Negative curvature from a cohomological viewpoint and cocycle superrigidity

In the framework of general negatively curved spaces, we present new superrigidity results and introduce new techniques based on bounded cohomology. This applies to irreducible lattices, and more generally to cocycles, of products of arbitrary locally compact groups. Together with a new vanishing result for higher rank groups, this also generalizes and unifies all previously known results in that direction. The non-vanishing results provide a large class of examples for our results on orbit equivalence rigidity (Monod and Shalom, Ann. of Math., in press). We prove the 'toy-case' of actions on trees.