TAILIEUCHUNG - Cycloidal normal subgroups of Hecke groups of finite index

Cycloidal normal subgroups, that is, subgroups with only one cusp, of finite index of Hecke groups, which have been introduced by E. Hecke, and can be thought of as some kind of generalisation of the well-known modular group, are studied. | Tr. J. of Mathematics 23 (1999) , 345 – 354. ¨ ITAK ˙ c TUB CYCLOIDAL NORMAL SUBGROUPS OF HECKE GROUPS OF FINITE INDEX ˙ Naci Cang¨ I. ul, Osman Bizim Abstract Cycloidal normal subgroups, that is, subgroups with only one cusp, of finite index of Hecke groups, which have been introduced by E. Hecke, and can be thought of as some kind of generalisation of the well-known modular group, are studied. It is shown that they correspond to cyclic quotients of Hecke groups and therefore have non-compact associated Riemann surfaces with a cusp. Finally, their total number have been formulated. 1. Introduction Let G be a Fuchsian group of the first kind. . let G be a subgroup of the group PSL(2,R) of orientation preserving isometries of the upper half plane. Let N be a normal subgroup of G having finite index µ. We know that N has a signature (g; m1 , . . . , mr ; t) as a Fuchsian group. Here g is a non-negative integer called the genus of the quotient surface on which N acts. m1 , . . . , mr are called the periods of N and are the orders of the finite ordered elements. Here each mk is an integer greater than 1. t is the parabolic class number of N which is the number of conjugacy classes of parabolic elements. In our case these are the elements of infinite order in N. When the quotient surface U /N is considered, the number t corresponds to so called cusps which can be thougt as the holes on the surface going towards infinity. Finally we define the level n of N to be the least positive integer so that T n N, where T is the parabolic generator having infinite order in the group G. It is known that these three numbers are connected to others by the relation 345 ¨ BIZ ˙ IM ˙ CANGUL, µ = . (1) Let Γ be the modular group of all linear fractional transformations Y (z) = az + b , cz + d (2) where a, b, c, d Z such that ad-bc=1. Γ is generated by R(z) = − 1 1 and S(z) = . z z+1 (3) Here R2 = S 3 =I. We take T=RS. Then T is of infinite order in Γ. Normal and .

• Toán học
• Cycloidal normal subgroups
• Hecke groups
• Well known modular group
• Associated Riemann
• Regular maps