Journal of the
Korean Mathematical Society JKMS

The purpose of this paper is to find expressions of the Fricke spaces of some basic surfaces which are a three-holed sphere $\Sigma(0,3)$, a one-holed torus $\Sigma(1,1)$, and a four-holed sphere $\Sigma(0,4)$. For this goal, we define the involutions corresponding to oriented axes of loxodromic elements and an inner product $\left\langle \ , \ \right\rangle$ which gives the information about locations of axes of loxodromic elements. The signs of traces of holonomy elements, which are calculated by lifting a representation from $\mathbf{PSL}(2,\mathbb{C})$ to $\mathbf{SL}(2,\mathbb{C})$, play a very important role in determining the discreteness of holonomy groups.

Keywords: Fricke space, involution, discrete holonomy group

MSC numbers: 57M05, 22E40, 20H10