Maximally stretched laminations on geometrically finite hyperbolic manifolds

Guéritaud, François¹ ; Kassel, Fanny¹

¹ Laboratoire Paul Painlevé, CNRS & Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France

Geometry & topology, Tome 21 (2017) no. 2, pp. 693-840.

Résumé

Let ${\Gamma }_0$ be a discrete group. For a pair $\left(j,\rho \right)$ of representations of ${\Gamma }_0$ 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 $PO\left(n,1\right)\times PO\left(n,1\right)$ 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 $PO\left(n,1\right)\times PO\left(n,1\right)$.

Détail

DOI : 10.2140/gt.2017.21.693

Classification : 20H10, 30F60, 32Q05, 53A35, 57S30
Keywords: hyperbolic manifold, geometrical finiteness, Lipschitz extension, proper action, group manifold, geodesic lamination

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