Syllabus:
This course is meant to progressively introduce the theory of Anosov representations, developped during the past fifteen years. We will be guided (...)
Read more
This course is meant to progressively introduce the theory of Anosov representations, developped during the past fifteen years. We will be guided by the following questions :
— When does a discrete subgroup Γ of a Lie group G admit defor- mations ? (i.e. when is there a non trivial one parameter family of morphisms ρt : Γ → G such that ρ0 is the inclusion ?)
— Which dynamical and geometric properties of the discrete group are preserved along such deformations ?
In order to illustrate the frequent rigidity phenomena in this context, we will start by proving a local rigidity theorem for lattices (Calabi—Weil’s theorem) and we will state other famous rigidity theorems (Mostow, Margulis. . . ).
As a counterpart to these rigidity results, we will then give many explicit examples of deformations of discrete groups. In particular, we will describe some aspects of character varieties of surface groups, which are a topic of active research.
Finally, we will inquire the geometric and dynamical properties of discrete groups that are preserved under deformation. This will lead us to introduce the notion of Anosov morphism, which gives a unified framework for the study of many deformations of geometric origin (quasi-Fuchsian representa- tions, divisible convex sets, globally hyperbolic anti-de Sitter varieties).
The topic of this course is closely related to that of the course « Variété des caractères et structures hyperboliques en dimension 3 », from the Master of Paris 6. Though the two courses are independent, it is worth following this course after the one of Antonin Guilloux.
Références
[1] Gérard Besson, Calabi–Weil rigidity, https ://www-fourier.ujf- grenoble.fr/sites/ifmaquette.ujf-grenoble.fr/files/Besson.pdf
[2] François Labourie, Anosov flows, surface groups and curves in projec- tive space, Inventiones Mathematicae, 165(1), 2006, p. 51-114.
[3] Olivier Guichard et Anna Wienhard, Anosov representations : do- mains of discontinuity and applications, Inventiones Mathematicae, 190(2), 2012, p. 357-438.