Real two-dimensional representations of the free product of two finite cyclic groups

Extract

Let Λ be the group of all real non-singular 2 × 2 matrices and let Ω be the group of all real 2 × 2 matrices with determinant 1 in which a matrix is identified with its negative. Then Ω is isomorphic to the group of all real linear fractional transformations of determinant 1. In a previous paper ((3)) the authors determined all faithful representations of the modular group (more generally of the Hecke groups) by a discrete subgroup of Ω, in which the representations were partitioned into conjugacy classes over Λ. In this paper we consider the question for the more general situation of the free product of any two cyclic groups of finite order. Our results parallel the results of (3) quite closely, but some significant differences in the details of the proofs arise. In particular Theorem 3 of section 3 below, which is purely group-theoretic, is of independent interest and should prove useful in other investigations.