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

Kaynakça

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2. 2 C.C. Sims, Graphs and Finite Permutation Groups, Math. Z. 95 (1967), 75-86.
3. 3 G.A. Jones,D. Singerman and K. Wicks, The Modular Group and Generalized Farey Graphs, London Math. Soc. Lecture Notes, CUP, Cambridge, 160 (1991), 316-338.
4. 4 N. L. Biggs and A. T. White, Permutation Groups and Combinatorial Structures , London Math. Soc. Lecture Notes 33, Cambridge University Press, Cambridge, (1979).
5. 5 M. Akbas, On Suborbital Graphs for The Modular Group, Bull. London Math. Soc., 33 (2001), 647-652.