Borel invariant for measurable cocycles of 3-manifold groups (Q6584687)

``Let \(\Gamma \le \mathrm{PSL}(2, \mathbb{C})\) be a torsion-free lattice. Let \((X, \mu_X )\) be a standard Borel probability \(\Gamma\)-space. Consider a measurable cocycle \(\sigma : \Gamma \times X \to \mathrm{PSL}(n,\mathbb{C})\). We first define a way to pullback bounded cohomology classes via the measurable cocycle \(\sigma\).'' Using the procedure given in [\textit{M. Bucher} et al., Duke Math. J. 167, No. 17, 3129--3169 (2018; Zbl 1417.22009)] the author defines the Borel invariant \(\beta_n(\sigma )\) associated to \(\sigma\). The main result of the paper is the following.\N\NFor any measurable cocycle \(\sigma : \Gamma \times X \to \mathrm{PSL}(n,\mathbb{C})\) we have \N\[\N|\beta_n(\sigma )|\le (n+1) \mathrm{Vol}(\Gamma \backslash\mathbb{H}^3)\N\] \Nand the equality holds if and only if the cocycle is cohomologous either to the irreducible representation \(\pi_n : \mathrm{PSL}(2, C)\to PSL(n, \mathbb{C})\) restricted to \(\Gamma\) or to its complex conjugated.