Positivity, Cross-ratios and the Collar Lemma

par François Labourie (Université Côte d'Azur)

lundi 9 oct. 2023, 14:00 → 15:15 Europe/Paris

Amphithéâtre Léon Motchane (IHES)

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.