Arithmetic and Topology of the Hexponential map

The modular group PSL₂(Z) acts on the upper-half plane HP with quotient the modular orbifold, uniformized by the function j HP→ C. We first show that second derived subgroup PSL₂(Z)'' corresponds to a Z² Z/6 Galois cover of the modular orbifold by a hexpunctured plane, uniformized by the hexponential map hexp HP → C (ω₀Z[j]), which is a primitive of Cη⁴ where ω₀∈ iR and C∈ R are explicit constants and η is Dedekind eta function. We describe the values of the cusp-compactification ∂ hexp QP¹→ ω₀ Z[j]. After defining the radial-compactification Shexp R → R/(2), we construct a simple section InSh R/(2) → S PSL₂(Z)' where S ⊂ RP¹ is a set of numbers whose continued fraction expansions arise from Sturmian sequences, which contains the set M of Markov quadratic irrationals as those numbers arising from periodic Sturmian sequences. Finally we show that the values of InSh are either Markov quadratic irrationals or transcendent.