Random character varieties

Consider a random group $\Gamma$ with $k$ generators and $r$ random relators of large length $N$. We study the geometry of the character variety of $\Gamma$ with values in $\SL(2,\C)$ or more generally any semisimple Lie group $G$. This is the moduli space of group homomorphisms from $\Gamma$ to $G$ up to conjugation. We are in particular able to determine its dimension, number of components and Galois group, with an excellent control on the probability of exceptions. The proofs use effective Chebotarev type theorems as well as new spectral gap bounds  for Cayley graphs of finite simple groups. They are also conditional on GRH. Joint work with Peter Varju and Oren Becker.