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

Data Seminario: 
26-Jun-2019
Ora Inizio: 
15:00
Ora Fine: 
16:00
Tian Yang
Texas A&M University

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.