Precisely invariant collars and the volume of hyperbolic 3-folds

Abstract

We give a lower bound for the volume of any hyperbolic 3-orbifold which admits an embedded tubular neighborhood of a closed geodesic. This bound depends only on the radius of this neighbourhood and not on the length of the geodesic. In the Kleinian group uniformizing such a hyperbolic 3-fold, this yields a lower bound on the co-volume purely in terms of the radius of a precisely invariant collar of a loxodromic axis. As an application of these results we obtain substantial improvements in the known volume bounds bounds of hyperbolic 3-manifolds and certain orbifolds. This paper forms part of our program to identify the minimal volume hyperbolic 3-fold. © 1998 Journal of Differential Geometry. © 1998 Applied Probability Trust.