J ul 2 00 7 SUPERRIGIDITY , GENERALIZED HARMONIC MAPS AND UNIFORMLY CONVEX SPACES

We prove several superrigidity results for isometric actions on Busemann non-positively curved uniformly convex metric spaces. In particular we generalize some recent theorems of N. Monod on uniform and certain non-uniform irreducible lattices in products of locally compact groups, and we give a proof of an unpublished result on commensurability superrigidity due to G.A. Margulis. The proofs rely on certain notions of harmonic maps and the study of their existence, uniqueness, and continuity. Ever since the first superrigidity theorem for linear representations of irreducible lattices in higher rank semisimple Lie groups was proved by Margulis in the early 1970s, see [M3] or [M2], many extensions and generalizations were established by various authors, see for example the exposition and bibliography of [Jo] as well as [Pan]. A superrigidity statement can be read as follows: Let • G be a locally compact group, • Γ a subgroup of G, • H another locally compact group, and • f : Γ → H a homomorphism. Then, under some certain conditions on G, Γ, H and f, the homomorphism f extends uniquely to a continuous homomorphism F : G → H. In case H = Isom(X) is the group of isometries of some metric space X, the conditions on H and f can be formulated in terms of X and the action of Γ on X. In the original superrigidity theorem [M1] it was assumed that G is a semisimple Lie group of real rank at least two * and Γ ≤ G is an irreducible lattice. It is not clear how to define a rank for a general topological group. One natural extension, although not a generalization, of the notion of higher rank is the assumption that G is a non-trivial product. Margulis [M1] also proved a superrigidity theorem for commensurability subgroups in semisimple Lie groups. The target in these superrigidity theorems was the group of isometries of a Riemannian symmetric space of non-compact type or an affine building. * Superrigidity theorems were proved later also for lattices in the rank one Lie groups SP(n, 1), F −20 4 see [Co] and [GS]. It seems however that the same phenomenon holds in these cases for different reasons. It was later realized in an unpublished manuscript of Margulis [M4] which was circulated in the 1990s (cf. [Jo]), that superrigidity for commensurability subgroups extends to a very general setting: a general locally compact, compactly generated …