PARAMETRIZATION OF CERTAIN GROUP ACTIONS

Abstract

We have investigated properties of c;:erlain groups by tooking al their actions on suitable spaces. These actions are studied by using a graphical technique now known as coset diagrams for the group c3 ,} (2, Z) . We have used these diagrams to establish a relat ionship between real and imaginary quadratic irrational numbers and the elements of the group. The aim of this research has been to study act ions of the group generated by the linear-fractional transformations ll: z --t z - I and z v: z ~ 2, which sati sfy the re lations u3 = \/3 = I on the projective line z+ 1 over the real, imaginary quadratic field and the finite field. We have shown that the coset diagram for the actions of G 3.3(2, Z) on the rational proj ect ive Ii.ne is connected and the action is transitive. Usi ng this we have shown that uJ = vJ = I are defining relations for the group, We have found out that if a is any real quadratic irrational number then the ambiguous numbers form a closed path in the coset diagram for the orbit aGo ),3 (2, Z) and it is the only closed path contained in it. Next we have parametrized the act ions of the group G'3,J(2,Z) = < II, II,' : u3 = 1/3 = ,2 = (u1)2 = (vl) l = I > on the projective line over the fi nite field F" , That is, each conj ugacy class of actions of C' 3,3 (2, Z) on PL(F,,) can be represented by a coset diagram D(8,q), where 8 E F" and q is a prime power. In particular, we have associated each conjugacy class of actions of the infin ite triangle groups b.(3,3,k) on PL(P,, ) with a coset diagram D(B,q).