The [omega]-Borel invariant for representations into SL

Let [tau] be the fundamental group of a completehyperbolic 3-manifold M with toric cusps. By following [3] we define the [omega]-Borelinvariant [mathematical expression not reproducible] associated to a representation [[rho].sub.[omega]]: [TAU] [right arrow] SL(n, [C.sub.[omega]]), where [C.sub.[omega] is a field introduced by [18] which can be constructed as a quotient of a suitable subset of [C.sup.N] with the data of a non-principal ultrafilter [omega] on IN and arealdivergent sequence [[lambda].sub.l] such that [[lambda].sub.l] [greater than or equal to] 1. Since a sequence of [omega] -bounded representations [rho]l into SL(n, C) determines a representation [[rho].sub.[omega]] into SL(n, [C.sub.[omega]]), forn = 2 we study the relation between the invariant [mathematical expression not reproducible] and the sequence of Borelinvariants [[beta].sub.2]([rho]l). We conclude by showing that if a sequence of re presentations [rho]l: [right arrow] SL(2, C) induces a representation [[rho].sub.[omega]]: [right arrow] SL(2, [C.sub.[omega]]) which determines areducible action on the asymptotic cone [C.sub.[omega]] ([H.sup.3], d/[[lambda].sub.l], O) with non-trivial length function, then it holds [mathematical expression not reproducible]. Keywords. Lattice, character variety, Borelinvariant, real tree, Morgan-Shalen compactification. Mathematics Subject Classification (2010). 57T10, 57M27, 53C35.