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

Let \Gamma be the fundamental group of a complete hyperbolic 3 -manifold M with toric cusps. By following [3] we define the \omega -Borel invariant \beta_n^\omega(\rho_\omega) associated to a representation \rho_\omega\colon \Gamma \rightarrow \mathrm{SL}(n,\mathbb C_\omega) , where \mathbb C_\omega is a field introduced by [18] which can be constructed as a quotient of a suitable subset of \mathbb C^\N with the data of a non-principal ultrafilter \omega on \N and a real divergent sequence \lambda_l such that \lambda_l \geq 1 . Since a sequence of \omega -bounded representations \rho_l into \mathrm{SL}(n,\mathbb C) determines a representation \rho_\omega into \operatorname{SL}(n,\mathbb C_\omega) , for n=2 we study the relation between the invariant \beta^\omega_2(\rho_\omega) and the sequence of Borel invariants \beta_2(\rho_l) . We conclude by showing that if a sequence of representations \rho_l\colon \Gamma \rightarrow \mathrm{SL}(2,\mathbb C) induces a representation \rho_\omega\colon \Gamma \rightarrow \operatorname{SL}(2,\mathbb C_\omega) which determines a reducible action on the asymptotic cone C_\omega(\mathbb{H}^3,d/\lambda_l,O) with non-trivial length function, then it holds \beta^\omega_2(\rho_\omega)=0 .