Farey Symbols for the Picard Group

Source abstract

We introduce Picard Farey symbols for finite-index subgroups of \(G=\PSL_2(\ZZ[i])\), using the Gaussian Farey tessellation of \(\HH^3\) by ideal octahedra. A symbol records finitely many Gaussian-rational octahedral occurrences, their local stabilizers, and ordered cooriented face pairings. We prove that a valid marked symbol reconstructs the finite Picard-cell action and hence the subgroup, and we work out a range of exact examples. For a torsion-free subgroup of index $n$, every fundamental domain formed from complete Farey octahedra contains $n/12$ octahedra; a spanning-tree construction gives one with at most $n/4+1$ side-pairing transformations. We also identify the Gaussian octahedral edge--face complex with the integral rank-two Steinberg presentation for \(\QQ(i)\). Thus the same decorated geometry directly determines a finite presentation of coefficient-valued Bianchi modular symbols; for $Γ_1(2+i)$ we give the complete $3\times4$ group-ring boundary matrix. Canonical reduction and an effective Hecke-compatible reduction remain open.