  We prove general superrigidity results for actions of irreducible lattices on
CAT(0) spaces; first, in terms of the ideal boundary, and then for the
intrinsic geometry (including for infinite-dimensional spaces). In particular,
one obtains a new and self-contained proof of Margulis' superrigidity theorem
for uniform irreducible lattices in non-simple groups. The proofs rely on
simple geometric arguments, including a splitting theorem which can be viewed
as an infinite-dimensional (and singular) generalization of the
Lawson-Yau/Gromoll-Wolf theorem.
