Hyperbolic-Group Boundaries Homeomorphic to Trees of $\mathbb{CP}^{2}$

Definitions and mathematical statement

Let $\mathbb{CP}^{2}$ be the complex projective plane, \[ \mathbb{CP}^{2}=({\mathbb{C}}^{3}\setminus\{0\})/\sim, \qquad z\sim w\iff w=\lambda z\text{ for some }\lambda\in{\mathbb{C}}\setminus\{0\}. \] A group $\Gamma$ is word-hyperbolic if it admits a finite generating set whose Cayley graph, with its graph metric, has $\delta$-thin geodesic triangles for some $\delta\geq 0$. Its Gromov boundary $\partial\Gamma$ is the space of equivalence classes of geodesic rays from a fixed vertex, where two rays are equivalent if their images have finite Hausdorff distance, equipped with the usual Gromov boundary topology.

Write $X(\mathbb{CP}^{2})$ for the tree of manifolds named in the supplied source record. The record does not reproduce its defining construction or fully specify its orientation conventions; this notation therefore retains its source-dependent meaning. The supplied background explicitly distinguishes this space from an analogous tree involving both orientations of $\mathbb{CP}^{2}$, so that analogous space must not be substituted for the target.

The question is whether there exist a finitely generated word-hyperbolic group $\Gamma$ and a homeomorphism \[ h\colon X(\mathbb{CP}^{2})\longrightarrow\partial\Gamma. \] A fully self-contained formulation still requires the precise definition of $X(\mathbb{CP}^{2})$ from the cited source.

Short English statement

Does some word-hyperbolic group have Gromov boundary homeomorphic to the particular tree of complex projective planes denoted by $X(\mathbb{CP}^{2})$ in the source? Its defining construction and orientation conventions require source verification. The supplied record labels this question open.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001467$, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record associates the problem with 2010 and supplies no named proposer.
• Non-positive Curvature and Geometric Structures in Group Theory (2010), Oberwolfach Reports, pp. 1165--1224, DOI 10.4171/owr/2010/20. Supplied source report and literature reference labeled "original OWR report": https://doi.org/10.4171/owr/2010/20.
• Supplied literature reference labeled "Swiatkowski, trees of manifolds": https://doi.org/10.2140/gt.2020.24.593.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports that the primary boundary-realization paper explicitly leaves $X(\mathbb{CP}^{2})$ as a question and discusses a cobordism obstruction, and reports realization of an analogous two-orientation tree rather than $X(\mathbb{CP}^{2})$ itself. This is supplied dated curation, not an independently verified literature assessment; its status and cited claims require independent verification.