Quadratic Diophantine equations and two generator Möbius groups

Abstract

In this paper, we study the set of rational μ in (-2, 2) for which the group Gμ generated by A = (1 μ 0 1), B = (1 0 μ 1) is not free by using quadratic Diophantine equations of the form ax2 - by2 = ±1. We give a new set of accumulation points for rational values of μ in (-2, 2) for which Gμ is not free, thereby extending the results of Beardon where he showed that 1/√N are accumulation points, where N is an integer which is not a perfect square. In particular, we exhibit an infinite set of accumulation points for μ between 1 and 2 including the point 1.