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 G3,3{G}_{3,3} generated by the linear fractional transformations a=1−1∕za=1-1/z and b=z+2b=z+2. This group is the unique subgroup of the modular group PSL(2,Z){\rm{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∕qp/q, to obtain an element in G3,3{G}_{3,3} with cusp point p∕qp/q. As a result, we say that the action of G3,3{G}_{3,3} on rational numbers is transitive.

Keywords

• continued fractions
• modular group
• cusp points
• 11a55
• 20h10