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.Authors: M. Burger, N. Monod