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 $\Gamma\rightarrow \mathrm{PSL}(n,\mathbb{C})$ 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.