Homomorphic Images of Generalized Triangle Subgroups of PSL(2, ℤ)

Abstract

The modular group generated by two linear fractional transformations, u : z 7􀀀! 􀀀1 z and v : z 7􀀀! z 􀀀 1 z , satisfying the relations u2 = v3 = 1 [46]. The linear transformation t : z 7! 1 z inverts u and v, i,e, t2 = (vt)2 = (ut)2 = 1 and extends PSL(2; Z) to PGL(2;Z). In [72] a condition for the existence of t is explained. G. Higman introduced coset diagrams for.PSL(2; Z) and PGL(2;Z) : Since then, these have been used in several ways, particularly for nding the subgroups which arise as homomorphic images or quotients of PGL(2;Z). The coset diagrams of the action of PSL(2;Z) represent permutation representations of homomorphic images. In these coset diagrams the three cycles of the homomorphic image of v, say v, are represented by small triangles whose vertices are permuted counter-clockwise, any two vertices which are interchanged by homomorphic image of u, say u, are joined by an edge , and t is denoted by symmetry along the vertical line. The xed points of u and v, if they exist are denoted by heavy dots. The xed points of t lies on the vertical line of symmetry. A real quadratic irrational eld is denoted by Q pd

, where d is a square free positive integer. If =

a1 + b1pd

c1 is an element of Q pd

, where a1; b1; c1; d; are integers, then has a unique representation such that a1; 􀀀 a2 1 􀀀 d

c1 and c1 are relatively prime. It is possible that ; and and its algebraic conjugate =

a1 􀀀 pd

c1 have opposite signs. In this case is called an ambiguous number by Q. Mushtaq in [69]. The coset diagrams of the action of PSL(2;Z) on Q pd

depict interesting re- sults. It is shown in [69] that for a xed value of d, there is only one circuit in the coset diagram of the orbit, corresponding to each . 4 Any homomorphism 1 : PGL(2;Z) ! PGL(2; q) give rise to an action on PL(Fq) : We denote the generators ( ) 1; ( ) 1 and (t) 1 by ; and t: If neither of the generators , and t lies in the kernel of 1; so that , and t are of order 2, 3 and 2 respectively, then 1 is said to be a non-degenerate homomorphism: In addition to these relations, if another relation ( )k = 1 is satis ed by it, then it has been proved in [74] that the conjugacy classes of non-degenerate homomorphisms of PGL(2;Z) into PGL(2; q) correspond into one to one way with the conjugacy classes of 1 and an element of Fq: That is, the actions of PGL(2;Z) on PL(Fq) are parametrized by the elements of Fq: This further means that there is a unique coset diagram, for each conjugacy class corresponding to 2 Fq. Finally, by assigning a parameter 2 Fq to the conjugacy class of 1, there exists a polynomial f( ) such that for each root i of this polynomial, a triplet ; ; t 2 PGL(2; q) satis es the relations of the triangle group (2; 3; k) = D ; ; t : 2 = 3 = ( t)2 = ( )k = ( t)2 = ( t)2 = 1 E : Hence, we can obtain the triangle groups (2; 3; k) through the process of parametrization. The generalized triangle group has the presentation

u; v : ur; vs;Wk

;where r; s; k are integers greater than 1, and W = u 1v 1 :::u kv k , where k > 1; 0 < i < r and 0 < i < s for all i. These groups are obtained by natural generalization of (r; s; k) de ned by the presentations D u; v : ur = vs = (uv)k = 1 E , where r; s and k are integers greater than one. It was shown in [37] that G is in nite if 1 r + 1 s + 1 k 1 provided r 3 or k 3 and s 6, or (r; s; k) = (4; 5; 2): This was generalized in [4], where it was shown that G is in nite whenever 1 r + 1 s + 1 k 1 . A proof of this last fact can be seen in [101]. 5 A generalized triangle group may be in nite when 1 r + 1 s + 1 k > 1. The complete classi cation of nite generalized triangle groups is given in 1995 by J. Howie in [39] and later by L. Levai, G. Rosenberger, and B. Souvignier in [57] which are fourteen in number. As there are fourteen, generalized triangle groups classi ed as nite [39], our area of interest is the set of groups which are homomorphic images or quotients of PSL(2;Z). Out of these fourteen only eight groups are quotients of the modular group. In this study, we have extended parametrization of the action of PSL(2;Z) on PL(Fp), where p is a prime number, to obtain the nite generalized triangle groups D 2 = 3 = 􀀀 2 3 = 1 E by this parametrization. By parametrization of action of PGL(2;Z) on PL(Fp) we have obtained the coset diagrams of D 2 = 3 = 􀀀 2 3 = 1 E for all 2 Fp. This thesis is comprised of six chapters. The rst chapter consists of some basic de nitions and concepts along with examples. We have given brief introduction of linear groups, the modular and the extended modular group, real quadratic irrational elds, nite elds, coset diagrams, triangle groups, and generalized triangle groups. In the second chapter, we show that entries of a matrix representing the element g = 􀀀 ( v)m1 􀀀 v2 m2 l where l 1 of PSL(2;Z) =

; v : 2 = v3 = 1

are denominators of the convergents of the continued fractions related to the circuits of type (m1;m2) ; for all m1;m2 2 N: We also investigate xed points of a particular class of circuits of type (m1;m2) and identify location of the Pisot numbers in a circuit of a coset diagram of the action of PSL(2;Z) on Q pd

[ f1g, where d is a non-square positive integer. 6 In the third chapter we attempt to classify all those subgroups of the homomor- phic image of PSL(2;Z) which are depicted by coset diagrams containing circuits of the type (m1; m2). In the fourth chapter we devise a special parametrization of the action of modular group PSL(2;Z) on PL(Fp), where p is prime, to obtain the generalized triangle groups D 2 = 3 = 􀀀 2 k = 1 E and by parametrization we obtain the coset diagrams of D 2 = 3 = 􀀀 2 k = 1 E for all 2 Fp. In the fth chapter we investigate the action of PSL(2;Z) on PL(F7n) for di¤erent values of n, where n 2 N, which yields PSL(2; 7). The coset diagrams for this action are obtained, by which the transitivity of the action is inspected in detail by nding all the orbits of the action. The orbits of the coset diagrams and the structure of prototypical D168 Schwarzite [48], are closely related to each other. So, we investigate in detail the relation of these coset diagram with the carbon allotrope structures with negative curvature D168 Schwarzite. Their relation reveals that the diagrammatic structure of these orbits is similar to the structure of hypothetical carbon allotrope D56 Protoschwarzite which has a C56 unit cell. In the last chapter, we investigate the actions of the modular group PSL(2;Z) on PL(F11m) for di¤erent values of m; where m 2 N and draw coset diagrams for various orbits and prove some interesting results regarding the number of orbits that occur.