Maximally stretched laminations on geometrically finite hyperbolic manifolds

Guéritaud, François; Kassel, Fanny

doi:10.2140/gt.2017.21.693

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Maximally stretched laminations on geometrically finite hyperbolic manifolds

François Guéritaud and Fanny Kassel

Geometry & Topology 21 (2017) 693–840

DOI: 10.2140/gt.2017.21.693

Abstract

Let $Γ_{0}$ be a discrete group. For a pair $(j, ρ)$ of representations of $Γ_{0}$ 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 $PO (n, 1) \times PO (n, 1)$ 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 $PO (n, 1) \times PO (n, 1)$ .

Keywords

hyperbolic manifold, geometrical finiteness, Lipschitz extension, proper action, group manifold, geodesic lamination

Mathematical Subject Classification 2010

Primary: 20H10, 30F60, 32Q05, 53A35, 57S30

References

Publication

Received: 5 July 2013

Accepted: 20 February 2015

Published: 17 March 2017

Proposed: Danny Calegari

Seconded: Jean-Pierre Otal, Benson Farb

Authors

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

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

Volume 21, issue 2 (2017)