The !-Borel invariant for representations into SL.n; C!/

Let be the fundamental group of a complete hyperbolic 3-manifold M with toric cusps. By following [3] we define the !-Borel invariant n!!/ associated to a representation !W ! SL.n; C!/, where C! is a field introduced by [18] which can be constructed as a quotient of a suitable subset of CN with the data of a non-principal ultrafilter ! on N and a real divergent sequence l such that l 1. Since a sequence of !-bounded representations l into SL.n; C/ determines a representation ! into SL.n; C!/, for n D 2 we study the relation between the invariant 2!!/ and the sequence of Borel invariants 2l/. We conclude by showing that if a sequence of representations lW ! SL.2; C/ induces a representation !W ! SL.2; C!/ which determines a reducible action on the asymptotic cone C!.H3; d=l; O/ with non-trivial length function, then it holds 2!!/ D 0.