#### Volume 21, issue 2 (2017)

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Maximally stretched laminations on geometrically finite hyperbolic manifolds

### François Guéritaud and Fanny Kassel

Geometry & Topology 21 (2017) 693–840
##### Abstract

Let ${\Gamma }_{\phantom{\rule{0.3em}{0ex}}0}$ be a discrete group. For a pair $\left(j,\rho \right)$ of representations of ${\Gamma }_{\phantom{\rule{0.3em}{0ex}}0}$ into $PO\left(n,1\right)=Isom\left({ℍ}^{n}\right)$ with $j$ geometrically finite, we study the set of $\left(j,\rho \right)$–equivariant Lipschitz maps from the real hyperbolic space ${ℍ}^{n}$ 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 $\Gamma$ of $PO\left(n,1\right)×PO\left(n,1\right)$ on $PO\left(n,1\right)$ by right and left multiplication: we give a double properness criterion for such actions, and prove that for a large class of groups $\Gamma$ the action remains properly discontinuous after any small deformation of $\Gamma$ inside $PO\left(n,1\right)×PO\left(n,1\right)$.

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