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

TL;DR

This paper introduces the $oldsymbol{ extomega}$-Borel invariant for representations into $SL(n,oldsymbol{ extomega})$, linking ultrafilter-based limits of classical invariants to geometric actions on asymptotic cones.

Contribution

It defines the $oldsymbol{ extomega}$-Borel invariant for ultrafilter-limited representations and explores its relation to classical invariants and geometric actions.

Findings

For $n=2$, the $oldsymbol{eta_2^ extomega( ho_ extomega)}$ relates to limits of classical Borel invariants.

If a sequence of representations induces a reducible action with non-trivial length, then the invariant vanishes.

The paper connects ultrafilter limits of invariants with geometric properties of asymptotic cone actions.

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 $\rho_\omega: \Gamma \rightarrow SL(n,\mathbb{C}_\omega)$, 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 $\lambda_l \geq 1$. 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 $\beta^\omega_2(\rho_\omega)$ and the sequence of Borel invariants $\beta_2(\rho_l)$. We conclude by showing that if a sequence of representations $\rho_l:\Gamma \rightarrow SL(2,\mathbb{C})$ induces a representation $\rho_\omega:\Gamma \rightarrow SL(2,\mathbb{C}_\omega)$ 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 $\beta^\omega_2(\rho_\omega)=0$.