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

If $\mathrm{\Gamma }<\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{C})$ is a lattice, we define an invariant of a representation $\mathrm{\Gamma }\to \mathrm{P}\mathrm{S}\mathrm{L}(n,\mathbb{C})$ using the Borel class $\beta(n)\in $\mathrm{H}$^3_$\mathrm{c}$($\mathrm{P}\mathrm{S}\mathrm{L}$(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{P}\mathrm{S}\mathrm{L}$(2,$\mathbb{C}$)\rightarrow $\mathrm{P}\mathrm{S}\mathrm{L}$(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 ${\mathrm{H}}_{\mathrm{c}}^3(\mathrm{S}\mathrm{L}(n,\mathbb{C}),\mathbb{R})$ as a normed space.