Geometry & Topology

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

Tian Yang

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We give counterexamples to a question of Bowditch that asks whether a nonelementary type-preserving representation ρ: π1(Σg,n) PSL(2; ) of a punctured surface group that sends every nonperipheral simple closed curve to a hyperbolic element must ρ 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.

Article information

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M05: Fundamental group, presentations, free differential calculus

mapping class group character variety type-preserving representations lengths coordinates


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.

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