Character varieties of random groups.

In this third lecture, I will talk about random groups and their linear representations. Introduced by Gromov in the 80s, random groups provide a wealth of examples of finitely presented groups. With O. Becker and P. Varju, we have undertaken the study of their character variety, that is the moduli space of their representations with values in  SL(2,C) or a more general semisimple algebraic group. I will show how new families of Zariski-dense rigid groups arise this way and what it tells us about the geometry of word maps. The method showcases a delicate interplay between characteristic zero and characteristic p and relies on GRH.