Continued fractions and the geometric decomposition of modular transformations

Abstract

It is known that the modular group Γ = PSL (2,Z) and the Farey tessellation F of the hyperbolic plane H by an ideal triangle Λ₀ = {0,1,∞} are connected by the relationship F = {g(I) : g ∈ Γ} where the Farey geodesic I represents the positive imaginary axis. We explore this relationship and note that each Farey geodesic in the Farey tessellation is uniquely associated with a pair of matrices in PSL (2,Z). By imposing an orientation on the Farey geodesics it is possible to identify each oriented geodesic uniquely with a matrix in