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October, 2009 Constructing geometrically infinite groups on boundaries of deformation spaces
Ken’ichi OHSHIKA
J. Math. Soc. Japan 61(4): 1261-1291 (October, 2009). DOI: 10.2969/jmsj/06141261

Abstract

Consider a geometrically finite Kleinian group G without parabolic or elliptic elements, with its Kleinian manifold M = ( H 3 Ω 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 Γ of quasi-conformal deformations of G such that there is a homeomorphism h from Int M to H 3 / Γ compatible with the natural isomorphism from G to Γ , the given laminations are unrealisable in H 3 / Γ , and the given conformal structures are pushed forward by h 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.

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Ken’ichi OHSHIKA. "Constructing geometrically infinite groups on boundaries of deformation spaces." J. Math. Soc. Japan 61 (4) 1261 - 1291, October, 2009. https://doi.org/10.2969/jmsj/06141261

Information

Published: October, 2009
First available in Project Euclid: 6 November 2009

zbMATH: 1195.57040
MathSciNet: MR2588511
Digital Object Identifier: 10.2969/jmsj/06141261

Subjects:
Primary: 30F40 , 57M50

Keywords: deformation space , geometrically finite group , Kleinian group

Rights: Copyright © 2009 Mathematical Society of Japan

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Vol.61 • No. 4 • October, 2009
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