Involutions of compact Klein surfaces

Let X be a compact Klein surface of topological genus g and k bounda ry components and let tp: X-+X be a dianalytic involution. We are interested in determining q0 up to topological conjugacy by a finite number of invariants mainly connected with Fix(q~), the fixed point set of qo. We shall also investigate corresponding inclusions between non-Eucl idean crystal lographic groups and use these to consider the subspaces of Teichmfiller space of Klein surfaces admitting involutions. As we shall see, Fix(cp) consists of (a) a finite number of isolated fixed points, (b) a finite number of simple closed curves. By analogy with the case of plane algebraic curves we shall call a fixed simple closed curve an oval. Ovals will be called twisted or untwisted according to whether they have M6bius band or annular ne ighbourhoods respectively. Of course, twisted ovals can only occur on non-orientable surfaces.