Bumping in the Deformation Spaces of Hyperbolic 3-Manifolds with Compressible Boundary.

Abstract

Let M be a compact, hyperbolizable 3-manifold with boundary and let AH(M) denote the space of discrete faithful representations of the fundamental group of $M$ into the group of orientation preserving isometries of hyperbolic 3-space. The components of the interior of AH(M) are well understood, with components enumerated by the space of marked homeomorphism types of oriented, compact irreducible 3-manifolds homotopy equivalent to M. When M has incompressible boundary, Anderson, Canary and McCullough characterized precisely when two components of the interior of AH(M) ``bump'', that is, when they have intersecting closures in AH(M). We introduce a construction that provides examples of bumping in the case when M has compressible boundary. Specifically, we exhibit bumping between components when the change in marked homeomorphism type corresponds to ``shuffling'' one end of an attached one handle around components of the boundary of M bordering a primitive essential annulus.