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

**Geometry/Topology**

Speaker: | Tian Yang, Stanford |

Location: | 2112 MSB |

Start time: | Tue, Sep 29 2015, 1:10PM |

We give counterexamples to a conjecture of Bowditch that if a non-elementary type-preserving representation ρ of a punctured surface group into PSL(2,R) 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. The main tool we use is Penner’s lengths coordinates of the decorated character spaces defined by Kashaev.