Sala Riunioni (Dip. Matematica).
We give counterexamples to a conjecture of Bowditch that if a non-elementary type-preserving representation ρ : π1(Σg,n) → P SL(2; R) of a punctured surface group sends every non-peripheral simple closed curve to a hyperbolic element, then ρ must be Fuchsian. The counterexamples come from relative Euler class ±1 representations of the four-punctured sphere group. As a related result, we show that the mapping class group action on each non-extremal component of the character space of type-preserving representations of the four-punctured sphere group is ergodic, confirming a conjecture of Goldman in this case. The main tool we use is the lengths coordinates of the decorated character spaces defined by Kashaev. At the end of the talk, I will also mention a recent joint work with Sara Maloni and Frederic Palesi on representations of the three-punctured projective plane group.