Volumes of <i>N</i>-Cusped Hyperbolic 3-Manifolds

Abstract

We prove that if M is an n-cusped hyperbolic 3-manifold of finite volume, then vol (M) ⩾ nv, where v is the volume of an ideal regular tetrahedron in H3. This generalizes the case of n = 1 which was previously known. Stated another way, this says that the Gromov norm of an n-cusped hyperbolic 3-manifold is at least n. We give all examples of manifolds for which this lower bound is realized when n = 1 or 2 and then prove that it cannot be realized for n ⩾ 3. Maximal cusp volumes in manifolds of low volume are then discussed followed by applications of these results to periods of homeomorphisms of finite order on 3-manifolds with toral boundary components.