Constructing geometrically infinite groups on boundaries of deformation spaces

Abstract

Consider a geometrically finite Kleinian group G without parabolic or elliptic elements, with its Kleinian manifold M=(H³∪Ω_G)⁄G. Suppose that for each boundary component of M, either a maximal and connected measured lamination in the Masur domain or a marked conformal structure is given. In this setting, we shall prove that there is an algebraic limit Γ of quasi-conformal deformations of G such that there is a homeomorphism h from IntM to H³⁄Γ compatible with the natural isomorphism from G to Γ, the given laminations are unrealisable in H³⁄Γ, and the given conformal structures are pushed forward by h to those of H³⁄Γ. Based on this theorem and its proof, in the subsequent paper, the Bers-Thurston conjecture, saying that every finitely generated Kleinian group is an algebraic limit of quasi-conformal deformations of minimally parabolic geometrically finite group, is proved using recent solutions of Marden’s conjecture by Agol, Calegari-Gabai, and the ending lamination conjecture by Minsky collaborating with Brock, Canary and Masur.