Geometry & Topology
- Geom. Topol.
- Volume 20, Number 2 (2016), 1213-1255.
On type-preserving representations of the four-punctured sphere group
We give counterexamples to a question of Bowditch that asks whether a nonelementary type-preserving representation of a punctured surface group that sends every nonperipheral simple closed curve to a hyperbolic element must be Fuchsian. The counterexamples come from relative Euler class 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.
Geom. Topol., Volume 20, Number 2 (2016), 1213-1255.
Received: 16 February 2015
Revised: 14 May 2015
Accepted: 4 July 2015
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 57M05: Fundamental group, presentations, free differential calculus
Yang, Tian. On type-preserving representations of the four-punctured sphere group. Geom. Topol. 20 (2016), no. 2, 1213--1255. doi:10.2140/gt.2016.20.1213. https://projecteuclid.org/euclid.gt/1510858977