Title: Mapping class group action on character varieties and the ergodicity

Abstract: Character varieties of a surface are central objects in several beaches of math-
ematics, such as low dimensional topology, algebraic geometry, differential geom-
etry and mathematical physics. On the character varieties, there is a tautological
action of the mapping class group – the group of symmetries of the surface, which is expected to be ergodic in certain cases. In this talk, I will review related results
toward proving the ergodicity and introduce two long standing and related conjectures: Goldman’s Conjecture and Bowditch’s Conjecture. It is shown by Marche and Wolff that the two conjectures are equivalent for closed surfaces. For punctured surfaces, we disprove Bowditch’s Conjecture by giving counterexamples, yet prove that Goldman’s Conjecture is still true in this case.