On type-preserving representations of thrice punctured projective plane group

Abstract

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.