Closed-form characterization of the Sol cut locus

Characterize the cut locus of the three-dimensional Sol geometry in closed form, so that the generic local-identifiability analysis for the Sol latent-space network model can be stated beyond the connected component containing the numerical witness.

Background

The paper studies identifiability of latent node positions from pairwise distances in the anisotropic Thurston geometries Nil, Sol, and the universal cover of the unit tangent bundle of the hyperbolic plane. Its generic local-identifiability theorem applies on the open set of configurations for which all node pairs avoid one another’s cut loci and are joined by suitable minimizing geodesics.

For Nil and the twisted hyperbolic geometry, the authors describe the relevant cut-locus structure sufficiently to discuss the connected components of the regular domain. For Sol, however, the cut locus is not available in closed form. Consequently, the theorem’s genericity conclusion for Sol is restricted to the connected component containing the selected numerical rank witness. A closed-form characterization would clarify the global domain on which the distance map is analytic and would strengthen the scope of the identifiability result beyond the witness component.

References

The theorem is conditional on a full-rank witness, its genericity statement is confined to the connected component of $U$ containing the witness (in $Nil$ and $$ the cut locus of a point lies in its fibre, a set of codimension two, and coincidences have codimension three, so $U$ has the two components of $\Omega_I$ distinguished by the sign of $\tilde\zeta$, which are exchanged by $$ and have the same rank; in $Sol$ the cut locus is not known in closed form and the statement is confined to the component of the witness), and its conclusion is local: an immersion excludes nearby non-congruent alternatives, not distant ones, and Supplement~\ref{supp:exp} finds distant alternatives at $N_0$ and $N_0+1$ in $$.

— Identifiability of Latent Space Network Models on Anisotropic Thurston Geometries  (2609.09236 - Papamichalis et al., 8 Sep 2026) in Section 4, subsection “Non-rigidity of finite configurations,” paragraph immediately following Theorem 3 (Generic local identifiability of the gauge-fixed positions)