Consider a geometrically finite Kleinian group without parabolic or elliptic elements, with its Kleinian manifold . Suppose that for each boundary component of , 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 of quasi-conformal deformations of such that there is a homeomorphism from to compatible with the natural isomorphism from to , the given laminations are unrealisable in , and the given conformal structures are pushed forward by to those of . 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.
"Constructing geometrically infinite groups on boundaries of deformation spaces." J. Math. Soc. Japan 61 (4) 1261 - 1291, October, 2009. https://doi.org/10.2969/jmsj/06141261