Coset Diagrams for the Modular Group and Adjacency Matrices

Abstract

TheringhomomorphismfromtheextendedrealquadraticirrationaleldQ(pn) = Q(pn)[ f1g to the projective line over the nite eld PL(Fp) lifts the action of the modular group PSL(2;Z) on PL(Fp). The intransitive action of PSL(2;Z) on Q(pn) has a closed path in each orbit of the action due to the existence of ambiguous numbers, which is sometimes called a circuit. The continued fractions are reconstituted as the composition of linear fractional transformations of G = PSL(2;Z), by which all the circuits of the orbits are obtained for the action G on Q(pn). The coset diagrams for the action of the modular group on PL(F5n) are seen as diagrammatic analogues of the fullerene C60. The characteristic polynomial of a truncated circuit is determined by expanding a graph. The adjacency matrix of the coset diagram C(n) gives rise to another adjacency matrix which can be construed as a bounded self-adjoint linear operator on the Hilbert space l20C(n). The three di⁄erent circuits of the action of V4 are depicted in adjacency matrices and a few spectral theoretic properties are observed. The merging of these circuits is then observed in the coset diagrams of the action of G on PL(Fp) which produces a result relying upon the cyclotomic extension of Q. In the end, adjacency matrices MCi for the Ci are constructed which merges into the adjacency matrices Mp Ci for (2;3;k). The thesis comprises of six chapters. The rst chapter contains some basic denitions and concepts along examples which include the quadratic elds, the modular group, triangle groups, group graphs and their relation with adjacency matrices, combinatorial and normalised Laplace matrices, and coset diagrams. In the second chapter, for any ij 2Q(pn), the path leading to the circuit Ci and the circuit itself in the orbit of the action of G on Q(pn), is obtained through its continued fractions. The interrelation of the structure of continued fractions with the types of circuits is established. The action of G on Q(p5) is chosen especially because a circuit of C(5) is related to the ratio of the Fibonacci numbers being the solution of the continued fractions of the Golden Ratio. The third chapter expounds the colligation of adjacency matrices of the abstract group PSL(2;5) with fullerene C60 due to the resemblance of its coset diagram with a description of PSL(2;5). The action of G on PL(F5n) yields PSL(2;5), so the adjacency matrices of PSL(2;5) are used to obtain coset diagrams, and to investigate various properties with reference to the intransitivity of the action and the number of orbits