We give counterexamples to a question of Bowditch that asks whether a nonelementary type-preserving representation of a punctured surface group that sends every nonperipheral simple closed curve to a hyperbolic element must be Fuchsian. The counterexamples come from relative Euler class representations of the four-punctured sphere group. We also show that the mapping class group action on each nonextremal component of the character space of type-preserving representations of the four-punctured sphere group is ergodic, which confirms a conjecture of Goldman for this case. The main tool we use are Kashaev and Penner’s lengths coordinates of the decorated character spaces.
Tian Yang. "On type-preserving representations of the four-punctured sphere group." Geom. Topol. 20 (2) 1213 - 1255, 2016. https://doi.org/10.2140/gt.2016.20.1213