Realisation of spherical and hyperbolic structures

Determine necessary and sufficient conditions for a representation into the spherical or hyperbolic subgeometry of real projective geometry to arise as the holonomy of a possibly branched structure in that subgeometry.

Background

The paper formulates a general geometrisation problem for subgeometries of real projective geometry: characterize representations into a subgroup G that occur as holonomies of possibly branched (G, X)-structures. Translation structures are described as fully characterized, whereas the corresponding spherical and hyperbolic cases are explicitly identified as unresolved.

References

Although the case of translation structures has been extensively studied and fully characterised, see , the cases of spherical and hyperbolic structures remain open.

— Branched real projective structures on surfaces and geometrisation of representations  (2609.18436 - Faraco et al., 16 Sep 2026) in Section 1, Subsection “Qualitative geometrisation and sub-geometries,” following Problem 1

The same statement is conjectured to hold for surfaces of any genus g\ge2.

— Branched real projective structures on surfaces and geometrisation of representations  (2609.18436 - Faraco et al., 16 Sep 2026) in Section 3, Subsection “Branched hyperbolic structures”