Homomorphic images of △(2,3,11)

Abstract

It is known that each conjugacy class of actions of P GL(2, Z) on F q ∪ {∞} can be represented by a coset diagram D( , q), where ∈ F q and q is a power of a prime p. In this paper, we are interested in parametrizing the conjugacy classes of actions of the infinite triangle group $(2, 3, 11) = x, y :depicting the conjugacy class of actions of $(2, 3, 11) on F q ∪{∞}. We have obtained conditions on and q which guarantee only those coset diagrams which depict homomorphic images of $(2, 3, 11) in P GL (2, q). We are interested in finding also when the coset diagrams for the actions of P GL(2, Z) on F q ∪ {∞} contain vertices on the vertical line of symmetry. It will enable us to show that for infinitely many values of q, the group P GL(2, q) has minimal genus, while also for infinitely many q, the group P SL(2, q) is an H * -group.