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

Abstract

Let $\Gamma$ be the fundamental group of a complete hyperbolic $3$-manifold $M$ with toric cusps. We define the $\omega$-Borel invariant ${\beta }_n^{\omega }({\rho }_{\omega })$ associated to a representation ${\rho }_{\omega }:\Gamma \to SL(n,{\mathbb{C}}_{\omega })$, where ${\mathbb{C}}_{\omega }$ is a field 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 $SL(n,\mathbb{C})$ determines a representation ${\rho }_{\omega }$ into $SL(n,{\mathbb{C}}_{\omega })$, 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 ${\rho }_l:\Gamma \to SL(2,\mathbb{C})$ induces a representation ${\rho }_{\omega }:\Gamma \to SL(2,{\mathbb{C}}_{\omega })$ 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 ${\beta }_2^{\omega }({\rho }_{\omega })=0$.

25 pages

References in corpus (2)

• Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds
• Volume rigidity at ideal points of the character variety of hyperbolic 3-manifolds