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The convex core of quasifuchsian manifolds with particles

Cyril Lecuire and Jean-Marc Schlenker

Geometry & Topology 18 (2014) 2309–2373

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.

quasifuchsian groups, cone singularities, conformal structures, bending laminations
Mathematical Subject Classification 2010
Primary: 20H10, 57M50
Received: 18 February 2013
Accepted: 5 October 2013
Published: 2 October 2014
Proposed: Benson Farb
Seconded: David Gabai, Jean-Pierre Otal
Cyril Lecuire
Institut de Mathématiques de Toulouse
Université Toulouse III
31062 Toulouse Cedex 9
Jean-Marc Schlenker
Mathematics Research Unit, BLG
University of Luxembourg, Campus Kirchberg
6 rue Richard Coudenhove-Kalergi
L-1359 Luxembourg