Farey Symbols for the Picard Group
Abstract
We develop Picard Farey symbols for finite-index subgroups of G=PSL2(ZZ[i]) from the Gaussian Farey tessellation of HH3 by ideal octahedra. The ambient geometry naturally involves two different index-two extensions of G: the orientation-preserving group PGL2(ZZ[i]) and a reflection extension. This distinction explains the two G-orbits of Farey faces, which merge into one PGL2-orbit, and the classical extended modular tessellation appears as a totally geodesic slice. A Picard Farey symbol records finitely many Gaussian-rational octahedral occurrences, their local stabilizers, and cooriented face-pairing transformations. We prove that such a symbol reconstructs the finite Picard-cell action and hence the subgroup; moreover, when subgroup membership is decidable, the symbol can be found by a terminating finite double-coset procedure. For torsion-free subgroups we obtain whole-octahedron fundamental domains. The marked symbol also supplies finite orbit relations for the established Gaussian Steinberg edge--face presentation, with its corresponding Bianchi modular-symbol relations. When the natural modular-to-Picard coset map is bijective, a side-connected Kulkarni special polygon lifts, using the same modular representatives, to a Picard fundamental polyhedron whose intersection with the invariant real plane is exactly that polygon; the level-(2+i)/level-$5$ family is worked out explicitly. Comparison examples show that octahedral boundary minimization, absolute geometric face minimization, and preservation of such a modular section are distinct problems. Canonical reduction and effective Hecke reduction remain open.
Related
- Reversibility and its asymptotic counting in Picard group
- The derived Picard group of Fukaya categories of closed surfaces
- Equivariant compactifications of a unipotent group by a smooth projective horospherical variety of Picard number one
- Joint partial equidistribution of Farey rays in negatively curved manifolds and trees
- Ampleness in the Farey graph
BibTeX
@misc{tiwari2026farey,
title = {{Farey Symbols for the Picard Group}},
author = {Devendra Tiwari and Helena Verrill},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.36629},
eprint = {2609.36629},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36629},
}
RIS
TY - UNPB
TI - Farey Symbols for the Picard Group
AU - Devendra Tiwari
AU - Helena Verrill
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.36629
UR - https://arxiv.org/abs/2609.36629
AB - We develop Picard Farey symbols for finite-index subgroups of G=PSL_2(ZZ[i]) from the Gaussian Farey tessellation of HH^3 by ideal octahedra. The ambient geometry naturally involves two different index-two extensions of G: the orientation-preserving group PGL_2(ZZ[i]) and a reflection extension. This distinction explains the two G-orbits of Farey faces, which merge into one PGL_2-orbit, and the classical extended modular tessellation appears as a totally geodesic slice. A Picard Farey symbol records finitely many Gaussian-rational octahedral occurrences, their local stabilizers, and cooriented face-pairing transformations. We prove that such a symbol reconstructs the finite Picard-cell action and hence the subgroup; moreover, when subgroup membership is decidable, the symbol can be found by a terminating finite double-coset procedure. For torsion-free subgroups we obtain whole-octahedron fundamental domains. The marked symbol also supplies finite orbit relations for the established Gaussian Steinberg edge--face presentation, with its corresponding Bianchi modular-symbol relations. When the natural modular-to-Picard coset map is bijective, a side-connected Kulkarni special polygon lifts, using the same modular representatives, to a Picard fundamental polyhedron whose intersection with the invariant real plane is exactly that polygon; the level-(2+i)/level-$5$ family is worked out explicitly. Comparison examples show that octahedral boundary minimization, absolute geometric face minimization, and preservation of such a modular section are distinct problems. Canonical reduction and effective Hecke reduction remain open.
ER -