Character variety and Deformation space of a Hyperbolic manifold

Character variety of a hyperbolic manifold plays an important role in the study of deformation space of the manifold. In fact, by a result of Weil, 1960, we know that locally the deformation space of hyperbolic structures of a manifold can be identified with the character variety of its fundamental group up to an equivalence. 

In this talk, I will first introduce the notions of hyperbolic structure, deformation space and character variety. I will go over some interesting results about the smoothness of the representation varieties. I will then outline a proof of Weil's theorem to illustrate how we can interpret small deformations of a hyperbolic structure in terms of deformations of its holonomy representation.