Positivity, Cross-ratios and the Collar Lemma

I will explain why positive representations of fundamental groups of surfaces satisfy a « collar lemma » similar to the classical collar lemma for hyperbolic geometry, and have associated positive cross-ratios. As a consequence, I will deduce that positive representations form closed subsets of the representation variety. I will spend some time recalling what a positive representation is, what the associated cross-ratios are, and explain the main new object that we shall use and that we call « photons ». This is joint work with Jonas Beyrer, Olivier Guichard, Beatrice Pozzetti and Anna Wienhard.