#### Volume 20, issue 2 (2016)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
On type-preserving representations of the four-punctured sphere group

### Tian Yang

Geometry & Topology 20 (2016) 1213–1255
##### 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 $±1$ 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.

 To William Goldman on the occasion of his sixtieth birthday
##### Keywords
mapping class group, character variety, type-preserving representations, lengths coordinates
Primary: 57M05