type (2 n − , • For [ A ] ∈ h [ R n ] , [ T n ] i , [ A ] is in Γ( X n ) if and only if Φ ( A ) satisfies thefollowings : (cid:26) Φ ( A ) , + Φ ( A ) n +1 , + Φ ( A ) n +1 , + Φ ( A ) n +1 ,n +1 ≡ , Φ ( A ) ,j + Φ ( A ) n +1 ,j ≡ j = 2 , · · · , n, n + 2 , · · · , n ) , • Γ( X n ) = (cid:10) [ R i n T n R − i n ] , [ R n n ] | i = 0 , , · · · , n − (cid:11) , • Γ( X n ) \ h [ R n ] , [ T n ] i = { [ I ] , [ R n ] , [ R n ] , · · · , [ R n − n ] } and • H / Γ( X n ) is an orbifold which has no genus, 2 n + 1 punctures and onecone point whose order is 2. >>> >>>>>> >>> Figure 13.
Example 8.5.
Let X d be the covering of P with degree d as Figure 14. Then X d is a Riemann surface of type ( d + 1 , d ). And , for [ A ] ∈ h [ R ] , [ T ] i , [ A ] is in Γ( X d )if and only if Φ d ( A ) ,j ≡ d )( j = 2 , , X d ). Here, • ♯ Rep is the index of Γ( X d ) in h [ R ] , [ T ] i , • ♯ Gen is a number of generators of Γ( X d ) by this calculation, • “genus” is the genus of H / Γ( X d ), • “puncture” is the number of punctures of H / Γ( X d ) and • “cone point (order)” is the number of cone points of H / Γ( X d ) and theirorders. Figure 14.
EECH GROUPS OF FLAT STRUCTURES ON RIEMANN SURFACES 19 d ♯ Gen ♯ Rep genus puncture cone point (order)2 11 12 0 11 03 29 32 1 24 04 87 96 8 58 05 142 156 24 68 6 (2,2,2,2,2,2)6 349 384 45 200 07 367 400 87 128 08 704 768 149 280 09 785 864 185 280 010 1704 1872 419 568 011 1353 1464 400 300 0
Example 8.6.
Let X d be the covering of P with degree d as Figure 15. Then X d is a Riemann surface of type ( d + 1 , d ). And, for [ A ] ∈ h [ R ] , [ T ] i , [ A ] is in Γ( X d ) ifand only if X i =1 (Φ d ( A ) i,j − Φ d ( A ) i, ) ≡ d )( j = 2 , , X d ). Figure 15. d ♯ Gen ♯ Rep genus puncture cone point (order)2 2 1 0 2 1 (4)3 29 32 1 24 04 5 4 0 5 05 142 156 24 68 6 (2,2,2,2,2,2)6 29 32 1 24 07 367 400 87 128 08 29 32 1 24 09 789 864 185 280 010 142 156 24 68 6 (2,2,2,2,2,2)11 1353 1464 400 300 012 115 128 11 76 013 2220 2380 682 416 14 (2,2,2,2,2,2,2,2,2,2,2,2,2,2)14 367 400 87 128 0
Acknowledgments
This work was supported by Global COE Program “Computationism as a Foun-dation for the Sciences”. The author thanks Professor Hiroshige Shiga for his valu-able suggestions and comments.
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Department of Mathematics Tokyo Institute of Technology 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, JAPAN
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