The bounded Borel class and 3-manifold groups

Abstract

If $\Gamma <PSL(2,\mathbb{C})$ is a lattice, we define an invariant of a representation $\Gamma \to PSL(n,\mathbb{C})$ using the Borel class $\beta \left(n\right)\in {\mathrm{H}}_{\mathrm{c}}^3(PSL(n,\mathbb{C}),\mathbb{R})$. 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 $\Gamma$ of the irreducible complex $n$-dimensional representation of $PSL(2,\mathbb{C})$ 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 ${\mathrm{H}}_{\mathrm{b}}^3(SL(n,\mathbb{C}),\mathbb{R})$ as a normed space.