Let G be an irreducible uniform lattice in a higher semi-simple rank Lie group or algebraic group. We prove that any G-action on the circle by C¹ diffeomorphisms is finite. This is achieved by showing that natural map from bounded to usual second cohomology is injective. The latter holds also for non-trivial unitary coefficients, and implies more finiteness results for G; for instance the stable commutator length vanishes. We prove the same theorems for certain lattices in products of trees.

Note: In case of rank one factors, see this Erratum.