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 $b=z+2$. This group is the unique subgroup of the modular group $\mathrm{PSL}(2,\mathbb{Z})$ 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.