Signature Cycles and Graphs

In this book, the main subject is to find some invariants of the signature of the normalizer by using suborbital graphs. In Chapter 1, the structure of Non-Euclidean Crystallographic groups is discussed and some properties of PSL(2,R), Modular group, congruence subgroups, normalizers of the congruence subgroups and also the preliminary definitions we require for discrete groups, Riemann surfaces, fundamental domains, graph theory and imprimitive action are given. In Chapter 2, the suborbital graphs of normalizer and number of orbits of congruence subgroup _0(N) of Modular group in extended rational number set are examined. Edge and circuit conditions on graphs arising from the action of congruence subgroup _0(N) on extended rational number set are determined. And , as the core of the thesis are found necessary and sufficient conditions for the suborbital graph F_(u,N) to be a forest. That is, it is shown that F_(u,N) suborbital graph is a forest if and only if it contains no n-gon , circuits of length n.