Volume 20, issue 2 (2016)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
On type-preserving representations of the four-punctured sphere group

Tian Yang

Geometry & Topology 20 (2016) 1213–1255
Abstract

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.

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