The action of Γ ^K (N) and its suborbital graphs

Abstract

For any natural number N, Γ K (N) where K is a subgroup of (Figure presented.) is a congruence subgroup of the modular group Γ acting on (Figure presented.). We determine orbits of the action of Γ K (N) on (Figure presented.) and present a Γ K (N) invariant equivalence relation by imprimitive action. Then, we investigate the suborbital graph (Figure presented.) arising from the action of Γ K (N) on the orbit of ∞ and give conditions for two adjacent vertices in graph. In addition, we obtain connectedness properties of the subgraph (Figure presented.).