## Journal of the Mathematical Society of Japan

### Constructing geometrically infinite groups on boundaries of deformation spaces

Ken’ichi OHSHIKA

#### Abstract

Consider a geometrically finite Kleinian group $G$ without parabolic or elliptic elements, with its Kleinian manifold $M=(\mbi{H}^{3}\cup \Omega_{G})/G$. 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 $\mbi{H}^{3}/\Gamma$ compatible with the natural isomorphism from $G$ to $\Gamma$, the given laminations are unrealisable in $\mbi{H}^{3}/\Gamma$, and the given conformal structures are pushed forward by $h$ to those of $\mbi{H}^{3}/\Gamma$. 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.

#### Article information

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

Dates
First available in Project Euclid: 6 November 2009

https://projecteuclid.org/euclid.jmsj/1257520507

Digital Object Identifier
doi:10.2969/jmsj/06141261

Mathematical Reviews number (MathSciNet)
MR2588511

Zentralblatt MATH identifier
1195.57040

#### 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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