Constructing geometrically infinite groups on boundaries of deformation spaces
arXiv:0809.1261
Abstract
Consider a geometrically finite Kleinian group without parabolic or elliptic elements, with its Kleinian manifold $M=(\H^3\cup Ω_G)/G$. 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 $\H^3/Γ$ compatible with the natural isomorphism from to , the given laminations are unrealisable in $\H^3/Γ$, and the given conformal structures are pushed forward by 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.
This is a revised version of my preprint which I disseminated some years ago