A comment on the bianchi groups

Transcription

1 6 JACM, No, 6-7 (08) Journal of Abstract and Computational Mathematics A comment on the bianchi groups Murat Beşenk Department of Mathematics, Pamukkale University, Denizli, Turkey Received: Jan 08, Accepted: 4 Feb 08 Published online: 8 Feb 08 Abstract: In this paper, we aim to discuss several the basic arithmetic structure of Bianchi groups In particularly, we study fundamental domain and directed orbital graphs for the group PSL(,O ) Keywords: Quadratic number field, bianchi groups, suborbital graphs, circuits Introduction It is known that the study of the group PSL(,C) : {T T : C C, T (z) az+b cz+d, a,b,c,d C and ad bc } consisting of all fractional linear transformations of the projective plane of points at infinity, with complex coefficients, was one of the major topics of mathematics in the 9th century and played an important role in the development of hyperbolic geometry The group PSL(,C) has a natural action on -dimensional hyperbolic space H which may be described in geometric terms An element T PSL(,C) induces iholomorphic map of the Riemann sphere C { } Jules Henri Poincare (854-9) observed that PSL(, C) can be identified with the group of orientation preserving isometries of H As in the study of the hyperbolic plane, the importance of finding discrete groups of hyperbolic isometries of H was emphasized Starting with an example by Emile Picard (856-94) and pursued by Luigi Bianchi (856-98) one particularly interesting class of discrete subgroups of PSL(,C) was considered It is constructed in the following arithmetic way: Denote by O d the ring of integers of an imaginary quadratic number field Q( d) with d < 0 and square free Then O d gives rise to a discrete group Γ d : PSL(,O d ) < PSL(,C) consisting of fractional linear transformations with coefficients in O d From [,] we can say that each subgroup τ of finite index in Γ d operates properly discontinuously on H and a fundamental domain for this τ action on hyperbolic -space in noncompact but of finite volume From various reasons, this class of arithmetically defined discrete groups of hyperbolic motions has gained new vivid interest in recent years Beside that the Bianchi groups Γ d are also of interest in their own group theoretical right In [, 4], the structures of directed and undirected graphs were investigated The relation between graphs and permutation groups was examined In [5], authors mentioned permutation groups, primitivity and their applications to graph theory In these papers [7,8,9,0] authors investigated some properties of suborbital graphs for the modular group, Picard group, the simple groups and SL(, Z) group And also in [6], the author discussed suborbital graphs for the normalizer of Γ 0 (N) in PSL(,R) group This paper is devoted to a computer aided analysis of subgoups of small index in Γ d, in particular, we will deal as properties with subgroups of small index in Γ d for d,,, 5, 7 There is a fruitful interaction between geometric-topological, group theoretical and arithmetic questions and methods The action of PSL(,C) on H Consider the hyperbolic -space H {(z,t) C R t > 0}, that is, the unique connected, simply connected Riemann manifold of dimension with constant sectional curvature - A standard model for H is the upper ( half space ) model H dx with the metric coming from the line element ds + dy + dt with z x+iy Every element of PSL(,C) t Corresponding author mbesenk@pauedutr c 08 BISKA Bilisim Technology

2 JACM, No, 6-7 (08) / wwwntmscicom/jacm 64 acts on H as an orientation preserving isometry via the formula, ( (az + b)(cz + d) + acy (z,t) cz + d + c y, y ) cz + d + c y where (z,t) H It is well-known that every orientation preserving isometry of H arises this waythe group of all isometries of H is generated by PSL(,C) and complex conjugation (z,t) (z,t), which is an orientation reversing involution Indeed, the group PSL(,C) acts transitively on the points of H so that the stabiliser of any point in H is conjugate to the stabiliser of (0,0,) Therefore H and its geometry can be obtained from SL(,C) as its symmetric space as SU(,C) is a maximal compact {( subgroup ) Likewise, the action } on the sphere at infinity is transitive so that point stabilisers are conjugate to ϒ 0 a a 0anda,b C Any finite subgroup of PSL(,C) must have a fixed point in H and so be conjugate to a subgroup of SO(,R) As such, it will either be cyclic, dihedral or conjugate to one of the regular solids groups and isomorphic to A 4, A 5 or S 4 The action of PSL(,C) on H leads to an action of SL(,C) α 0 Lemma The group SL(,C) is generated by the elements ζ, ζ 0 where α C These generators 0 operate on H as follows Proof Suppose T α (z,r) (z + α,r), 0 SL(,C) If c 0, we have and for c 0 we obtain the factorization 0 0 ( z (z,r) z + r, r ) z + r ac 0 c 0 dc c 0 a 0 a b 0 a 0 a 0 We thus conclude that the matrices ζ,ζ together with the matrices W η : ( η 0 0 η ), η C \ {0}, generate SL(,C) But the elements W η may be represented as products of the matrices ζ,ζ as is evident from the following formulas η 0 η η 0 η 0 0 η 0 η, 0 0 β ( 0 β )( The first mathematical works on non-euclidean geometry were published by N Lobachevski (89) and J Bolyai (8) E Beltrami (868) introduced the upper space model, the unit ball model and the projective disc model of n dimensional hyerbolic geometry, Milnor (98) These models were well-known to the classical writers who used them in their investigations on discontinuous groups, eg Bianchi (95), Fricke (94), Klein (897), Humbert (99), Picard (978), Poincare (96) The upper-space model of three dimensional hyperbolic geometry is presented by Beardon ), c 08 BISKA Bilisim Technology

3 65 Murat Beşenk: A comment on the bianchi groups (98) and Magnus (974) The formulas of elementary hyperbolic geometry become particularly nice if we represent the points of hyperbolic -space by matrices in SL(,C) It is know that the following formalism is due to H Helling Bianchi groups and fundamental domain Definition A quadratic number field is a field F C such that F has dimensional as a vector space over Q Such a field takes the form F Q( n) {p + q n : p,q Q and n > square-free} If n is positive then F is a real quadratic number field, and if n is negative then F is an imaginary quadratic number field The trace function of F is the additive homomorphism tr : F Q, tr(α) α + α The norm function of F is the multiplicative homomorphism N : F Q, N(α) αα Specifically, N(p + q n) p q n The integers of the quadratic field F Q( n) are n The discriminant of F is F 4n + n if n (mod4), O F Z[g], g n if n,(mod4) if n (mod4), if n,(mod4) Thus we see that every quadratic number field is of the form Q( d), where d is any square free integer For any natural number m, define O d,m : Z + mωz where For simplicity, denote O d, by O d We also define O : Z + d if d (mod4), ω : d if d (mod4) d ω N(p + qω) trace Elements o f norm i p + q p ±,±i p + q p ± + p + pq + q p ±,±ω,±ω 5 5 p + 5q p ± p + pq + q p ± Let d is rational number We examine Q( d) {p+q d : p,q Q} C Clearly if d Q or d, then Q( d) Q For d, Q( ) Q(i) {p + q : p,q Q} is called Gaussian numbers Also we may say that rational numbers within the root can be arranged For instance, Q( ) Q( ) Q( ) Q( ) Definition Let d < 0, square free integer and let O d denote the ring of integers in Q( d) The groups Γ d PSL(,O d ) are called as the Bianchi groups When d,,, 7, the rings O d have an Euclidean algorithm and the Γ d groups are known as the Euclidean Bianchi groups Theorem Let K be an imaginary quadratic field of discriminant K < 0, and let O d be its ring of integers Then the group PSL(,O d ) has following properties: (i)psl(,o d ) is a discrete subgroup of PSL(,C) (ii)psl(,o d ) has finite covolume, but is not cocompact (iii)psl(,o d ) is a geometrically finite group (iv)psl(,o d ) is finitely presented c 08 BISKA Bilisim Technology

4 JACM, No, 6-7 (08) / wwwntmscicom/jacm 66 We fix an imaginary quadratic number field K may construct now a fundamental domain F K H for group PSL(,O d ) where O d is the ring of integers in K Lemma The fundamental domain F Q(i) for PSL(,O ) is described by F Q(i) {z +t j H 0 Rez, 0 Imz, zz +t } F Q(i) is a hyperbolic pyramid with one vertex at and the other four vertices in the points P + j, P + j, P + i + j and P 4 + i + j For PSL(,O ), the Picard group, the region exterior to all isometric spheres, is the region exterior to all unit spheres whose centres lie on the integral lattice in C The stabiliser PSL(,O ) is an extension of the translation subgroup by a rotation of order about the origin We thus obtain the fundamental region shown in figure Fig : Fundamental domain for Γ The Bianchi group Γ d defined by Γ d : PSL(,O d ) SL(,O d )/{±I} is a discrete subgroup of PSL(,C) viewed as the group of orientation preserving isometries of the -dimensional hyperbolic space H Hence Γ d acts properly discontinuously on H Fundamental domains for this action can analyze for small values of d Lemma The fundamental domain F Q( ) for PSL(,O ) is described by F Q( ) {z +t j H 0 Rez, Rez Imz, Imz ( Rez)} {z C 0 Imz, Rez Imz Rez} The picture of F Q( ) is closed quadrangle with corners in the points Q 0, Q 6, Q + 6 and Q 4 Now, we give examples about generator of Bianchi groups We note that in here Γd ab commutator factor group of Γ d c 08 BISKA Bilisim Technology

5 67 Murat Beşenk: A comment on the bianchi groups Example (I) The group Γ is generated by the matrices 0 A 0, B 0, C where i A presentation of Γ is given by the following relations i, E 0 B (AB) ACA C (AE) I, E (CE) (BE) (CBE) I and we have Γ ab (II) The group Γ is generated by the matrices 0 A 0, B 0, C where ω A presentation of Γ is given by the following relations ω 0 B (AB) ACA C (BC BC) I i 0 0 i and again we can compute Γ ab (III) The group Γ is generated by the matrices 0 ω A 0, B 0, C 0 Where ω + A presentation of Γ is given by the following relations B (AB) ACA C (ACBC B) I, (ACBC B) A C BCBC BC BCB I and then one may finds Γ ab (IV) The group Γ 7 is generated by the matrices 0 ω A 0, B 0, C 0 Where ω + 7 A presentation of Γ 7 is given by the the following relations and one computes Γ ab 7 Z Z B (AB) ACA C (BAC BC) I 4 Congruence subgroups and digraphs It is known that number theoretic interest in the Bianchi groups has centered primarily on the congruence subgroups and the congruence subgroup property There has also been a considerable amount of work on quadratic forms with entries in O d and their relation to Γ d If σ is an ideal in O d then the principal congruence subgroup modσ, Γ d (σ) consists of those transformations in Γ d corresponding to matrices in SL(,O d ) congruent to ±Imodσ Γ d (σ) {±T : T SL(,O d ),T Imodσ} Γ d (σ) can also be described as the kernel of the natural this map ψ : SL(,O d ) SL(,O d /σ) modulo ±I Thus each principal congruence subgroup is normal and of finite index A congruence subgroup is a subgroup which contains a principal congruence subgroup Notice that in the Euclidean cases each ideal is principal and a formula in M Newman allows us to compute the index of a principal congruence subgroup Namely if α O d, d {,,, 7, } then c 08 BISKA Bilisim Technology

6 JACM, No, 6-7 (08) / wwwntmscicom/jacm 68 Γ d : Γ d (α) ρ α p α ( ) where p runs over the primes dividing α and ρ if α or ρ p otherwise Now we say some subgroup { } W 0 : SL(,C) c a a C, b,c R, W : W : { a b SL(,C) c a { i SL(,C) ci a } a C, b,c R, } a C, b,c R The group SL(,C) acts on W 0 by ω MωM for ω W 0, M SL(,C) Similarly, the group SL(,C) acts on W and W The full congruence subgroup is { } α β α β 0 Γ (σ) : SL(,O γ δ d ) mod σ γ δ 0 where σ is a non zero ideal in the ring of integers O d Note that is a cusp of Γ (σ) Another subgroup is { } α β Γ 0 (σ) : SL(,O γ δ d ) γ 0 mod σ Some of the groups Bianchi are subgroups of discrete groups which can be generated by hyperbolic reflections The relationship of the Bianchi to reflection groups is studied in Shvartsman (987), Shaikheev (987), Vinberg (987) and Ruzmanov (990) We can give applications for PSL(,O ), hence we will obtain an extension of suborbital graphs Theorem PSL(,O ) acts transitively on Σ : Q( ) { } We represent as ± ε 0 where ε is or i Proof We can show that the orbit containing is Σ If x y Σ, then as (x,y), there exist α,β Z[i] with αy βx α x Then the element of PSL(,O β y ) sends 0 to x y Lemma 4 The stabilizer of 0 in PSL(,O ) is the set PSL(,O ) 0 : Γ,0 { } 0 λ Z[i] λ Proof The stabilizer of a point in Σ is a infinite cyclic group As the action is transitive, ( stabilizer )( of ) any( two ) points are conjugate Hence it is enough to examine the stabilizer of 0 in PSL(,O ) T and so b 0 0 Then b 0, d and as dett, a Therefore c λ Z[i] Thus T Indeed this shows d λ that the set PSL(,O ) 0 is equal to ( 0 ) Definition PSL(,O ) 0 (N) : Γ 0 (N) {T PSL(,O ) b 0(modN), N Z[i]} c 08 BISKA Bilisim Technology

7 69 Murat Beşenk: A comment on the bianchi groups It is clear that Γ,0 < Γ 0 (N) < Γ We may use an equivalence relation induced on Σ by Γ Now let r s, x Σ Corresponding to these, there are two matrices y r x S : s, S : y in Γ for which S (0) r s, S (0) x y Therefore r s x if and only if y s r x sx ry S S Γ y 0 (N) So sx ry 0(modN) Definition 4 Let G be a graph and a sequence v,v,,v k of different vertices Then form v v v k v, where k > and k positive integer, is called a directed circuit in G Definition 5 Let (Γ,Σ) be transitive permutation group Then Γ acts on Σ Σ by Θ : Γ (Σ Σ) Σ Σ, Θ(T,(α,α )) (T (α ),T (α )), where T Γ and α,α Σ The orbits of this action are called suborbitals of Γ Now we investigate the suborbital directed graphs or digraphs for the action Γ on Σ We say that the subgraph of vertices form the block [ { x } [ ] : 0] y Σ x (modn), y 0(modN) is denoted by F u,n : F( 0, N u ) where (u,n) Theorem There is an edge r s x y in F u,n if and only if there exists a unit ε Z[i] such that x ±εur (modn), y ±εus(modn) and ry sx εn Proof Suppose that there exists an edge r s x y F u,n Hence there exist some T Γ such that sends the pair (, u N ) to the pair ( r s, x y ) Clearly T ( ) r s and T ( N u ) x az+b y For T (z) cz+d we have that a c r au+bn s and cu+dn x y Then there exist the units ε 0,ε ) Z[i] such ) that a ε 0 r, c ε 0 s and au + bn ε x, cu + dn ε y So, we can write ( )( u 0 N ( ε0 r ε x ε 0 s ε y from the determinant ry sx εn is achieved Finally, taking with ε ε 0 ε, we get that x ±εur (modn), y ±εus(modn) And also Conversely, we can take the plus sign Therefore there exist b,d Z[i] such that ( x )( εur +bn, ) ( y εus+dn ) If we choose u εr x a εr and c εs, then we find x au+bn and y cu+dn So T (, N u ) As ε(ry sx) N 0 N εs y we have ad bc and T Γ Hence r s x y in F u,n Similarly, minus sign case may shown Theorem 4 F u,n contains directed hyperbolic triangles if and only if there exists a unit ε Z[i] such that ε u εu+ 0(modN) and ε u + εu + 0(modN) Proof We suppose that F u,n contains a directed hyperbolic triangle Since the transitive action, the form of the directed hyperbolic triangle can written like this 0 u N r N 0 As the edge condition in above the theorem, we have to be provided for the second edge N u N r, that is u r ε and r ±εu (modn) Then we have εr ±ε u (modn) Therefore ε u + εr 0(modN) and εu εr ε ± are obtained Hence there are two cases The first case is ε u + εr 0(modN) and εr εu The second case is ε u +εr 0(modN) and εr εu+ Finally we may say that for ε Z[i] these equations ε u εu+ 0(modN) and c 08 BISKA Bilisim Technology

8 JACM, No, 6-7 (08) / wwwntmscicom/jacm 70 ε u +εu+ 0(modN) are satisfied Conversely, we can solve these equations only with special conditions Let ε Z[i] be a unit such that ε u εu + 0(modN) Above the theorem implies that there is a directed hyperbolic triangle 0 u N u ε N 0 in F u,n Similarly, we can find another directed hyperbolic triangle for ε u + εu + 0(modN), in this case in F u,n 0 u N u + ε N 0 Now we get example for first equation Example Let N + 6i If we take u + i, then this equation ( + 4i)ε ( + i)ε + 0mod( + 6i) is achieved for ε i Hence we get this directed hyperbolic triangle as 0 + i + 6i + 6i 0 in F +i,+6i And also, it is clear that ( + i, + 6i) and ry sx εn Similarly, let N i Again we choose u, then ε +ε + 0mod(i) is held if and only if ε i So, directed triangle is i + i in F,i Fig : Circuits in F +i,+6i and F,i c 08 BISKA Bilisim Technology

9 7 Murat Beşenk: A comment on the bianchi groups Corollary The transformation ψ : εu ε u εu+ εn which is defined by means of the congruence ε u εu + εn εu 0(modN) is an( elliptic ) element of order Obviously that detψ, ψ( I and ) tr(ψ) Moreover, it is easily seen u u u that ψ, ψ 0 N : ψ ε u, ψ 0 : ψ κ 0 N 0 εu ε u ±εu εn u Corollary The transformation η : has detη and tr(η) ± Furthermore, η, εn εu ± 0 N u u η ε u u, η ε u u, η ε u u, η 4 5ε u u F n,, η n F n+ ε, where n 0 positive integer and F n Fibonacci numbers It is known that F 0 0, F and F n+ F n + F n+ That is, two directed paths 0 u N u ε N u ε N u ε N u 5ε F u n F n+ ε N N are obtained Example Let N 4 and u 7 We consider first case If we take ε then we have this directed path References [] Grunewald F, Schwermer J, Subgroups of Bianchi groups and arithmetic quotients of hyperbolic -space, Transactions of The American Mathematical Society, Vol:5, 47-78, 99 [] Maclachlan C, Reid AW, The arithmetic of hyperbolic -manifolds, Springer - Verlag, New York, 00 [] Sims CC, Graphs and finite permutation groups, Mathematische Zeitschrift, Vol:95, 76-86, 967 [4] Jones GA, Singerman D and Wicks K, The modular group and generalized Farey graphs, London Mathematical Society, Vol:60, 6-8, 99 [5] Biggs NL, White AT, Permutation groups and combinatorial structures, Cambridge University Press, Cambridge, 979 [6] Beşenk M, Signature cycles and graphs, Lambert Academic Publishing, Saarbrücken, Germany, 0 [7] Akbaş M, On suborbital graphs for the modular group, Bulletin London Mathematical Society, Vol:, , 00 [8] Beşenk M, Güler BÖ, Köroğlu T, Orbital graphs for the small residue class of PSL(, 5), General Mathematics Notes, Vol:7, 0-, 06 [9] Beşenk M, Güler BÖ, Değer AH, Kesicioğlu Y, Circuit lengths of graphs for the Picard group, Journal of Inequalities and Applications, Vol:, 06-, 0 [0] Beşenk M, Suborbital graphs for a special subgroup of the SL(,Z), Filomat, Vol:0, 59-60, 06 c 08 BISKA Bilisim Technology