Non-Hitchin representations in prescribed strata

Determine necessary and sufficient conditions for a non-Hitchin representation of a closed surface group into the projective group to be realized as the holonomy of a branched real projective structure in a prescribed stratum \(\mathcal{RP}^2(\mu)\), assuming \(\mathrm{w}_2(\rho)\equiv\deg(\mu)\pmod 2\).

Background

The paper proves that the parity relation between the second Stiefel–Whitney class of a representation and the total branching degree is necessary for realization in a prescribed stratum. It also gives a complete realization result for Hitchin representations of even total degree, but explicitly states that no complete characterization is known for non-Hitchin representations.

References

There is no complete characterisation for non-Hitchin representations, and the following question remains to be addressed.

— Branched real projective structures on surfaces and geometrisation of representations  (2609.18436 - Faraco et al., 16 Sep 2026) in Section 1, Subsection “Strata of branched projective structures,” Problem \ref{prob:geometrisation strata}

For non-Hitchin representations, however, the minimal degree required for geometrisation remains an open question:

— Branched real projective structures on surfaces and geometrisation of representations  (2609.18436 - Faraco et al., 16 Sep 2026) in Section 1, Subsection “Strata of branched projective structures,” immediately before Problem \ref{prob:min degree}

In the case of the trivial representation, the geometrisation problem thus reduces directly to the realisability of branched coverings over 2 with a prescribed branching data (or branch profile), a fundamental question in topological surface theory that, in general, remains an open problem.

— Branched real projective structures on surfaces and geometrisation of representations  (2609.18436 - Faraco et al., 16 Sep 2026) in Section 1, Subsection “Uniform lower bound for the minimal degree,” Problem following the discussion of trivial representations