Journal of the Mathematical Society of Japan

Constructing geometrically infinite groups on boundaries of deformation spaces

Ken’ichi OHSHIKA

Full-text: Open access

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.

Article information

Source
J. Math. Soc. Japan, Volume 61, Number 4 (2009), 1261-1291.

Dates
First available in Project Euclid: 6 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1257520507

Digital Object Identifier
doi:10.2969/jmsj/06141261

Mathematical Reviews number (MathSciNet)
MR2588511

Zentralblatt MATH identifier
1195.57040

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds 30F40: Kleinian groups [See also 20H10]

Keywords
Kleinian group deformation space geometrically finite group

Citation

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


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