This talk is based on joint work with Masoud Kamgarpour and GyeongHyeon Nam. Character varieties are geometric objects associated to maps from a finitely generated group to a reductive group. Seminal work by Hausel and Rodriguez-Villegas studied topological properties of character varieties by counting points over finite fields. They considered twisted GL(n) character varieties of the fundamental group of certain surfaces. 

In this talk, we will discuss character varieties associated to an orientable surface with punctures, and a reductive group with connected centre. We give a sufficient condition for their smoothness. Assuming regular semisimple, and regular unipotent  monodromy, we give an explicit, polynomial point count. This allows us to determine the Euler characteristic and the number of connected components.