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

TL;DR

This paper constructs counterexamples to a conjecture about type-preserving representations of the four-punctured sphere group and proves ergodicity of the mapping class group action on certain character space components.

Contribution

It provides the first counterexamples to Bowditch's question and confirms Goldman's ergodicity conjecture for the four-punctured sphere case.

Findings

Counterexamples to Bowditch's question using relative Euler class representations.

Ergodicity of the mapping class group action on non-extremal components of the character space.

Application of Kashaev-Penner length coordinates to analyze the character space.

Abstract

We give counterexamples to a question of Bowditch that if a non-elementary type-preserving representation $\rho:\pi_1(\Sigma_{g,n})\rightarrow PSL(2;\mathbb R)$ of a punctured surface group sends every non-peripheral simple closed curve to a hyperbolic element, then must $\rho$ be Fuchsian. The counterexamples come from relative Euler class $\pm1$ representations of the four-punctured sphere group. We also 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, which confirms a conjecture of Goldman for this case. The main tool we use is Kashaev-Penner's lengths coordinates of the decorated character spaces.