The -Borel invariant for representations into
Alessio Savini
Università di Bologna, Italy
![The $\omega$-Borel invariant for representations into $\mathrm{SL}(n,\mathbb{C}_\omega)$ cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-ggd-volume-13-issue-3.png&w=3840&q=90)
Abstract
Let be the fundamental group of a complete hyperbolic -manifold with toric cusps. By following [3] we define the -Borel invariant associated to a representation , where is a field introduced by [18] 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. 981–1006
DOI 10.4171/GGD/511