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

Yang, Tian

doi:10.2140/gt.2016.20.1213

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

Tian Yang

Geometry & Topology 20 (2016) 1213–1255

DOI: 10.2140/gt.2016.20.1213

Abstract

We give counterexamples to a question of Bowditch that asks whether a nonelementary type-preserving representation $ρ : π_{1} (Σ_{g, n}) \to PSL (2; ℝ)$ 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 $\pm 1$ 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.

To William Goldman on the occasion of his sixtieth birthday

Keywords

mapping class group, character variety, type-preserving representations, lengths coordinates

Mathematical Subject Classification 2010

Primary: 57M05

References

Publication

Received: 16 February 2015

Revised: 14 May 2015

Accepted: 4 July 2015

Published: 28 April 2016

Proposed: David Gabai

Seconded: Benson Farb, Gang Tian

Authors

Tian Yang
Department of Mathematics Stanford University Stanford, CA 94305 USA

Volume 20, issue 2 (2016)