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 $M=(\mbi{H}^{3}\cup \Omega_{G})/G$. 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 $\Gamma$ of quasi-conformal deformations of $G$ such that there is a homeomorphism $h$ from $\mathrm{Int}M$ to $\mbi{H}^{3}/\Gamma$ compatible with the natural isomorphism from $G$ to $\Gamma$, the given laminations are unrealisable in $\mbi{H}^{3}/\Gamma$, and the given conformal structures are pushed forward by $h$ to those of $\mbi{H}^{3}/\Gamma$. 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.