Geometry & Topology

The convex core of quasifuchsian manifolds with particles

Cyril Lecuire and Jean-Marc Schlenker

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Abstract

We consider quasifuchsian manifolds with “particles”, ie cone singularities of fixed angle less than π going from one connected component of the boundary at infinity to the other. Each connected component of the boundary at infinity is then endowed with a conformal structure marked by the endpoints of the particles. We prove that this defines a homeomorphism between the space of quasifuchsian metrics with n particles (of fixed angle) and the product of two copies of the Teichmüller space of a surface with n marked points. This extends the Bers double uniformization theorem to quasifuchsian manifolds with “particles”.

Quasifuchsian manifolds with particles also have a convex core. Its boundary has a hyperbolic induced metric, with cone singularities at the intersection with the particles, and is pleated along a measured geodesic lamination. We prove that any two hyperbolic metrics with cone singularities (of prescribed angle) can be obtained, and also that any two measured bending laminations, satisfying some obviously necessary conditions, can be obtained, as in Bonahon and Otal [Ann. of Math. 160 (2004) 1013–1055] in the nonsingular case.

Article information

Source
Geom. Topol., Volume 18, Number 4 (2014), 2309-2373.

Dates
Received: 18 February 2013
Accepted: 5 October 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732863

Digital Object Identifier
doi:10.2140/gt.2014.18.2309

Mathematical Reviews number (MathSciNet)
MR3268778

Zentralblatt MATH identifier
1334.57015

Subjects
Primary: 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx] 57M50: Geometric structures on low-dimensional manifolds

Keywords
quasifuchsian groups cone singularities conformal structures bending laminations

Citation

Lecuire, Cyril; Schlenker, Jean-Marc. The convex core of quasifuchsian manifolds with particles. Geom. Topol. 18 (2014), no. 4, 2309--2373. doi:10.2140/gt.2014.18.2309. https://projecteuclid.org/euclid.gt/1513732863


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