On word structure of the modular group over finite and real quadratic fields Original Research Article

Let Ω denote the projective line over the real quadratic field and δ denote the projective line over the finite field Fq with q elements. Coset diagrams for the orbits of the modular group G acting on Ω and δ give some interesting information. By using these diagrams we determine a condition for the existence of an orbit of G on Ω containing a circuit of a given type. If such a circuit exists, we find a condition under which the orbit contains a real quadratic irrational number α along with its algebraic conjugate ᾱ. As there are two projections from Ω to δ we are interested in the case when G acts on δ and we determine necessary and sufficient conditions for the existence of two orbits of G: one containing α along with 1/α and the other containing α together with 1/ᾱ.