Multiple bumping of components of deformationspaces of hyperbolic 3-manifolds

Abstract

Let M be a hyperbolizable 3-manifold with nonempty incompressible boundary of negative Euler characteristic. Suppose that B 1 ,..., B k is a collection of components of the interior of the space of complete, marked hyperbolic 3-manifolds homotopy equivalent to M , such that for any i, j , [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. We prove that there is a geometrically finite hyperbolic structure on int ( M ) which is in the closure of each B i . Moreover, we show that this structure can be constructed so as to admit quasiconformal deformations which also lie in the closure of every B i .