Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 61, Number 4 (2009), 1261-1291.
Constructing geometrically infinite groups on boundaries of deformation spaces
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.
J. Math. Soc. Japan, Volume 61, Number 4 (2009), 1261-1291.
First available in Project Euclid: 6 November 2009
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OHSHIKA, Ken’ichi. Constructing geometrically infinite groups on boundaries of deformation spaces. J. Math. Soc. Japan 61 (2009), no. 4, 1261--1291. doi:10.2969/jmsj/06141261. https://projecteuclid.org/euclid.jmsj/1257520507