Geometric properties of maximal representations in SO(2, 3)
Nicolas Tholozan (École Normale Supérieure, Paris)
Aula Beltrami - Tuesday, October 25, 2016 h.15:00
Abstract. The Lie group SO(2, 3) is a Hermitian Lie group, and one can thus define the Toledo invariant of a representation of the fundamental group of a closed surface of genus g in this Lie group. In this talk, I will describe geometric properties of the action of maximal representations (i.e. representations whose Toledo invariant is 4g−4) on the pseudo-Riemannian symmetric space H2,2 = SO(2, 3)/SO(2, 2) and its boundary, the Einstein space Ein2,1. In particular, I will show that these representations preserve a unique maximal space-like surface in H2,2, which gives an alternate proof of a recent theorem of Labourie and Collier.
The results I will present use an interesting interplay between the theory of Anosov representations and the theory of Higgs bundles. It is a joint work in preparation with Jérémy Toulisse.
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