Orientation Reversing Involutions on Hyperbolic Tori and Spheres

This article studies the relationship between simple closed geodesics and orientation reversing involutions on one-holed hyperbolic tori. 1. Simple closed geodesics on the one holed torus We consider topological surfaces of signature (g, n) (where g is the underlying genus and n is the number of simple boundary curves) with negative Euler characteristic and endowed with a hyperbolic metric such that the boundary curves are simple closed geodesics or cusps. A surface of signature (1, 1) will be called a one holed torus, and for the remainder of the article T will denote such a surface. The boundary geodesic of T will always be denoted η. If the boundary geodesic is a cusp, then in place of a true boundary geodesic, by considering a small enough horocycle neighborhood around the cusp, one obtains the same properties than in the case where the boundary curve is a simple closed geodesic. Notably, the notion of distance to a cusp can be introduced. Geodesics will generally be considered non-oriented and primitive, and the notation for a path or a geodesic will not be distinguished in order to simplify notation. Cutting T along an interior simple closed geodesic γ gives a surface of signature (0, 3), generally referred to as a pair of pants or a Y -piece. It is given up to isometry by the lengths of its three boundary geodesics. By cutting along the unique perpendicular geodesic path dη between the two copies of γ, one now obtains a hyperbolic rectangle with the boundary geodesic η. In figure 1 four additional geodesic curves have been drawn which decompose the rectangle intro four isometric right angled hyperbolic pentagons. The length of hη is determined by the lengths of γ and η is through the formula for hyperbolic pentagons sinh( dη 2 ) sinh( γ 2 ) = cosh( η 4 ). For every simple closed geodesic γ, there is a unique simple geodesic path from η to η and perpendicular to η, which does not cross γ. These path is labeled hγ in Date: January 10, 2006. 2000 Mathematics Subject Classification. Primary 30F45; Secondary 30F20, 53C23.