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November 2021 On type-preserving representations of thrice punctured projective plane group
Sara Maloni, Frédéric Palesi, Tian Yang
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J. Differential Geom. 119(3): 421-457 (November 2021). DOI: 10.4310/jdg/1635368618


In this paper we consider type-preserving representations of the fundamental group of the three-holed projective plane into the Projective Linear Group $\operatorname{PGL}(2,\mathbb{R})$ and study the connected components with non-maximal euler class. We show that in euler class zero for all such representations there is a simple closed curve which is non-hyperbolic, while in euler class $\pm 1$ we show that there are $6$ components where all the simple closed curves are sent to hyperbolic elements and $2$ components where there are simple closed curves sent to non-hyperbolic elements. This answers a question asked by Brian Bowditch. In addition, we show also that in most of these components the action of the mapping class group on these non-maximal component is ergodic. An important tool that we use in this work is an extension of Kashaev’s theory of decorated character varieties to the context of non-orientable surfaces.


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Sara Maloni. Frédéric Palesi. Tian Yang. "On type-preserving representations of thrice punctured projective plane group." J. Differential Geom. 119 (3) 421 - 457, November 2021.


Received: 16 September 2018; Accepted: 2 November 2019; Published: November 2021
First available in Project Euclid: 1 November 2021

Digital Object Identifier: 10.4310/jdg/1635368618

Rights: Copyright © 2021 Lehigh University


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Vol.119 • No. 3 • November 2021
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