  Let Gamma_0 be a discrete group. For a pair (j,rho) of representations of
Gamma_0 into PO(n,1)=Isom(H^n) with j geometrically finite, we study the set of
(j,rho)-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 Teichm\"uller space to a geometrically finite setting and to higher
dimension. Another application is to actions of discrete subgroups Gamma of
PO(n,1)xPO(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 Gamma the action remains properly discontinuous after any small
deformation of Gamma inside PO(n,1)xPO(n,1).
