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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 is a congruence subgroup of the modular group Γ acting on . We determine orbits of the action of Γ K (N) on and present a Γ K (N) invariant equivalence relation by imprimitive action. Then, we investigate the suborbital graph 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 .