A Class of Generalized Triangle Groups as Quotients of 𝑷𝑮𝑳(𝟐, 𝑍)

Abstract

It is well known that the modular group PSL(2, Z) is generated by the linear fractional transformations 𝑥 ∶ 𝑧 → −1 𝑧 and 𝑦 : 𝑧 → 𝑧−1 𝑧 which satisfy the relations x2 = y3 = 1. An additional relation in a group converts it into a quotient of the group. If the additional relation is simply the power of the product of two generators x and y then it turns out to be a triangle group Δ(𝑙, 𝑚, 𝑛) = < 𝑥, 𝑦 ; 𝑥𝑙 = 𝑦𝑚 = (𝑥𝑦)𝑛 = 1 >. The triangle groups Δ(2,3, 𝑛) are especially important for being homomorphic images of the modular group PSL(2,Z). If an additional relation is of the form (𝑤(𝑥, 𝑦))𝑛 where 𝑤(𝑥, 𝑦) = 𝑥𝑝1𝑦𝑞1𝑥𝑝2𝑦𝑞2 . . . 𝑥𝑝𝑘𝑦𝑞𝑘 then the group converts into a generalized triangle group Δ∗(𝑙, 𝑚, 𝑛) = < 𝑥, 𝑦 ; 𝑥𝑙 = 𝑦𝑚 = (𝑤(𝑥, 𝑦))𝑛 = 1 >. It is known that the generalized triangle group is infinite when and finite when . J. Howie, V. Metaftsis and R. M. Thomas proved a very important result about the classification of finite generalized triangle groups. A word 𝑤 is defined as a finite sequence 𝑥1 𝜀1𝑥2 𝜀2 . .. 𝑥𝑘 𝜀𝑘 , where for each 𝒊, 𝒙𝒊 belongs to the set of generators and each 𝜺𝒊 is either 1 or -1. The third relator leads to a word which is of special interest in this thesis. Several group theorists discussed one relator quotients of various groups. A considerable number of them concentrated on one relator quotients of the modular group. M. D. E. Conder is one of them who found quotients of the modular group by inserting additional relations as words up to length 24. Y. T. Ulutas and I. N. Cangul, by using a different technique, investigated one relator quotients of the modular group by inserting additional relations as words up to length 21. Later on, a number of researchers followed both the techniques, but all were restricted by considering additional relations as words of finite lengths. In the entire discussion of one relator quotients, length of the additional relation as a word is the centre point of our concern. In this dissertation, our aim is to study a class of generalized triangle groups as quotients of the modular group. Since, modular group is a two generator group, we insert an additional relation of the form 𝑤(𝑥, 𝑦) = 𝑥𝑝1𝑦𝑞1𝑥𝑝2𝑦𝑞2 … 𝑥𝑝𝑘 𝑦𝑞𝑘 in the finite presentation of the group. We consider powers of the generators as terms of Fibonacci sequence of numbers. That is we consider groups < 𝑥, 𝑦 ; 𝑥2 = 𝑦3 = 𝑤(𝑥, 𝑦) = 1 > which are one relator quotient of the modular group and a class of generalized triangle groups. There are two major parts to investigate in this class of groups. Firstly, we determine additional relations for all lengths k, that is, the length of word 𝑤(𝑥, 𝑦)- which varies from 1 to infinity. Secondly, we insert these (infinite) number of additional relations in finite presentation of the modular group and investigate the quotient groups thus obtained. This thesis comprises five chapters. In chapter one, we mentions some basic concepts related to one relator quotients. This chapter contains finite presentations of groups, quotient of a group, group action on suitable sets, coset diagrams, projective general linear group, projective special linear group, triangle groups, generalized triangle groups, Fibonacci sequence, words, reduced words, equivalent words, syllable of a word, Tietze transformations, finite fields and projective lines over the finite fields. In chapter two, there is a comprehensive survey of one relator quotients generally and one relator quotients of the modular group particularly. This study not only explains the results but also stresses upon the methodology adopted by various researchers. One relator quotients of the modular group are of special importance due to the interesting features of this group. In chapter three, we generate words of all syllables. We use Fibonacci sequence of numbers in the powers of the generators in the additional relation for generating words of all syllables. We develop an algorithm by which we generate words. This algorithm gives four outputs; words of all syllables, reduced form of the words, count the number of x and y in words, and in their respective reduced forms. In the end, we divide words in classes on the basis of Fibonacci sequence. In chapter four, we find one relator quotients of the modular group related to Fibonacci sequence of numbers. The words obtained in chapter three are used as additional relation in the modular group so that they can later be investigated as quotient groups. Finally, to identify these quotients we use Tietze transformation and in certain cases ‘Groups, Algorithms and Programming’ (GAP). It is a class of generalized triangle groups which we investigate as quotients of the modular group. Furthermore, from this class of quotients we choose one quotient, which is the alternating group of degree 4, that is, A4 and by taking action of A4 on the projective line over the finite field F257, that is PL(F257) we construct an algebraic substitution box (S-box). By investigating the security strength parameters of this S-box, we conclude that this S-box is highly secure for the communication and highly preferable for cryptographic applications. In chapter five, we determine number of all one relator quotients of the modular group for each syllable by considering all possible additional relations. Furthermore, we proved a number of results by which we find the number of cyclically reduced nonequivalent words for each syllable k. The one relator quotients corresponding to these cyclically reduced non-equivalent words are sufficient instead of finding all but equivalent quotients. In this chapter, we also view the additional relations as circuits (close paths) and find some interesting relationships between them. From the circuits point of view, if we consider all the possibilities of the additional relation then there are two types of circuits; one type consists of circuits having all triangles with one vertex inside or all triangles with one vertex outside of the circuit and the second type consists of circuits containing some (at least one) triangles with one vertex in side and some (at least one) triangles with one vertex outside the circuit. First type depicts triangle groups as quotients of the modular group and the other type depicts generalized triangle groups as quotients of the modular group. The study of one relator quotients provides a mechanism to determine all one relator quotients of any two-generator group.