A MEASURABLE CARTAN THEOREM AND APPLICATIONS TO DEFORMATION RIGIDITY IN COMPLEX HYPERBOLIC GEOMETRY

Abstract. We establish a version for measurable maps of a theo-rem of E. Cartan [10] according to which a bijection of the bound-ary 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 [16] with a sketch of a proof using bounded cohomology techniques and then proven by Koziarz and Maubon in [17] 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, [13]