In this paper we consider type-preserving representations of the fundamental group of the three--holed projective plane into $\mathrm{PGL}(2, \R) =\mathrm{Isom}(\HH^2)$ and study the connected components with non-maximal euler class. We show that in euler class zero for all such representations there is a one 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 answer 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. In this work, we use an extension of Kashaev's theory of decorated character varieties to the context of non-orientable surfaces.

中文翻译：

---

三次穿孔射影平面群的保型表示

在本文中，我们考虑将三孔射影平面的基本群的类型保持表示为 $\mathrm{PGL}(2, \R) =\mathrm{Isom}(\HH^2)$ 并研究具有非最大欧拉类的连通分量。我们证明，对于所有此类表示，在欧拉零类中，有一条简单的非双曲线闭合曲线，而在欧拉类 $\pm 1$ 中，我们展示了所有简单闭合曲线都发送到的 $6$ 组件双曲线元素和 $2$ 组件，其中有简单的闭合曲线发送到非双曲线元素。这回答了 Brian Bowditch 提出的问题。此外，我们还表明，在大多数这些组件中，映射类组对这些非最大组件的作用是遍历的。在这项工作中，我们使用 Kashaev 的扩展