Investigation of Γ-Invariant Equivalence Relations of Modular Groups and Subgroups

Abstract

In [3], the modular group, the movement of an element of the modular group on Q ̂ (extended set of rational numbers) in hyperbolic geometry, and Farey graph, G_(u,n) and F_(u,n) were investigated. Furthermore, it is indicated that the fixed of any two points is conjugated in Γ, and the element of the modular group that leaves constant an element on Q ̂ is infinite period. Hence, the element of the modular group that leaves the ∞ element constant is symbolized as Γ_∞. In the same study, H, the subgroups of Γ of containing Γ_∞ are obtained and its invariant equivalence relations are generated on Q ̂. Taking these points into account, in this study, we show that, the element that fixed x/y in modular group based on the choice of x/y for x,y∈Z and (x,y)=1, instead of a special value of set Q ̂, such as ∞. Similarly, we study subgroups H containing Γ_(x/y) and we examine its the invariants equivalence relations on Q ̂.

Keywords

• Infinite period
• Invariant equivalence relations
• Modular group

References

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