We define domains of discontinuity for mapping class group actions on the character variety X=Hom(F_3, SL(2,C))//SL(2,C) of the free group F_3 of rank 3. In particular, we consider F_3 as the fundamental group of the four-holed sphere S, and of the three-holed projective plane N, and consider the associated mapping class group actions on X.
We describe the combinatorial view point we adopt using trace functions on simple closed curves, and we sketch the main ideas of the proof. Time permitting, we will also discuss the real case.
(This is joint work with F. Palesi and S. P. Tan.)