Let G be a complex semisimple Lie group, and let M be a topological space. The **G-character variety of S****,** roughly speaking, is the space of conjugacy classes of representations from pi_1(M) to G. In the case where M is a surface, we get an action of Out(pi_1(M)) on the character variety. We will survey various results about this action, with the following motivating question: to what extent does Out(pi_1(M)) act properly discontinuously on the character variety? This talk will involve some topology, some algebra, and some analysis, which will likely either satisfy or upset the entire audience.

### Graduate Student Geometry-Topology Seminar

Monday, September 12, 2022 - 4:30pm

#### Yi Wang

UPenn