Arithmetic Fuchsian Groups Derived from Tesselations {4g,4g} and ...

Arithmetic Fuchsian Groups Derived from Tesselations {4g, 4g} and Identified in Quaternion Orders Vandenberg Lopes Vieira,

Reginaldo Palazzo J´ unior

Depto Matem´atica, Estat´ıstica e Computa¸c˜ao, DMEC, UEPB 58104-000, Campina Grande, PB E-mail: [email protected],

Departamento de Telem´atica, FEEC, UNICAMP Cidade Universit´aria Zeferino Vaz CP 6101 - 13083-970, Campinas, SP E-mail: [email protected].

Mercio Botelho Faria Universidade Federal de Vi¸cosa, UFV Departamento de Matem´atica - Campus UFV 36570-000, Vi¸cosa, MG E-mail: [email protected].

1

Introduction

quaternion orders O of a quaternion algebra over extensions K of Q of the form [K : Q] = 2α , α a positive integer number. Therefore, the aim of this paper is to propose a systematic procedure for constructing arithmetic Fuchsian groups Γ derived from a quaternion algebra A over an algebraic number field K, such that [K : Q] = 2n , for g = 2n ; [K : Q] = 2n+1 , for g = 3 · 2n ; and [K : Q] = 2n+2 , for g = 5 · 2n , where 2n , 2n+1 and 2n+2 denote, respectively, the degree of the extension of the rational field Q. Consequently, generalizing the previous results considered in [6], [5]. These arithmetic Fuchsian groups are identified as the group of units of O, denoted by O1 , [5]. The generators of Γ (hyperbolic transformations) realize the edge-pairings of a regular hyperbolic polygon P4g (fundamental region of Γ4g ), such that the hyperbolic area of H/Γ is µ (H/Γ) = 4π (g − 1), for each g in consideration, where g is the genus of a orientable surface compact H/Γ, Γ ' Γ4g , [6]-[3]. Therefore, for each g, the group Γ4g is geometrically finite, [3].

Fuchsian groups are discrete subgroups of the projective special linear group consisting of 2 × 2 matrices whose elements are in R, that is, P SL(2, R), or equivalently, the set of M¨obius transformations, [6]. The elements of P SL(2, R) are isometries which act by homeomorphisms on the upper-half plane H = {z ∈ C : Im (z) > 0}. If Γ is a finite co-area Fuchsian group acting on H, then it is possible to endow the quotient space H/Γ (the space of the Γ-orbits) with a structure of a Riemann surface, [1]. Katok, [6], and Johansson, [5], proposed an arithmetic procedure to obtain Fuchsian groups from Takeuchi’s results, [7]. These groups are known as arithmetic Fuchsian groups. Johansson [5] showed that the elements of an arithmetic Fuchsian group are identified with the elements of an order O of a quaternion algebra A over a quadratic extension K of Q, that is, [K : Q] = 2, whose generators of Γ are of the form √¢ ¶ √ ¡ µ 1 This paper is organized as follows. In Sect ¢ r1 c + √ d t a¡ + b √ G= , tion 2 we present some basic considerations a−b t 2 r2 c − d t on the model of the hyperbolic plane and on with a, b, c, d ∈ Z [θ], where Z [θ] is the ring of Fuchsian groups for the development of this pa√ integers of Q√ ( m), m > 0, r1 = −r2 ∈ Z, per. In Section 2.1 we show the model of the t ∈ Z [θ] and t 6∈ Z [θ]. fundamental region (geometry of the surface) The problem we are faced with is the iden- considered in the construction of the Fuchsian tification of arithmetic Fuchsian groups Γ in groups Γ4g . In Section 3 we revise the concept

a regular hyperbolic polygon P4g , to be considered in Section 5, leads to an oriented compact surface H/Γ, with genus g, where Γ ' Γ4g (Γ4g and Γ are isomorphic) is the Fuchsian group associated with a self-dual hyperbolic tessellation {4g, 4g}, see Section 2.1. Therefore, for each g, the Fuchsian group is co-compact, [1]. Hence, the hyperbolic area of P4g , µ (P4g ) = µ (H/Γ), is finite. Consequently, the group Γ4g contains no parabolic elements, that is, M¨obius transformations T ∈ P SL(2, R) such that their traces are equal to 2, tr(T ) = |a + d| = 2. We also assume that the group Γ4g contains no elliptic elements, T ∈ P SL(2, R) such that tr(T ) = |a + d| < 2, this assumption is sufficient for the projection π : H → H/Γ to be a covering application. Hence, Γ4g con2 Preliminaries tains only hyperbolic elements, T ∈ P SL(2, R) We consider the upper-half plane H2 = {z ∈ such that tr(T ) = |a + d| > 2. Consequently, C : Im (z) > 0} equipped with a Riemannian µ (H/Γ) = 4π(g − 1), where g is the genus of the surface H/Γ, [6] and [1]. metric p 2 2 dx + dy ds = . y 2.1 Fundamental region of quaternion algebra and present some basic results regarding hyperbolic lattices both relevant to the development of the paper. In Section 4 we present the Fuchsian groups Γ4g . In Section 5 we consider the identification of the arithmetic Fuchsian groups with the quaternion orders for the case g = m · 2n , for any n ∈ N, where N is the set of natural number and m = 1, 3 and 5. The reason we consider m = 1, 3, and 5 is that for these values of m the trace of each element in Γ is an algebraic integer number, see Theorem 3.2, whereas for some other odd values of m the trace is an algebraic number. Finally, in Section 6 the conclusions are drawn.

With this metric, H is a model of the hyperbolic plane. Let P SL(2, R) be the set of all M¨obius transformations, T : C → C, given by T (z) = az+b cz+d , where a, b, c, d ∈ R and ad − bc = 1. To this transformation the following pair of matrices are associated µ ¶ a b AT = ± . c d Hence, P SL(2, R) ' SL(2, R)/{±I2 }, where SL(2, R) is the group of real matrices with determinant equal to 1 and I2 denotes the 2 × 2 identity matrix. A Fuchsian group Γ is a discrete subgroup of P SL(2, R), that is , Γ consists of orientation preserving isometries T : H → H, acting on H by homeomorphisms, [6] and [1]. Associated with a Fuchsian group Γ there is a fundamental region P 1 , resulting from the action of Γ on H. This fundamental region has a polygonal shape containing 4g edges. Therefore, we may endow the quotient space H/Γ with a metric of a Riemann surface. As it is well known, every compact Riemann surface with genus g ≥ 2 may be modeled in the hyperbolic plane, [3]. The pairings of the 4g edges of

The Fuchsian groups Γ4g that we will consider are obtained of regular hyperbolic polygons with 4g edges, fundamentals regions. Furthermore, 4g other polygons meet at each vertex of P4g . Hence, the corresponding self-dual tessellations of the hyperbolic plane is denoted by {4g, 4g}.The regular polygon P4g tessellates the hyperbolic plane H. Hence, for every g, P4g is a fundamental region characterizing the surface H/Γ, where Γ4g ' Γ. This is the reason we consider in this section the concept of fundamental region of a Fuchsian group. To what follows, X is a metric space and G is the homeomorphism group acting on X . Thus, we may now consider the concept of a fundamental region. Definition 2.1 Let X be a metric space, and Γ a group of homeomorphisms acting properly discontinuously2 on X . A closed subset P ⊂ X and P 6= ∅, is called a fundamental region of Γ if: 1. ∪ T (P ) = X T ∈Γ ◦

1

Including the edges of the polygon P and its interior.

◦

2. P ∩ T (P ) = ∅ for T ∈ Γ. 2

See [3].

◦

Example 3.1 Consider H = (−1, −1)R , Hamiltonian algebra, and H1 = {x ∈ H : N rd(x) = 1}. Then, given x = x0 + x1 i + x2 j + x3 k ∈ H1 , we obtain N rd(x) = x20 + x21 + x22 + x23 = 1. From this, it follows that T rd(x) = 2x0 ∈ [−2, 2], since |x0 | ≤ 1. Hence, 1 az+b Definition 2.2 Let T (z) = cz+d ∈ P SL(2, R) T rd(H ) = [−2, 2]. be an isometry of H2 , with c 6= 0. The set Consider ϕ as an embedding of the algebra A = (a, b)K in the algebra of matrices CT = {z ∈ C : |cz + d| = 1}, √ M (2, K( a)) where is called an isometric circle of T . ¶ µ ¶ µ √ a 0 1 0 √ , ϕ(1) = , ϕ(i) = 0 − a 0 1 Let Γ be a Fuchsian group whose elements are orientation preserving isometries in the unit µ ¶ µ √ ¶ 0 1 a 0 disc D2 = {z ∈ C : |z| < 1}, that is, √ ϕ(j) = , ϕ(k) = . b 0 −b a 0 az + b , |a|2 − |b|2 = 1, T ∈ Γ. T (z) = ¯ Since ϕ(i2 ) = aI2 , ϕ(j 2 ) = bI2 , and ϕ(i)ϕ(j) = bz + a ¯ −ϕ(j)ϕ(i), it follows that ϕ is an isomorThe unit disc D2 equipped wit a Riemannian phism of A = (a, b)K in the subalgebra of √ p M (2, K( a)). Each element of A = (a, b)K 2 dx2 + dy 2 , z = x + yi, ds = is identified with 1 − (x2 + y 2 ) µ √ √ ¶ x0 + x1 a x2 + x3 a √ √ 3 ϕ : x 7→ . (1) is also a model of the hyperbolic b (x2 − x3 a) x0 − x1 a ¯ ¯plane . Conˇ ) = {z ∈ C : ¯bz + a¯ > 1}, then sider I(T The set P denotes the interior of P . The family {T (P ) : T ∈ Γ} is called a tessellation of X . The fundamental region considered in this paper is the region R0 known as the Ford region and it is established as follows.

ˇ ) ∩ D2 establishes a Ford region, 3.1 R0 = ∩ I(T T ∈Γ that is, a fundamental region for Γ, where X denotes the closure of the set X, [6]. Thus, R0 is a fundamental region consisting of the points in D2 outside of the isometric circles of all the isometries of Γ. We are going consider only regular region of the form R0 .

3

Quaternion algebra

Hyperbolic lattices

Let A = (a, b)K be a quaternion algebra over K and R be a ring of K. An R-order O in A is a subring of A containing 1, equivalently, it is a finitely generated R-module such that A = KO. We also call an R-order O a hyperbolic lattice due to its identification with an arithmetic Fuchsian group.

Example 3.2 Let A = (a, b)K and OK the ring of integers of K, where a, b ∈ O∗K = Let A = (a, b)K be a quaternion algebra over a field K with basis {1, i, j, k} satisfying i2 = a, OK − {0}. Then j 2 = b, and k = ij = −ji, where a, b ∈ K. O = {x0 +x1 i+x2 j+x3 k : x0 , x1 , x2 , x3 ∈ OK }, If x ∈ A, let us say x = x0 + x1 i + x2 j + x3 k, with x0 , x1 , x2 , x3 , ∈ K, then x = x0 − is an order in A denoted by O = (a, b) . OK x1 i − x2 j − x3 k is called conjugate of x. The reduced norm of x and the reduced trace of x, For each order O in A, consider O1 as the denoted, respectively, by N rd(x) and T rd(x), set are defined as N rd(x) = xx = x20 − ax21 − bx22 + O1 = {x ∈ O : N rd(x) = 1}. abx23 and T rd(x) = x + x = 2x0 . Obviously, Note that O1 is a multiplicative group. We observe that the Fuchsian groups may N rd(x · y) = N rd(x) · N rd(y) be obtained by the isomorphism ϕ as in (1). In 3 In this paper, Γ and Γp will represent discreet fact, if x ∈ O1 , then N rd (x)¡ = det 2 2 ¢ (ϕ (x)) = 1. groups of isometries acting on H and D , respectively 1 is a subgroup 2 2 From this, it follows that ϕ O and Γ ' Γp . Moreover, H /Γ and D /Γp represent the of SL(2, R). Therefore, the derived group from same surface (g-torus).

the quaternion algebra A = (a, b)K whose order Without loss of any generality, let us assume is O, denoted by Γ(A, O), is given by P4g is centered at the origin of D2 . Let us also consider the edges of P4g is ordered in a cyclic ¡ ¢ ϕ O1 SL(2, R) Γ(A, O) = < ' P SL(2, R). form as follows: {±I2 } {±I2 } u1 , v1 , u01 , v10 , . . . , ug , vg , u0g , vg0 , (2) Theorem 3.1 [6] Γ(A, O) is a Fuchsian and the hyperbolic isometries T1 , S1 , . . . , Tg , group. Sg , the generators of the Fuchsian group Γ4g , If Γ is a subgroup of Γ(A, O) with finite in- [3], are such that, dex, then Γ is a Fuchsian group derived from a Tk (uk ) = u0k and Sj (vj ) = vj0 , (3) quaternion algebra A. By definition, the Fuchsian group derived from an algebra A is arith- k, j = 1, . . . , g. From these pairings an orimetic, [6]. Consequently, the arithmetic Fuch- ented compact surface D2 /Γ4g with genus g sian groups may be characterized by the set of results. Therefore, there is associated with a traces of its elements as shown in the next the- self-dual tessellation {4g, 4g} a regular hyperbolic polygon which in turn is associated with a orem. Fuchsian group Γ4g . From this, it follows that group Theorem 3.2 [6] Let Γ be a¡ Fuchsian the hyperbolic area is given by, [6], ¢ with finite hyperbolic area µ H2 /Γ . Then Γ ¡ ¢ µ (P4g ) = µ D2 /Γ4g = 4π (g − 1) . is derived from a quaternion algebra A over a totally real algebraic number field K if and only Let us consider now an elliptic transformaif Γ satisfies the following conditions: tion TC with order 4g whose associated matrix is ! Ã iπ 1. Let K1 = Q (tr (T ) : T ∈ Γ). Then K1 is 0 e 4g , (4) C= an algebraic number field of finite degree, −iπ 4g 0 e 4 and T r (Γ) belongs to the integer ring of K1 , OK1 . such that TC (u1 ) = v1 and rk is the power of TC such that 2. Let ϕ be an embedding of K1 in C such TC rk (u1 ) ∈ {uk , vk , u0k , vk0 , k = 1, . . . , g}. (5) that ϕ 6= Id, where Id denotes the identity. Then ϕ (tr (Γ)) is bounded in C. Hence, we may write the generators of Γ4g as conjugations of T1 by means of powers of C. 4 Fuchsian Groups Γ4g For instance, suppose we want to determine T2 such that T2 (u2 ) = u02 . We know that T1 (u1 ) = In this section the Fuchsian group Γ4g is pre- u0 and TC 4 (u1 ) = u2 since u2 is 4 edges apart 1 sented. In what follows, the model of the hy- from u1 , see (2). Hence, TC −4 (u2 ) = u1 . Thus, perbolic plane to be used is the Poincar´e disc TC 4 (T1 (TC −4 (u2 ))) = TC 4 (u01 ) = u02 , D2 , that is, D2 = {z ∈ C :| z |< 1} equipped with a Riemannian metric that is, T 4 ◦ T ◦ T −4 (u ) = u0 . Hence, it C

ds2 =

4(dx2

dy 2 )

+ , [1 − (x2 + y 2 )]2

where z = x + Im · y and Im is an imaginary number. Let us consider a self-dual tessellation {4g, 4g}, g ≥ 2, and let P4g be the regular hyperbolic polygon with 4g edges associated with this tessellation. The polygon P4g tessellates the hyperbolic plane D2 , such that at each vertex there are 4g other polygons P4g . Hence, for each g, P4g is a fundamental region of Γ4g . 4

Or a ring R in K such that OK1 ⊂ R ⊂ K1 .

1

C

2

2

suffices to consider T2 = TC 4 ◦ T1 ◦ TC −4 . So, by using (3) and (5) we obtain Ak = C 4λ A1 C −4λ

and Bj = C β A1 C −β , (6) wher Ak and Bj are matrices corresponding to the transformations Tk and Sj , respectively, with k, j = 1, . . . , g, λ = k − 1 and β = 4j − 3. Therefore, once we have the generator T1 the remaining ones may be obtained by conjugations. But, T1 (z) = ¯az+b , where a and b are bz+¯ a given by, [2], arg(a) =

(g−1)π 2g ,

|a| = tan (2g−1)π 4g

However, for n = 1, we have 2 + 2 cos π4 = √ r³ 2 + 2. Assume, by induction, that the result ´ |b| = − 1, arg(b) = −(2g+1)π tan2 (2g−1)π . holds for n. Thus, 4g 4g q √ π π = = 2 cos 2 + 2 cos 2+θ n+2 n+1 2 2 The remaining hyperbolic transformations 0 0 Tk (uk ) = uk and Sj (vj ) = vj generating the contains n radicals. Fuchsian group Γ4g and also realizing the reConsider the edges of P4g are ordered as in maining edge-pairings are obtained by the fol- (2), such that T (u ) = u0 and S (v ) = v 0 as j k j k lowing conjugations in (3). Thus, We may express and

Tk = TC rk ◦ T1 ◦ TC −rk ,

 2(m−1) A1 C −2(m−1) , m odd  C

Sj = TC rj ◦ T1 ◦ TC −rj . Am =

5

Identification of the Groups Γ4g in Orders for g = m · 2n , with m = 1, 3, and 5



, C (2m−3) A1 C −(2m−3) , m even

(7) where the Am ’s are the transformation matrices associated with the generators of Γ4g , with m = 1, . . . , 2g and In this section we consider the process of idenà iπ ! e 4g 0 tifying arithmetic Fuchsian groups Γ4g derived C= , −iπ from a quaternion algebra over a totally real al0 e 4g gebraic number field K, for the following cases: 1) [K : Q] = 2n , with g = 2n ; 2) [K : Q] = is the matrix corresponding to the elliptic 2n+1 , with g = 3 · 2n ; and [K : Q] = 2n+2 , transformation with order 4g. Now, we consider the isometry from the with g = 5 · 2n , where g denotes the genus of a 2 upper-half plane to the unit disc, that is, surface D /Γ4g . We begin with the following: f : H → D2 , given by f (z) = zi+1 z+i . Then, n Proposition 5.1 If g = m · 2 , where n is any Γ = f −1 Γ4g f is a subgroup of P SL(2, R), that is, Γ is isomorphic to Γ4g (Γ ' Γ4g ). natural number, and m = 1, 3, and 5, then Let Gl = f −1 Al f be the generators of Γ, for (2g−1)π (g−1)π 2 tan 4g cos 2g = 2 + θ, l = 1, . . . , m · 2n+1 . Using Proposition 5.1 and the equalities in (7), we obtain where √ √ ¶ µ 1 s xl + yl √θ zl + wl√ θ r , (8) Gl = q 2 −zl + wl θ xl − yl θ √ θ = 2 + 2 + ··· + 2 + 2 where xl , yl , zl , wl ∈ Z [θ] and θ is as in Proposition 5.1. contains n radicals for m = 1, Now, our objective is to show, for each g = s r q m · 2n , with m = 1, 3, 5 and n any natural √ number, that the group Γ4g is derived from a θ = 2 + 2 + ··· + 2 + 3 quaternion algebra A over a field K, when the degrees of the extensions are [K : Q] = 2n , contains n + 1 radicals for m = 3, and for m = 1, [K : Q] = 2n+1 , for m = 3, and v s u r [K : Q] = 2n+2 , for m = 5. With this objective, u √ √ t 5 we consider the proposition, whose demonstraθ = 2 + 2 + · · · + 2 + 10+2 2 tion we omit. contains n + 2 radicals for m = 5.

Proposition 5.2 [8] Let Γ be a finitely generated arithmetic Fuchsian group. Suppose that Proof: Let us consider only the case g = 2n , the generators G1 , . . . , Gl of Γ are written in since the cases g = 3 · 2n and g = 5 · 2n are the form treated similarly. If g = 2n , then √ √ ¶ µ xk + yk √θ zk + wk√ θ Gk = , π 2 tan (2g−1)π cos (g−1)π = 2 + 2 cos 2n+1 . −zk + wk θ xk − yk θ 4g 2g

√ k = 1, . . . , l, where Gk ∈ M (2, K( θ)) and θ, xk , yk , zk , wk ∈ K, being K a field. Then any element T ∈ Γ assumes the same form of the generators of Γ.

formula5 ϕ(x0 +x1 i+x2 j +x3 k) = x0 ·M0 +x1 · M1 + x2 · M2 + x3 · M3 , this is, ϕ(x) = T, where i2 = θ, j 2 = −1, k = ij, xl , yl , zl , wl ∈ Z[θ] and K = Q(θ). Therefore, each element of the Fuchsian group Γ4g is identified, through of the The hyperbolic lattices associated to the isomorphism ϕ, with an √ element √ of the order n groups Γ4g , g = m · 2 , are described in the O = (θ, −1)R and {1, θ, Im, θ Im} is an RTheorem 5.1. basis of O.

Theorem 5.2 For each g = m · 2n , with m = Proposition 5.3 [8] Let A = (a, b)K be a 1, 3, 5 and6 n any natural number, the Fuchsian quaternion algebra with a basis {1, i, j, k} , r ∈ group Γ ' Γ4g , associated with a regular hyN∗ , with r fixed, and R be the set perbolic polygon P4g , is derived from a quatero nα nion algebra A = (θ, −1)K over a number field : α ∈ OK and m ∈ N , R= K = Q(θ), when the degrees of the extensions rm are [K : Q] = 2n , for m = 1, [K : Q] = 2n+1 , where OK is the ring of integers of the field K. for m = 3, and [K : Q] = 2n+2 , for m = 5, Then where θ is as in Proposition 5.1. O = {x0 + x1 i + x2 j + x3 k : x0 , x1 , x2 , x3 ∈ R} Proof: We shall present only the proof of the case g = 2n , once the cases g = 3 · 2n and is an order in A. g = 5 · 2n are treated similarly. Consider g = 2n . By Proposition and (8), Theorem 5.1 For each g = m · 2n , with m = any element T ∈ Γ is written in the form 1, 3, 5 and n any natural number, the elements √ √ ¶ µ 1 xl + yl √θ zl + wl√ θ of a Fuchsian group Γ ' Γ4g are identified, via , T = s −zl + wl θ xl − yl θ 2 isomorphisms, with the elements of the group of units O1 in order O = (θ, −1)R , where where s ∈ N e x , y , z , w ∈ Z [θ]. Thus, by l

l

l

l

l Theorem 5.1, there is x = 2xsl + 2ysl i+ 2zsl j + w 2s k ∈ and (9) O = (θ, −1)R such that ϕ(x) = T . Therefore, l tr (T ) = T rd(x) = 2x 2s ∈ R. Hence, we have and√θ is as√in Proposition 5.1. Consequently, tr(Γ) ⊂ R. On the other hand, {1, θ, Im, θ Im} is an R-basis for the lattice K = Q (tr (T1 ) : T ∈ Γ) = Q (θ) , O, where Im denotes an imaginary unit. is a consequence of the fact that tr(T ) ∈ Proof: Consider the following matrices M , Q(tr(T1 )), ∀ T ∈ Γ and tr(T1 ) = 2 + θ. From 0 √ M1 , M2 and M3 in M (2, Q( θ)), given by this, it follows that the first condition in Theorem 3.2 is satisfied. Since we may choose any µ ¶ µ √ ¶ 1 0 θ 0 one of the 2n monomorphisms ϕ : K → R, √ M0 = , M1 = , 0 1 0 − θ let ϕ2 : K → R be a homomorphism given by ϕ2 (θ) = −θ, which we extend to an isomor√ ¶ µ ¶ µ θ 0 1 0√ phism, ψ2 : L → C, defined by . M2 = , M3 = √ √ −1 0 θ 0 ψ2 (x + y θ) = ϕ2 (x) + ϕ2 (y)i θ, x, y ∈ K, √ If T ∈ Γ, then using Proposition and (8), we where L = K( θ) and [L : K] = 2. Consider obtain now a quaternion algebra A [Γ] over K = Q (θ), √ √ ¶ µ [6], 1 xl + yl √θ zl + wl√ θ ½ λ ¾ T = s , (10) P −zl + wl θ xl − yl θ 2 A [Γ] = ai Ti : ai ∈ K, Ti ∈ Γ .

n α R= : α ∈ Z[θ] 2m

o m∈N ,

i=1

where s ∈ N e xl , yl , zl , wl ∈ Z [θ]. Hence, T is 5 Similar to the isomorphism of (1). In this case, identified with the following element x ∈ O = ϕ(A) ⊂ M (2, K(√θ)) with A = (θ, −1) and K = K yl xl zl wl (θ, −1)R , x = 2s + 2s i + 2s j + 2s k, through Q(θ). 6 of the isomorphism ϕ : A → ϕ(A) defined by If m = 1, then n > 0.

Hence, using the representation of the genera- 6 Conclusions tors of Γ in (8), we have ½µ ¶ ¾ In this paper we have proposed a procedure a1 b1 to construct arithmetic Fuchsian groups Γ4g A [Γ] = : a1 , b 1 ∈ L , −b01 a01 from regular hyperbolic polygons P4g with 4g 0 n 0 where a1 , b1 are the conjugates of a1 and b1 in edges, when g = m · 2 , for m = 1, 3, 5, L, respectively. Let Ψ : A [Γ] → M (2, C) be an where n is any natural number. The groups Γ4g were identified with orders O = (θ, −1)R embedding defined by µ ¶ of a quaternion algebra A over totally real alψ2 (a1 ) ψ2 (b1 ) gebraic number fields K, [K : Q] = 2n , for Ψ (α) = . −ψ2 (b01 ) ψ2 (a01 ) g = 2n , [K : Q] = 2n+1 , for g = 3 · 2n , and Thus, [K : Q] = 2n+2 , for g = 5·2n with the objective ¶ ¾ to determine the hyperbolic lattices associated ½µ a b Aψ2 = Ψ (A [Γ]) = : a, b ∈ ψ2 (L) .to the groups Γ4g . −b a Hence Aϕ2 ⊗ R ' H, [6]. On the other hand, if T is any element of Γ and tr (T ) = a + a0 , References then by Example 3.1, ψ2 (a) + ψ2 (a0 ) ∈ [−2, 2]. [1] A. Beardon. The Geometry of Discret However, since a + a0 ∈ K = Q(θ), we have Groups. Springer-Verlag, New York, 1983. ¡ ¢ ¡ ¢ ¡ ¢ ψ2 (a) + ψ2 a0 = ψ2 a + a0 = ϕ2 a + a0 , [2] E.D. Carvalho. Construction and Labeling that is, ϕ2 (a + a0 ) ∈ [−2, 2]. Therefore, of Geometrically Uniform Signal Constelϕ2 (tr (Γ)) is bounded in C. Hence, the seclations in Euclidean and Hyperbolic Spaces. ond condition of Theorem 3.2 is also satisDoctoral Dissertation, FEEC-UNICAMP, fied. We conclude that Γ is derived from a December 2001 (in Portuguese). quaternion algebra A over K = Q (θ), where r q p √ [3] M. Firer. Fuchsian Groups. Class Notes, θ = 2 + 2 + · · · + 2 + 2. IMECC-UNICAMP, 2001 (in Portuguese). As an application of the previously established concepts, we derive the generators of the [4] Stewart, I.N. e Tall, D.O. Algebraic Number Theory. Chapman and Hall, 1996. Fuchsian groups Γ ' Γ4g . Due to space limitation we present only the case g = 4 with only [5] S. Johansson. On fundamental domains of the generator G1 . arithmetic Fuchsian groups. Example 5.1 Let g = 4, hence www.math.chalmers.se/˜sj/forskning.html. √   √ √ √ 4 x1 (1+i(1+ 2)) −( 2+iy1 ) 2+ 2 [6] S. Katok. Fuchsian Groups. The University 2 √ √ 2 √ . A1 =  √ 4 of Chicago Press, 1992. −( 2−iy1 ) 2+ 2 x1 (1−i(1+ 2)) 2

2

By using the equalities in (7) and the fact that Gl = f −1 Al f , for l = 1, . . . , 8, we obtain  √ √ √ √  4 4 G1 = 

x1 −y1

2+ 2 2√ √ 4 −z1 −w1 2+ 2 2

z1 −w1

2 √ 4

2+ 2

,

[7] K. Takeuchi. A characterization of arithmetic Fuchsian groups. J. Math. Soc. Japan, vol. 27, pp. 600-612, 1975.

[8] Vieira, V. L. Arithmetic Fuchsian Groups Identified in Quaternion Orders for the 2 √ Construction of Signal Constellations. ¡where√x¢1 = 2 + θ, y1 = 2 + √2 + 2θ, z1 = Doctoral Dissertation, FEEC-UNICAMP, 2. Therefore, 1 + 2 (2 + θ) and w1 = February 2007 (in Portuguese). the order associated with the arithmetic Fuchsian group Γ16 is O = (θ, −1)Rp , where R = √ ©α ª 2 + 2 and 2m : α ∈ Z[θ] and m ∈ N , θ = ½ q ¾ q √ √ 4 4 1, 2 + 2, Im, 2 + 2 Im x1 +y1

√ 2+ 2

is an R-basis of O, according to Theorem 5.1.