Continued fractions related to a group of linear fractional transformations

There are strong relations between the theory of continued fractions and groups of linear fractional transformations. We consider the group G3,3 generated by the linear fractional transformations a = 1 - 1/z and b = z + 2. This group is the unique subgroup of the modular group PSL(2, iL ) with index 2. We calculate the cusp point of an element given as a word in generators. Conversely, we use the continued fraction expansion of a given rational number p/q, to obtain an element in G(3,3) with cusp point p/q. As a result, we say that the action of G(3,3) on rational numbers is transitive.