Maximally stretched laminations on geometrically finite hyperbolic manifolds

TL;DR

This paper investigates maximally stretched geodesic laminations on geometrically finite hyperbolic manifolds, extending Thurston's work to higher dimensions and providing criteria for proper discontinuity of certain group actions.

Contribution

It establishes the existence of maximally stretched laminations in higher dimensions and generalizes Thurston's asymmetric metric to a broader geometric setting.

Findings

Existence of maximally stretched laminations for minimal Lipschitz constant ≥ 1

Extension of Thurston's asymmetric metric to higher dimensions

Properness criteria for group actions on PO(n,1)

Abstract

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).