ON THE TRIANGLE GROUP A (2,3,9)
DOI:
https://doi.org/10.26692/surj-ss.v34i2.7241Keywords:
Triangle groups Coset diagrams Fragments, Projective general linear groups, Modular group, and Projective lines over Galois fieldsAbstract
The infinite triangle group A (2,3,9)=<x,y: x=y=(xy)=1> has a fundamental domain consisting of two copies of a hyperbolic triangle with angles??? 39 2 Let a, b, c be the vertices of the hyperbolic triangle and R, represent hyperbolic-reflection in hyperbolic sides ac, eb and ab. Let x= R, Randy = R, R,, so that yx = R.R,R, RRR. Then. R, R, is an anticlockwise hyperbolic-rotation or a about a, R, R, is an anticlockwise hyperbolic-rotation of 2 about b, and R, R, is an anticlockwise 3 hyperbolic-rotation of 2n about c. Hence, x²= y = (xy)" = 1 that A (2,3.9) acts on, is hyperbolic. 1 which is 4 (2, 3.9). The space It is known that conjugacy classes of actions of PGL (2. Z) on F, U (can be represented by coset diagrams D (0, q) where 8 €F, and q is a power of a prime p. In this paper we have obtained conditions on 6 and q which guarantee only those coset diagrams which depict homomorphic images of the infinite triangle group A (2, 3, 9) in PGL. (2. q.
