A Measurable Cartan Theorem and Applications to Deformation Rigidity in Complex Hyperbolic Geometry

Abstract

We establish a version for measurable maps of a theorem of E. Cartan [10] according to which a bijection of the boundary of complex hyperbolic plane mapping chains into chains comes from an isometry.As an application, we prove a global rigidity result which was originally announced in [5] and [18] with a sketch of a proof using bounded cohomology techniques and then proven by Koziarz and Maubon in [19] using harmonic map techniques.As a corollary one obtains that a lattice in SU(p, 1) cannot be deformed nontrivially in SU(q, 1), q ≥ p, if either p ≥ 2 or the lattice is cocompact.This generalizes to noncocompact lattices a theorem of Goldman and Millson,[14].

Locations

• Pure and Applied Mathematics Quarterly - View - PDF
• CiteSeer X (The Pennsylvania State University) - View - PDF