Twist & Shout: Exploring Isometries of Hyperbolic Surfaces

at each point z 2 H, making it a model of the hyperbolic plane. Consider the closed compact surface = H= . We aim to show that its isometry group is …nite, and furthermore that we can achieve a sharp upper bound directly proportional to the genus of the surface. Consider, by comparison, the isometries of R=Z, the Euclidean torus T . Any Euclidean translation of R descends to an isometry of T , yielding an uncountablely large isometry group. Recall that a homeomorphism, di¤eomorphism, or isometry descends to a surface from its covering space when there exists an equivalent morphism on such that, if is the covering map, = . We equivalently refer to as a lift of . We will show that the isometry group is …nite ultimately by considering its action on special points whose property is preserved under isometry. Given an element of the unit tangent bundle, there exists a unique complete geodesic realized by parallel transport (this is equivalent to saying that two geodesics passing through the same point on a surface must be the same or else be proceeding in di¤erent directions). We record this as Fact 0. We will restrict our attention to elements of the unit tangent bundle on whose associated geodesic is closed, of length L, and whose assosciated geodesic has a double point at its assosciated point of our surface; recall that a double point occurs where a geodesic crosses itself transversely.