Volumes of hyperbolic 3-manifolds. Notes on a paper of Gabai, Meyerhoff, and Milley

We present a new approach and improvements to the recent results of Gabai, Meyerhoff and Milley concerning tubes and short geodesics in hyperbolic 3 3 -manifolds. We establish the following two facts: if a hyperbolic 3 3 -manifold admits an embedded tubular neighbourhood of radius r 0 > 1.32 r_0>1.32 about any closed geodesic, then its volume exceeds that of the Weeks manifold. If the shortest geodesic of M M has length less than ℓ 0 > 0.1 _0>0.1 , then its volume also exceeds that of the Weeks manifold.