Closed paths in the coset diagrams for y,t:y6=t6=1 acting on real quadratic fields

We are interested in linear-fractional transformations 6 y,t satisfying the relations y(6) = t(6) =1, with a view to studying ail action of the subgroup H = <y,t > on Q(rootn) boolean OR {infinity} by using coset diagrams. For a fixed non-square positive integer n, if an element alpha = a+rootn/c and its algebraic conjugate have different signs, then alpha is called an ambiguous number. They play an c important role in the study of action of the group H on Q(rootn) boolean OR {infinity}. In the action of H on Q(rootn) boolean OR {infinity}, Stab(alpha) (H) are the only non-trivial stabilizers and in the orbit alphaH; there is only one (up to isomorphism). We classify all the ambiguous numbers in the orbit and use this information to see whether the action is transitive or not.