The bounded Borel class and 3-manifold groups

Abstract

If Γ<PSL(2,C) is a lattice, we define an invariant of a representation Γ→PSL(n,C) using the Borel class β(n)∈Hc3(PSL(n,C),R). We show that this invariant satisfies a Milnor–Wood type inequality and its maximal value is attained precisely by the representations conjugate to the restriction to Γ of the irreducible complex n-dimensional representation of PSL(2,C) or its complex conjugate. Major ingredients of independent interest are the study of our extension to degenerate configurations of flags of a cocycle defined by Goncharov, as well as the identification of Hb3(SL(n,C),R) as a normed space.