INVOLUTIONS AND THE FRICKE SPACES OF SURFACES WITH BOUNDARY

Abstract

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 ( , ) 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, C) to SL(2, C), play a very important role in determining the, discreteness of holonomy groups.