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

MSC Classifications:

46L05, 11A55, 11B57, 46L55, 37E05, 82B20 show english descriptions General theory of $C^*$-algebras
Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15]
Farey sequences; the sequences ${1^k, 2^k, \cdots}$
Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
Maps of the interval (piecewise continuous, continuous, smooth)
Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 46L05 - General theory of $C^*$-algebras
11A55 - Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15]
11B57 - Farey sequences; the sequences ${1^k, 2^k, \cdots}$
46L55 - Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
37E05 - Maps of the interval (piecewise continuous, continuous, smooth)
82B20 - Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

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