Systoles of hyperbolic 3-manifolds

Abstract

Let M be a complete hyperbolic n-manifold of finite volume. By a systole of M we mean a shortest closed geodesic in M. By the systole length of M we mean the length of a systole. We denote this by sl (M). In the case when M is closed, the systole length is simply twice the injectivity radius of M. In the presence of cusps, injectivity radius becomes arbitrarily small and it is for this reason we use the language of ‘systole length’.In the context of hyperbolic surfaces of finite volume, much work has been done on systoles; we refer the reader to [2, 10–12] for some results. In dimension 3, little seems known about systoles. The main result in this paper is the following (see below for definitions):