Algebraic limits of Kleinian groups which rearrange the pages of a book

Abstract

In this paper, we give examples of two new phenomena in Kleinian groups. We rst exhibit a sequence of homeomorphic marked hyperbolic 3-manifolds whose algebraic limit is not homeomorphic to any element in the sequence. We then use this construction to exhibit situations where the space of convex co-compact representations of a given 3-manifold group has many components but its closure is connected. Let M be a compact, irreducible, oriented 3-manifold and let D( 1(M)) denote the space of all discrete, faithful representations of 1(M) into PSL2(C). A sequence of representations { n} ⊂ D( 1(M)) converging to ∈ D( 1(M)) gives rise to a sequence {N n = H= n( 1(M))} of hyperbolic 3-manifolds, each of which is homotopy equivalent to M . The sequence {N n} is said to converge algebraically to N = H= ( 1(M)). (See [7, 13, 14] for more information about algebraic convergence of Kleinian groups.) In many situations (see [1, 6, 15, 24, 25, 27]), it has been shown that N n must be homeomorphic to N for all large enough n, and we had suspected that this would always be the case. In this paper, we give a collection of examples where N n is not homeomorphic to N for any n. Our sequences are quite well-behaved: the n( 1(M)) are convex co-compact and mutually quasiconformally conjugate, and the algebraic limit ( 1(M)) is geometrically nite. In our examples, M is obtained by gluing a collection of I -bundles to a solid torus along a family of parallel annuli. These manifolds are particularly simple examples of books of I -bundles (see [9]) where, to explain the terminology, one should think of the solid torus as the binding and the I -bundles