On type-preserving representations of the four-punctured sphere group

Abstract

We give counterexamples to a question of Bowditch that asks whether a nonelementary type-preserving representation $\rho :{\pi }_1\left({\Sigma }_{g,n}\right)\to PSL\left(2;ℝ\right)$ of a punctured surface group that sends every nonperipheral simple closed curve to a hyperbolic element must $\rho$ be Fuchsian. The counterexamples come from relative Euler class $\pm 1$ 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.