Beatrice POZZETTI — Higher rank Teichmüller theories

Let $\Gamma$ be the fundamental group of a compact surface $S$ with negative Euler characteristic, and $G$ denote $PSL(2,R)$, the group of isometries of the hyperbolic plane. Goldman observed that the Teichmüller space, the parameter space of marked complex structures on $S$ can be identified with a connected component of the character variety $Hom(\Gamma ,G)/G$, which can be selected by means of a characteristic invariant. Thanks to the work of Labourie, Burger-Iozzi-Wienhard, Fock-Goncharov, Guichard-Wienhard we now know that, surprisingly, this is a much more general phenomenon: there are many higher rank semisimple Lie groups $G$ admitting components of the character variety only consisting of injective homomorphisms with discrete image, the socalled higher Teichmüller theories. The richness of these theories is partially due to the fact that, as for the Teichmüller space, truly different techniques can be used to study them: bounded cohomology, Higgs bundles, positivity, harmonic maps, incidence structures, geodesic currents, real algebraic geometry... In my talk I will overview a number of recent results in the field (following Labourie, Burger-Iozzi-Wienhard, Fock-Goncharov, Guichard-Wienhard, Bonahon-Dreyer, Li, Zhang, Martone-Zhang, Bargaglia, Alessandrini-Li, Collier-Tholozan-Toulisse.)