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Continued Fractions Related to a Group of Linear Fractional Transformations by Bilal Demir is a Mathematics article available to read on EtoBox.

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## 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.

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Author

Bilal Demir

Publisher

Walter de Gruyter GmbH

Published

2023

Field

Mathematics (Physical Sciences)