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We prove general superrigidity results for actions of irreducible lattices onCAT(0) spaces; first, in terms of the ideal boundary, and then for theintrinsic geometry (including for infinite-dimensional spaces). In particular,one obtains a new and self-contained proof of Margulis' superrigidity theoremfor uniform irreducible lattices in non-simple groups. The proofs rely onsimple geometric arguments, including a splitting theorem which can be viewedas an infinite-dimensional (and singular) generalization of theLawson-Yau/Gromoll-Wolf theorem.Comment: Improved version of earlier preprint. Definitions 3, 5 and proof of  Theorem 55 modifie

Superrigidity for irreducible lattices and geometric splitting