2.1 (Leitner) — Limits between Thurston geometries

Definitions and mathematical statement

The eight Thurston model geometries in dimension three are \[ S^3,\quad {\mathbb{R}}^3,\quad \mathbb H^3,\quad S^2\times{\mathbb{R}},\quad \mathbb H^2\times{\mathbb{R}},\quad \mathrm{Nil},\quad \mathrm{Sol},\quad \widetilde{\mathrm{SL}_2({\mathbb{R}})}. \] Here $\mathrm{Nil}$ is the three-dimensional Heisenberg geometry, $\mathrm{Sol}$ is the standard solvable three-dimensional geometry, and the last model is the universal-cover geometry associated with $\mathrm{SL}_2({\mathbb{R}})$. Each is understood with its Thurston geometric structure, not merely as an underlying topological space.

The source calls a geometric transition a continuous path of geometries that abruptly changes type at a limit. Schematically, write $\mathcal G\rightsquigarrow\mathcal H$ if a family of structures of type $\mathcal G$ for $t>0$ converges, in the chosen transition framework, to a structure of type $\mathcal H$ as $t\downarrow0$. This is a directed relation. The topology, ambient model, and permitted coordinate changes or rescalings are essential to its meaning and are not specified by the excerpt.

Problem (Leitner). Describe all geometric transitions between the eight Thurston geometries, including which directed limits $\mathcal G\rightsquigarrow\mathcal H$ occur and which cannot occur. More generally, develop criteria to determine when one geometry is a limit of another, and identify restrictions that any limiting geometry must satisfy. The request to understand all transitions may include more than the existence relation between model types; the equivalence convention for classifying distinct transition families is also unspecified.

The source stresses that constructing a transition proves existence, whereas proving that a limit cannot occur is generally harder. The registry summary reports projective-geometric results for limits of hyperbolic geometry: Euclidean, Nil, and Sol possibilities, and exclusions of $\mathbb H^2\times{\mathbb{R}}$ and $\widetilde{\mathrm{SL}_2({\mathbb{R}})}$. It also reports many hyperbolic/anti-de Sitter transitions, a related framework involving a geometry outside the eight listed above. These framework-dependent reports do not by themselves define the unrestricted limiting relation in the question.

Short English statement

Classify the possible transitions among Thurston's eight three-dimensional geometries and find general tests and obstructions for one geometry to arise as a limit of another. The notion of convergence and allowed rescaling must be fixed.

Sources

• Local UnsolvedMath registry, permanent ID 10900006, source code catalogue entry; attributed to Leitner, with no proposal year supplied.
• Delp, Hoffoss and Manning, Problems In Groups, Geometry, and Three-Manifolds (2015), subsection 2.1 (Leitner), PDF page 2; supplied locator https://arxiv.org/pdf/1512.04620#page=2, and https://arxiv.org/abs/1512.04620. The provenance records source-TeX extraction checked against rendered PDF text and access on 2026-07-29.
• The registry research summary reports projective-geometric limit classifications and hyperbolic/anti-de Sitter transitions without supplying individual primary references or full hypotheses.