On suborbital graphs for the extended modular group Γ̂

Abstract

In this paper, we show that the extended modular group ?? acts on ?? transitively and imprimitively. Then the number of orbits of ??0(N) on ?? is calculated and compared with the number of orbits of ?0(N) on ??. Especially, we obtain the graphs ?n,N of ??0(N) on ??, for each N ? ? and each unit u ? UN, then we determine the suborbital graph F?u,N. We also give the edge conditions in ?u,N and the necessary and sufficient conditions for a circuit to be triangle in F?u,N. © 2012 Springer.