Let be the fundamental group of a complete hyperbolic -manifold with toric cusps. By following  we define the -Borel invariant associated to a representation , where is a field introduced by  which can be constructed as a quotient of a suitable subset of with the data of a non-principal ultrafilter on and a real divergent sequence such that .
Since a sequence of -bounded representations into determines a representation into , for we study the relation between the invariant and the sequence of Borel invariants . We conclude by showing that if a sequence of representations induces a representation which determines a reducible action on the asymptotic cone with non-trivial length function, then it holds .
Cite this article
Alessio Savini, The -Borel invariant for representations into . Groups Geom. Dyn. 13 (2019), no. 3, pp. 981–1006DOI 10.4171/GGD/511