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

Yang, Tian¹

¹ Department of Mathematics, Stanford University, Stanford, CA 94305, USA

Geometry & topology, Tome 20 (2016) no. 2, pp. 1213-1255.

Résumé

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 $\pm 1$ 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.

Détail

DOI : 10.2140/gt.2016.20.1213

Classification : 57M05
Keywords: mapping class group, character variety, type-preserving representations, lengths coordinates

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