INVOLUTIONS AND THE FRICKE SPACES OF SURFACES WITH BOUNDARY

INVOLUTIONS AND THE FRICKE SPACES OF SURFACES WITH BOUNDARY

The purpose of this paper is to find expressions of the Fricke spaces of some basic surfaces which are a three-holed sphere ${\sum}$(0, 3), a one-holed torus ${\sum}$(1, 1), and a four-holed sphere ${\sum}$(0, 4). For this goal, we define the involutions corresponding to oriented axes of loxodromic elements and an inner product <,> 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 PSL(2, $\mathbb{C}$) to SL(2, $\mathbb{C}$), play a very important role in determining the discreteness of holonomy groups.