Acta Mathematica

Non-realizability and ending laminations: Proof of the density conjecture

Hossein Namazi and Juan Souto

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Abstract

We give a complete proof of the Bers–Sullivan–Thurston density conjecture. In the light of the ending lamination theorem, it suffices to prove that any collection of possible ending invariants is realized by some algebraic limit of geometrically finite hyperbolic manifolds. The ending invariants are either Riemann surfaces or filling laminations supporting Masur domain measured laminations and satisfying some mild additional conditions. With any set of ending invariants we associate a sequence of geometrically finite hyperbolic manifolds and prove that this sequence has a convergent subsequence. We derive the necessary compactness theorem combining the Rips machine with non-existence results for certain small actions on real trees of free products of surface groups and free groups. We prove then that the obtained algebraic limit has the desired conformal boundaries and the property that none of the filling laminations is realized by a pleated surface. In order to be able to apply the ending lamination theorem, we have to prove that this algebraic limit has the desired topological type and that these non-realized laminations are ending laminations. That this is the case is the main novel technical result of this paper. Loosely speaking, we show that a filling Masur domain lamination which is not realized is an ending lamination.

Article information

Source
Acta Math., Volume 209, Number 2 (2012), 323-395.

Dates
Received: 17 April 2010
Revised: 8 May 2012
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892663

Digital Object Identifier
doi:10.1007/s11511-012-0088-0

Mathematical Reviews number (MathSciNet)
MR3001608

Zentralblatt MATH identifier
1258.57010

Rights
2012 © Institut Mittag-Leffler

Citation

Namazi, Hossein; Souto, Juan. Non-realizability and ending laminations: Proof of the density conjecture. Acta Math. 209 (2012), no. 2, 323--395. doi:10.1007/s11511-012-0088-0. https://projecteuclid.org/euclid.acta/1485892663


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