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