TY  - JOUR
T1  - Investigation of Γ-Invariant Equivalence Relations of Modular Groups and Subgroups
AU  - Gökcan, İbrahim
AU  - Değer, Ali Hikmet
PY  - 2020
DA  - December
Y2  - 2020
JF  - Conference Proceedings of Science and Technology
PB  - Murat TOSUN
WT  - DergiPark
SN  - 2651-544X
SP  - 110
EP  - 114
VL  - 3
IS  - 1
LA  - en
AB  - 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 ̂.
KW  - Infinite period
KW  - Invariant equivalence relations
KW  - Modular group
CR  - 1 B. Schoeneberg, Elliptic Modular Functions , Springer-Verlag, Berlin, Heidelberg, New York,(1974).
CR  - 2 C.C. Sims, Graphs and Finite Permutation Groups, Math. Z. 95 (1967), 75-86.
CR  - 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.
CR  - 4 N. L. Biggs and A. T. White, Permutation Groups and Combinatorial Structures , London Math. Soc. Lecture Notes 33, Cambridge University Press, Cambridge, (1979).
CR  - 5 M. Akbas, On Suborbital Graphs for The Modular Group, Bull. London Math. Soc., 33 (2001), 647-652.
UR  - https://dergipark.org.tr/tr/pub/cpost/issue//774513
L1  - https://dergipark.org.tr/tr/download/article-file/1218330
ER  -