Branched real projective structures on surfaces and geometrisation of representations

Abstract: 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.