Branched real projective structures on surfaces and geometrisation of representations

We introduce and study branched real projective structures on compact surfaces. Our main result shows that every representation of the fundamental group of a compact surface into $PSL(3,\mathbb{R})$ arises as the holonomy representation of a branched real projective structure, regardless of whether the surface is orientable. We also consider the realisation problem with prescribed branching data, relating the branching degree to the second Stiefel–Whitney class of the representation. The results obtained in this direction suggest several interesting avenues for future research.