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We define, associated to a given a representation π: Γ → H of a finitely generated group into a topological group, invariants defined in terms of bounded cohomology classes. In the case H = SU(1, n) we illustrate, among others and without proof, rigidity results which generalize a theorem of Goldman and Millson ([14]). In the case H = Homeo+(S1), the group of orientation preserving homeomorphisms of the circle, we give a new complete proof of a rigidity result of Matsumoto ([17]), stating that any two representations with maximal Euler number are semiconjugate.

Bounded cohomology, boundary maps, and rigidity of representations into homeo+(S¹) and SU(1,n)