Dehn filling of cusped hyperbolic 3-manifolds with geodesic boundary

We define for each g > 2 and k > 0 a set Mg,k of orientable hyperbolic 3manifolds with k toric cusps and a connected totally geodesic boundary of genus g. Manifolds in Mg,k have Matveev complexity g+k and Heegaard genus g+1, and their homology, volume, and Turaev-Viro invariants depend only on g and k. In addition, they do not contain closed essential surfaces. The cardinality of Mg,k for a fixed k has growth type g. We completely describe the non-hyperbolic Dehn fillings of eachM in Mg,k, showing that, on any cusp of any hyperbolic manifold obtained by partially fillingM , there are precisely 6 non-hyperbolic Dehn fillings: three contain essential discs, and the other three contain essential annuli. This gives an infinite class of large hyperbolic manifolds (in the sense of Wu) with ∂-reducible and annular Dehn fillings having distance 2, and allows us to prove that the corresponding upper bound found by Wu is sharp. If M has one cusp only, the three ∂-reducible fillings are handlebodies. MSC (2000): 57M50 (primary), 57M20 (secondary). 1 Definition and statements In this paper we introduce certain classesMg,k of compact 3-manifolds, we determine many topological and geometric invariants of the elements of Mg,k, and we analyze their Dehn fillings, answering in particular a question raised by Wu [21] on the distance between non-hyperbolic fillings of a large 3-manifold. We also show that #Mg,k grows very fast as g goes to infinity. Definition of Mg,k All the manifolds considered in this paper will be viewed up to homeomorphism, and will be connected and orientable by default. Let ∆ denote the standard tetrahedron, and let ∆̇ be ∆ with its vertices removed. An