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Non-differentiability of limit sets in anti-de Sitter geometry
Amphithéâtre Léon Motchane (IHES)
Amphithéâtre Léon Motchane
Le Bois Marie
35, route de Chartres
The study of Anosov representations deals with discrete subgroups of Lie groups that have a nice limit set, meaning that they share the dynamical properties of limit sets in hyperbolic geometry. However, the geometry of these limits sets is different: while limit sets in hyperbolic geometry have a fractal nature (e.g. non-integer Hausdorff dimension), some Anosov groups have a more regular limit set (e.g. $C^1$ for Hitchin representations).
My talk will focus on quasi-Fuchsian subgroups of SO(n,2), and show that the situation is intermediate: their limit sets are Lipschitz submanifolds, but not $C^1$. I will discuss the two main steps of the proof. The first one classifies the possible Zariski closures of such groups. The second uses anti-de Sitter geometry in order to determine the limit cone of such a group with a $C^1$ limit set.