We define the Toledo invariant of a G-Higgs bundle on a Riemann surface,where G is a real semisimple group of Hermitian type, and we prove aMilnor-Wood type bound for this invariant when the bundle is semistable. We prove rigidity results when the Toledo invariant is maximal, establishingin particular a Cayley correspondence when the symmetric space defined by G isof tube type. This gives a new proof of the Milnor-Wood inequality of Burger-Iozzi-Wienhardfor representations of the fundamental group of a Riemann surface into G.Compared to previous results using Higgs bundles, it uses general theory andavoids any case by case study.
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