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Maximally stretched laminations on geometrically finite hyperbolic manifolds

François Guéritaud and Fanny Kassel

Geometry & Topology 21 (2017) 693–840

Let Γ0 be a discrete group. For a pair (j,ρ) of representations of Γ0 into PO(n,1) = Isom(n) with j geometrically finite, we study the set of (j,ρ)–equivariant Lipschitz maps from the real hyperbolic space n to itself that have minimal Lipschitz constant. Our main result is the existence of a geodesic lamination that is “maximally stretched” by all such maps when the minimal constant is at least 1. As an application, we generalize two-dimensional results and constructions of Thurston and extend his asymmetric metric on Teichmüller space to a geometrically finite setting and to higher dimension. Another application is to actions of discrete subgroups Γ of PO(n,1) × PO(n,1) on PO(n,1) by right and left multiplication: we give a double properness criterion for such actions, and prove that for a large class of groups Γ the action remains properly discontinuous after any small deformation of Γ inside PO(n,1) × PO(n,1).

hyperbolic manifold, geometrical finiteness, Lipschitz extension, proper action, group manifold, geodesic lamination
Mathematical Subject Classification 2010
Primary: 20H10, 30F60, 32Q05, 53A35, 57S30
Received: 5 July 2013
Accepted: 20 February 2015
Published: 17 March 2017
Proposed: Danny Calegari
Seconded: Jean-Pierre Otal, Benson Farb
François Guéritaud
Laboratoire Paul Painlevé
CNRS & Université Lille 1
59655 Villeneuve d’Ascq Cedex
Fanny Kassel
Laboratoire Paul Painlevé
CNRS & Université Lille 1
59655 Villeneuve d’Ascq Cedex