On type-preserving representations of thrice punctured projective plane group

In the first part of the talk we will discuss famous topological and dynamical questions and conjectures about character varieties and the associated action of the mapping class group. In the second part of the talk we will discuss joint work with F. Palesi and T. Yang about type-preserving representations of the fundamental group of the three-holed projective plane N into PGL(2, R). First, we prove Kashaev’s conjecture on the number of connected components with non-maximal euler class. Second, we discuss Bowditch's question about which of these components have the following property: there is a a simple closed curve sent to a non-hyperbolic element. Finally, we show, in most cases, that the action of the pure mapping class group Mod(N) on these non-maximal components is ergodic, proving Goldman conjecture. Time permitting we will discuss a work in progress with Palesi where we expand these results to all five surfaces (orientable and non-orientable) of characteristic -2.