A study on trace functions of closed curves on projective orbifolds

Abstract

For the study about 3-manifolds, N. Dunfield and P. Thurston used trace functions of closed curves on orbifolds. An orbifold is locally modeled on $\mathbb{R}^n$ quotient by some finite group which acts on $\mathbb{R}^n$. An orbifold with an $(\mathbb{RP}^2,PGL(3, \mathbb{R}))$ -structure is called a projective orbifold. A trinion is a 2-orbifold which has a sphere as the underlying space with three cone-points. And this orbifold has $(\mathbb{RP}^2,PGL(3,\mathbb{R}))$ -structures. In this paper, there are some properties of extensions of trace functions to $\mathbb{R}^* \times \mathbb{R}^*$ about five closed curves on trinoins which have a negative Euler characteristic. A trace function of a closed curve on an orbifold is defined through an element of an deformatioin space of $\mathbb{RP}^2$-structures. So, basic definitions for an deformatioin space are introduced. In fact, for the work of N. Dunfield and P. Thurston, we need so many images of extensions of trace functions to $\mathbb{C}^* \times \mathbb{C}^*$.