The bounded Borel class and complex representations of 3-manifold groups

If $Gamma<mathrm{PSL}(2,mathbb{C})$ is a lattice, we define an invariant of a representation $Gammarightarrow mathrm{PSL}(n,mathbb{C})$ using the Borel class $beta(n)in mathrm{H}^3_mathrm{c}(mathrm{PSL}(n,mathbb{C}),mathbb{R})$. We show that the invariant is bounded and its maximal value is attained by conjugation of the composition of the lattice embedding with the irreducible complex representation $mathrm{PSL}(2,mathbb{C})rightarrow mathrm{PSL}(n,mathbb{C})$. Major ingredients of independent interest are the extension to degenerate configuration of flags of a Goncharov cocycle and its study, as well as the identification of $mathrm{H}^3_mathrm{c}(mathrm{SL}(n,mathbb{C}),mathbb{R})$ as a normed space.