An Af Algebra Associated with the Farey Tesselation

To the Farey tesselation of the upper half-plane we associate an AF algebra A encoding the ‘cutting sequences’ that define vertical geodesics. The Effros-Shen AF algebras arise as quotients of A. We define a unital completely positive map E : A → A and an endomorphism ρ : A→ A such that Eρ = idA , and which can be regarded as noncommutative analogs of the Farey map F : [0, 1] → [0, 1] acting on the continued fraction decomposition of irrational numbers as F ([a1, a2, . . .]) = [a1 − 1, a2, . . .], and of its two left inverses. We also construct, for each τ ∈ ( 0, 1 4 ] , projections (En)n in A such that EnEn±1En ≤ τEn. Introduction The semigroup S generated by the matrices A = [ 1 0 1 1 ] and B = [ 1 1 0 1 ] is isomorphic to F + 2 , the free semigroup on two generators. A geometric representation of this fact, which is intimately connected to the continued fraction algorithm, is provided by the Farey tesselation {gG : g ∈ S} of H depicted in Figure 1, where G = { 0 ≤ Rz ≤ 1, ∣z − 12 ∣∣ ≥ 1 2 } (cf., e.g., [20]).