Mathematics – Geometric Topology
Scientific paper
2008-09-08
Mathematics
Geometric Topology
This is a revised version of my preprint which I disseminated some years ago
Scientific paper
Consider a geometrically finite Kleinian group $G$ without parabolic or elliptic elements, with its Kleinian manifold $M=(\H^3\cup \Omega_G)/G$. 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 $\H^3/\Gamma$ compatible with the natural isomorphism from $G$ to $\Gamma$, the given laminations are unrealisable in $\H^3/\Gamma$, and the given conformal structures are pushed forward by $h$ to those of $\H^3/\Gamma$. 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.
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