# The $ω$-Borel invariant for representations into $SL(n,C_{ω})$

### Alessio Savini

Università di Bologna, Italy

## Abstract

Let $Γ$ be the fundamental group of a complete hyperbolic $3$-manifold $M$ with toric cusps. By following [3] we define the $ω$-Borel invariant $β_{n}(ρ_{ω})$ associated to a representation $ρ_{ω}:Γ→SL(n,C_{ω})$, where $C_{ω}$ is a field introduced by [18] which can be constructed as a quotient of a suitable subset of $C_{N}$ with the data of a non-principal ultrafilter $ω$ on $N$ and a real divergent sequence $λ_{l}$ such that $λ_{l}≥1$.

Since a sequence of $ω$-bounded representations $ρ_{l}$ into $SL(n,C)$ determines a representation $ρ_{ω}$ into $SL(n,C_{ω})$, for $n=2$ we study the relation between the invariant $β_{2}(ρ_{ω})$ and the sequence of Borel invariants $β_{2}(ρ_{l})$. We conclude by showing that if a sequence of representations $ρ_{l}:Γ→SL(2,C)$ induces a representation $ρ_{ω}:Γ→SL(2,C_{ω})$ which determines a reducible action on the asymptotic cone $C_{ω}(H_{3},d/λ_{l},O)$ with non-trivial length function, then it holds $β_{2}(ρ_{ω})=0$.

## Cite this article

Alessio Savini, The $ω$-Borel invariant for representations into $SL(n,C_{ω})$. Groups Geom. Dyn. 13 (2019), no. 3, pp. 981–1006

DOI 10.4171/GGD/511