Continued fractions related to a group of linear fractional transformations

Abstract

There are strong relations between the theory of continued fractions and groups of linear fractional transformations. We consider the group G 3,3 generated by the linear fractional transformations a=1-1/z and b = z + 2 b=z+2. This group is the unique subgroup of the modular group PSL (2, Z)) 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.