An AF algebra associated with the Farey tessellation

To the Farey tessellation of the upper half-plane we associate an AF algebra encoding the cutting sequences that define vertical geodesics. The Effros-Shen AF algebras arise as quotients of our algebra. Using the path algebra model for AF algebras we construct for each $\tau \in (0,1/4\big]$, projections $(E_n)$ in this algebra such that $E_n E_{n\pm 1}E_n \leq \tau E_n$.