  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.
