Maximally stretched laminations on geometrically finite hyperbolic manifolds

Abstract

Abstract. Let Γ0 be a discrete group. For a pair (j, ρ) of representations of Γ0 into PO(n, 1) = Isom(H n ) with j geometrically finite, we study the set of (j, ρ)-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 Γ of PO(n, 1) × PO(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 Γ the action remains properly discontinuous after any small deformation of Γ inside PO(n, 1) × PO(n, 1).

Dominating surface group representations and deforming closed anti-de Sitter 3–manifolds

“…Thus the theorem is optimal. This proposition is proved by Guéritaud and Kassel in a slightly different context [12]. For completeness, we include a proof in the appendix of this article.…”

“…The arguments are not new and the proof essentially follows the one of Guéritaud-Kassel [12] (where they assume that the target space is the hyperbolic space). As we said before, the tricky part will be to prove lower semi-continuity at a parabolic representation ρ.…”

“…Theorem (Guéritaud-Kassel, [12]). Let S be a closed oriented surface of negative Euler characteristic, j a Fuchsian representation of π 1 (S) into PSL(2, R), and ρ a representation of π 1 (S) into the isometry group of a Riemannian CAT(−1) space.…”

“…In [12], Guéritaud and Kassel study the minimal Lipschitz constant Lip(j, ρ) and its relation with the comparison of the length spectra of j and ρ in a broader context. Namely, they consider j a discrete and faithful representation of the fundamental group of a geometrically finite hyperbolic n-manifold into Isom(H n ).…”

Margulis spacetimes via the arc complex

“…Suppose that is finitely generated. By [26,31], a necessary and sufficient condition for the action of ρ, j on AdS 3 to be properly discontinuous is that (up to switching the two factors) j be injective and discrete and ρ be "uniformly shorter" than j, in the sense that there exists a ( j, ρ)-equivariant Lipschitz map H 2 → H 2 with Lipschitz constant <1; in the case that ρ is convex cocompact, this is equivalent to…”