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

Abstract

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

Several typos have been corrected

References in corpus (1)

• Duality and invariants of representations of fundamental groups of 3-manifolds into PGL(3,C)

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