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

Abstract

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 ${\rho }_{\omega }:\Gamma \to \mathrm{SL}(n,{\mathbb{C}}_{\omega })$, where ${\mathbb{C}}_{\omega }$ is a field introduced by [18] which can be constructed as a quotient of a suitable subset of ${\mathbb{C}}^{\mathbb{N}}$ with the data of a non-principal ultrafilter $\omega$ on $\mathbb{N}$ and a real divergent sequence ${\lambda }_l$ such that ${\lambda }_l\ge 1$.

Since a sequence of $\omega$-bounded representations ${\rho }_l$ into $\mathrm{SL}(n,\mathbb{C})$ determines a representation ${\rho }_{\omega }$ into $\mathrm{SL}(n,{\mathbb{C}}_{\omega })$, for $n=2$ we study the relation between the invariant ${\beta }_2^{\omega }({\rho }_{\omega })$ and the sequence of Borel invariants ${\beta }_2({\rho }_l)$. We conclude by showing that if a sequence of representations ${\rho }_l:\Gamma \to \mathrm{SL}(2,\mathbb{C})$ induces a representation ${\rho }_{\omega }:\Gamma \to \mathrm{SL}(2,{\mathbb{C}}_{\omega })$ 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 ${\beta }_2^{\omega }({\rho }_{\omega })=0$.