The \(\omega\)-Borel invariant for representations into \(\text{SL}(n,\mathbb{C}_\omega)\) (Q2327666)

Let \(\Gamma\) be the fundamental group of a finite-volume hyperbolic \(3\)-manifold \(M\). If a sequence of representations \(\Gamma\to \mathrm{PSL}(n,\mathbb{C})\) diverges to an ideal point of the character variety, then \textit{A. Parreau} [Math. Z. 272, No. 1--2, 51--86 (2012; Zbl 1322.22022)] constructed an associated action on the euclidean building associated to \(\mathrm{PSL}(n,\mathbb{C})\) and in particular a representation \(\Gamma\to\mathrm{PSL}(n,\mathbb{C}_\omega)\), where \(\mathbb{C}_\omega\) is the ultrametric asymptotic cone of \(\mathbb{C}\) associated to a non-principal ultrafilter \(\omega\) and a sequence of scaling constants \(\lambda_k\). (The ultrametric asymptotic cone is defined as the ultralimit of the sequence of metrics \(d^{1/\lambda_k}\) rather than \(d/\lambda_k\).) The bounded continuous cohomology of \(\mathrm{PSL}(n,\mathbb{C})\) is generated by the Borel classes \(\beta_n\in H^{2n-1}_{bc}(\mathrm{PSL}(n,\mathbb{C})\). For a representation \(\rho\) of \(\Gamma=\pi_1M\), its Borel invariant is defined by \(\beta(\rho):=\langle\rho^*b_n,\left[M,\partial M\right]\rangle\) using the isomorphisms \(H^*_b(\Gamma)\cong H^*_b(M)\cong H^*_b(M,\partial M)\). \textit{M. Bucher} et al. [Duke Math. J. 167, No. 17, 3129--3169 (2018; Zbl 1417.22009)] showed how to compute \(\beta(\rho)\) from a boundary map (with a formula involving cross ratios and Bloch-Wigner dilogarithms) and proved this way the upper bound \(\beta(\rho)\le\frac{(n+1)n(n-1)}{6}\mathrm{vol}(M)\). The paper under review constructs an analogously defined Borel invariant for representations \(\Gamma\to\mathrm{PSL}(n,\mathbb{C}_\omega)\). For \(n=2\) this is done using cross ratios in \(P^1\mathbb{C}_\omega\) and by extending the Bloch-Wigner dilogarithm to \(\mathbb{C}_\omega\) via \(\omega\)-limits. The step from \(n=2\) to \(n2\) is done using boundary maps in an analogous way to the construction of Bucher-Burger-Iozzi. Several properties of the classical Borel invariant extend to this new invariant, e.g., the invariant is compatible with the inclusion \(\mathrm{PSL}(n,\mathbb{C}_\omega)\to\mathrm{PSL}(n+1,\mathbb{C}_\omega)\), and it has the same upper bound by \(\frac{(n+1)n(n-1)}{6}\mathrm{vol}(M)\). The final section is dedicated to the case \(n=2\). Under a mild nondegeneracy condition for the boundary map, one obtains from the definitions that for a diverging sequence of representations to \(\mathrm{PSL}(2,\mathbb{C})\) the Borel invariant of the limiting \(\mathrm{PSL}(2,\mathbb{C}_\omega)\)-representation indeed agrees with the \(\omega\)-limit of the sequence of Borel invariants. Moreover, the author proves a vanishing result for limiting representations that have a reducible action with non-trivial length function on the asymptotic cone of hyperbolic space.