Abstract:
In Part I of this paper we develop a theory of harmonic mappings into non-
positively curved metric spaces. The main application of the theory, which is presented
in Part II, is to provide a new approach to the study of p-adic representations of
lattices in noncompact semisimple Lie groups. The celebrated work of G. Margulis
[Mar] establishes "superrigidity'" for lattices in groups of real rank at least two. The
fact that superrigidity fails for lattices in the isometry groups of the real and complex
hyperbolic spaces is known. In fact, Margulis deduced as a consequence of superrigidity
the conclusion that lattices are necessarily arithmetic in groups of rank at least two.
Arithmeticity of lattices was conjectured and proved in some cases by A. Selberg (see
[Se] for discussion). Constructions of nonarithmetic lattices in the real hyperbolic case
were given by Makarov [Mak], Vinberg [V], and Gromov-Piatetski-Shapiro [GPS].
For the complex hyperbolic case, nonarithmetic lattices have been constructed in low
dimensions by G. D. Mostow [Mos] and Deligne-Mostow [DM]. In this paper we
establish p-adic superrigidity and the consequent arithmeticity for lattices in the
isometry groups of Quaternionic hyperbolic space and the Cayley plane
