Modular group acting on real quadratic fields

Abstract

Coset diagrams for the orbit of the modular group G = 〈x, y: x2 = y3 = 1〉 acting on real quadratic fields give some interesting information. By using these coset diagrams, we show that for a fixed value of n, a non-square positive integer, there are only a finite number of real quadratic irrational numbers of the form , where θ and its algebraic conjugate have different signs, and that part of the coset diagram containing such numbers forms a single circuit (closed path) and it is the only circuit in the orbit of θ.