Generic Simplicity of Hyperbolic Dirichlet Tilings

Definitions and mathematical statement

Let $\mathbb H^3$ be hyperbolic three-space with distance $d$, and let $\Gamma<\operatorname{PO}(3,1)$ be a discrete torsion-free subgroup acting by hyperbolic isometries. For $x\in\mathbb H^3$ and $\gamma\in\Gamma$, define the closed Dirichlet cell \[ P_\gamma(x)=\{z\in\mathbb H^3:d(z,\gamma x)\le d(z,\eta x)\text{ for every }\eta\in\Gamma\}. \] The Dirichlet tiling $D_x$ consists of these cells and their faces. For a face $F$ of this tiling, define its number of incident three-dimensional cells by \[ n_x(F)=\#\{\gamma\in\Gamma:F\subseteq P_\gamma(x)\}. \] The supplied question asks whether, for every such $\Gamma$, the tiling $D_x$ is simple for a generic base point $x\in\mathbb H^3$. If the source's word "facets" denotes the maximal three-dimensional cells of the tiling, its stated incidence requirements become \[ \begin{aligned} n_x(e)&=3 &&\text{for every one-dimensional face }e\text{ of }D_x,\\ n_x(v)&=4 &&\text{for every zero-dimensional face }v\text{ of }D_x. \end{aligned} \] Here the faces under consideration lie in $\mathbb H^3$; no compactification or conditions at ideal vertices are supplied.

Two qualifications remain unresolved. First, the source does not specify the meaning of "generic," so it does not determine whether the exceptional base points must have measure zero, form a meagre set, or satisfy another condition. Second, "facet" can instead mean a two-dimensional face of a three-dimensional cell. Counting those faces is different from counting incident three-dimensional cells. The displayed formulation is therefore conditional on the maximal-cell interpretation; the supplied wording alone does not select a fully precise formulation.

Short English statement

For every discrete torsion-free group of hyperbolic three-space isometries, ask whether a generic base point produces a Dirichlet tiling with three tiles incident to every edge and four tiles incident to every vertex. This interpretation requires clarification of the source's word "facets" and of "generic." The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002527$, title "Generic Simplicity of Hyperbolic Dirichlet Tilings," source problem code catalogue entry, in the collection "Oberwolfach Reports Open Problems."
• Supplied source citation: Mini-Workshop: Kähler Groups (2014), Oberwolfach Reports, pp. 531--553, DOI 10.4171/owr/2014/09. https://doi.org/10.4171/owr/2014/09. The record attributes the problem to this workshop report and supplies no individual proposer.
• Supplied literature reference: "Kapovich, 2013." https://doi.org/10.1007/s00222-013-0453-4. No article title or specific theorem supporting the assessment is supplied.
• Supplied literature reference: "Oberwolfach Report 9, 2014." https://doi.org/10.4171/owr/2014/09.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the unrestricted generic simplicity conjecture open and reports only partial results under extra hypotheses. This is supplied dated curation, not an independently verified literature assessment; the record explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied record labels its statement corrected_verified, explaining that mathematical content was preserved while notation, typography, or omitted definitions were repaired. This is a registry qualification, not mathematical verification of the present draft.