Paradoxical Decompositions, 2-generator Kleinian Groups, and Volumes of Hyperbolic 3-manifolds

The 2-thin part of a hyperbolic manifold, for an arbitrary positive number 2, is defined to consist of all points through which there pass homotopically non-trivial curves of length at most 2. For small enough 2, the 2-thin part is geometrically very simple: it is a disjoint union of standard neighborhoods of closed geodesics and cusps. (Explicit descriptions of these standard neighborhoods are given in Section 1.) If 2 is small enough so that the 2-thin part of M has this structure then 2 is called a Margulis number of M. There is a positive number, called a 3-dimensional Margulis constant, which serves as a Margulis number for every hyperbolic 3-manifold. The results of this paper provide surprisingly large Margulis numbers for a wide class of hyperbolic 3-manifolds. In particular we obtain the following result, which is stated as Theorem 10.3: