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 = ( H 3 ⋃ 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 Int M to H 3 / Γ compatible with the natural isomorphism from G to Γ , the given laminations are unrealisable in H 3 / Γ , and the given conformal structures are pushed forward by h to those of H 3 / 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.