Geometry of Continued Fractions — Further open questions

Definitions and mathematical statement

In the upper half-plane model of $\mathbb H^2$, the Farey tessellation is the ideal-triangle tessellation with vertices ${\mathbb{Q}}\cup\{\infty\}$ at infinity. Reduced fractions $p/q$ and $r/s$, with $\infty=1/0$, are joined by a hyperbolic geodesic precisely when \[ |ps-qr|=1. \] The tessellation is invariant under $\operatorname{PSL}_2({\mathbb{Z}})$.

Karpenkov's Problem 19 asks for a natural generalization of this tessellation to higher-dimensional hyperbolic geometry, that is, constructions in $\mathbb H^d$ for $d\geq3$ together with an explanation of how they generalize the Farey construction. This is a construction and comparison problem rather than a fully quantified conjecture. The source does not specify which features are required: ideal vertices, a rational or arithmetic boundary structure, a determinant-type incidence criterion, a symmetry group, a tessellation by simplices versus other cells, or compatibility across dimensions. These possibilities identify missing choices; none is imposed here as an additional source hypothesis.

The supplied research summary reports partial progress through Farey-type tessellations of $\mathbb H^3$, arithmetic hyperbolic tessellations, and relations between Klein sails and cone-Hilbert geometry. It says no canonical universally agreed generalization in all dimensions has been established. No explicit construction, reference, or uniqueness criterion is supplied for these claims. Consequently "natural" and "canonical" cannot presently be converted into a single existence or uniqueness assertion from this record alone.

Short English statement

The task is to extend the arithmetic ideal-triangle Farey tessellation of the hyperbolic plane to higher-dimensional hyperbolic spaces. Several related constructions are reported, but the source does not specify which Farey features a satisfactory generalization must preserve.

Sources

• Local UnsolvedMath registry, permanent ID 1900019, exact source code catalogue entry, "Geometry of Continued Fractions --- Further open questions"; status partially solved, classification PARTIAL-PROGRESS.
• Karpenkov, Open problems in geometry of continued fractions (2017), Problem 19, https://arxiv.org/abs/1712.01450. The record identifies extraction from public arXiv source TeX, accessed July 29, 2026, and says the source presented the item as open.
• The supplied research summary mentions Farey-type tessellations of $\mathbb H^3$, arithmetic hyperbolic tessellations, and Klein-sail/cone-Hilbert connections as partial responses. It gives no named author, explicit construction, or authoritative citation for these claims, and its assessment about a universally agreed all-dimensional construction is not a precise theorem.