PreprintCC BY-NC-ND 4.0arXiv
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,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.
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BibTeX
@misc{faraco2026branched,
title = {{Branched real projective structures on surfaces and geometrisation of representations}},
author = {Gianluca Faraco and Nicholas Rungi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18436},
eprint = {2609.18436},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18436},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Branched real projective structures on surfaces and geometrisation of representations
AU - Gianluca Faraco
AU - Nicholas Rungi
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18436
UR - https://arxiv.org/abs/2609.18436
AB - 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,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.
ER -