Groups and Dynamics Seminar

Date: December 8, 2016

Time: 3:30PM - 4:30PM

Location: BLOC 220

Speaker: Tian Yang, Stanford University

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

Abstract: We give counterexamples to a conjecture of Bowditch that if a non-elementary type-preserving representation ρ : π1(Σg,n) → PSL(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. The main tool we use is Penner’s lengths coordinates of the decorated character spaces defined by Kashaev.