Maximally stretched laminations on geometrically finite hyperbolic manifolds

Abstract

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 $PO\left(n,1\right)\times PO\left(n,1\right)$.