Continued fractions related to a group of linear fractional transformations

Abstract

Abstract There are strong relations between the theory of continued fractions and groups of linear fractional transformations. We consider the group G 3 , 3 {G}_{3,3} generated by the linear fractional transformations a = 1 − 1 ∕ z a=1-1/z and b = z + 2 b=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 ∕ q p/q , to obtain an element in G 3 , 3 {G}_{3,3} with cusp point p ∕ q p/q . As a result, we say that the action of G 3 , 3 {G}_{3,3} on rational numbers is transitive.