Pleating-Ray Tilings of the Representation-Space Complement

Definitions and mathematical statement

Put ${\mathbb{C}}^*={\mathbb{C}}\setminus\{0\}$. For $\omega\in{\mathbb{C}}^*$, consider the ordered pair of generators \[ A=\begin{pmatrix}1&1\\0&1\end{pmatrix},\qquad B_\omega=\begin{pmatrix}1&0\\\omega&1\end{pmatrix}. \] Let $\mathcal R\subseteq{\mathbb{C}}^*$ denote the Riley slice associated with these marked generators, as designated in the supplied source, and set \[ X={\mathbb{C}}^*\setminus\overline{\mathcal R}. \] The closure may be taken in ${\mathbb{C}}$ or relatively in ${\mathbb{C}}^*$, since these give the same set $X$. The index set ${\mathbb{Q}}/2{\mathbb{Z}}$ identifies rational numbers whose difference is an even integer.

Conjecture, as supplied. There exists a natural tiling of $X$ by quadrilaterals $\mathcal R_c(r)$, indexed by every $r\in{\mathbb{Q}}/2{\mathbb{Z}}$, such that each $\mathcal R_c(r)$ contains the extended pleating ray of slope $r$ as a diagonal. Each tile arises from restrictions of holonomy representations of hyperbolic Dehn fillings of \[ S^3\setminus\left(K(r^*)\cup\alpha_r\right). \] Here the hyperbolic qualification is retained from the supplied original formulation.

Unresolved specifications. The supplied record does not give a membership definition of the Riley slice, including the precise representation and marking conventions. It also does not define the extended pleating rays, the operation $r\mapsto r^*$, the objects $K(r^*)$ and $\alpha_r$, or the subscript $c$. The permitted Dehn fillings, the groups to which holonomy representations are restricted, and the identification of those restrictions with parameters $\omega$ are unspecified. Finally, the meanings of natural, quadrilateral, diagonal, and tiling, including boundary membership and allowed intersections, are not specified. Consequently this is the source's conjectural assertion with its unresolved terminology preserved; the supplied record alone does not determine a fully specified tiling theorem.

Short English statement

The supplied conjecture asks whether the complement of the closed Riley slice in ${\mathbb{C}}^*$ has quadrilateral tiles indexed by rational slopes modulo $2$, with extended pleating rays as diagonals and a construction using restricted hyperbolic Dehn-filling holonomy representations. Several objects and boundary conventions remain undefined in the supplied record, so a precise final formulation requires mathematical source review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004414$, source problem code catalogue entry, titled "Pleating-Ray Tilings of the Representation-Space Complement," in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2020; no individual proposer is supplied.
• Supplied report citation: Low-dimensional Topology (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/8, pp. 465--516. https://doi.org/10.4171/owr/2020/8. The background describes these reports as records of mathematical workshops and their open-problem discussions. The supplied literature-source label for this same link is "Oberwolfach Report 8/2020." The differing year labels are retained without reconciliation.
• Supplied literature reference, labeled "Jørgensen and others, Riley-slice pleating": https://arxiv.org/abs/2111.03230. The record supplies no fuller bibliographic attribution for this reference.
• Literature assessment dated exactly 2026-08-22: the supplied status is open, and the supplied assessment states that Riley-slice work captures neighborhoods of rational pleating rays and refines the exterior, but that the proposed global quadrilateral tiling has not been proved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked, with the Riley slice, matrix generators, indexing, and Dehn-filling source replacing the original phrase "the indicated representations." This verification claim is supplied provenance; no independent source visit or verification is asserted here.