3rd INTERNATIONAL E-CONFERENCE ON MATHEMATICAL ADVANCES AND ITS APPLICATIONS, İstanbul, Turkey, 24 - 27 June 2020, pp.106
In the study published by Jones, Singerman and Wicks (1991), the modular group, the movement of an
element of the modular group on ℚ̂ (extended set of rational numbers) in hyperbolic geometry and,
Farey graph, 𝐺𝑢,𝑛 and 𝐹𝑢,𝑛 were studied. Also, it is showed that the fixed of any two
points is conjugated in 𝛤, and the element of the modular group that leaves constant an element on
ℚ̂ is infinite period. So, 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
invariant equivalence relations are generated on ℚ̂. In this study, we show that, the element that
fixed 𝑥 in modular group according to based on
the choice of 𝑥 for 𝑥, 𝑦 ∈ ℤ and (𝑥, 𝑦) = 1, instead of a special value of set ℚ̂, such as ∞.
Similarly, we study subgroups
𝐻 containing 𝛤𝑥 and we examine that the invariants equivalence relations on ℚ̂.
Keywords: Modular group, Infinite period, Invariant equivalence relations.